Monday, October 4, 2021


Welcome to Musing

Continued at Musing2

Students should not reinvent the wheel. But, they should practice and overlearn factual and procedural knowledge so that they stick in long-term memory. Also, students can avoid the roadblocks of a limited working memory by using worked examples (models) to solve problems. Students with similar mathematical abilities should be grouped. 

Note: Another scheme is dropping F's in the grading system so that the lowest grade is 50%; however, in my opinion, it is nothing more than lowering expectations, along with tactics like grade inflation and online credit recovery. Also, grades or evaluations should be based on merit, not race or ethnic identity. (Note: Abbot's lecture was shamefully canceled by MIT but will be given at Princeton, where thousands of students have already signed up for the Zoom lecture in late October. Kudos to Princeton.)  

Note: The best way to study and learn is self-testing. Make flashcards for instant feedback at home and school. "Self-testing improves learning and retention." (Test Thyself from Stanislas Dehaene, How We Learn, 2020; What Works! from Dunlosky, Rawson, Marsh, Nathan, and Willingham, Scientific American Mind, September/October 2013)

Advanced math students should be placed with other advanced math students, not in mixed classrooms where every child gets the same math for equity. Topical redundancy drives smart kids to boredom. As a result, our best math students grossly underachieve. These kids need challenging math content, a faster pace, and topical acceleration. In short, they should experience an alternative curriculum different from average students, starting early in elementary school, perhaps as soon as 1st grade. Students who are above average in mathematical ability don't need topical redundancy.

Some children are above average in mathematical ability. (That's life!) I call these students advanced math students. In my algebra lessons, very young children learn to manage abstraction, but those students who grasp abstract ideas better or faster than others are, in my opinion, advanced math students. I can identify these students through my Teach Kids Algebra project, grades 1 to 5. 

War on Excellence!

In the New York Post, Michael Benjamin writes, "Accelerated learning programs with like-minded and equally abled classmates provided the academic stimulus, challenge, and competition that made us better students." 😇

🐸 No matter, the mayor of NYC is phasing out gifted and talented programs, mixing high-achieving kids with low-achieving ones in the same classroom. Mixing happens in almost every elementary school classroom in almost every state. Educators pay attention to struggling students but not to excelling students or the academic elite. Almost all classroom teachers don't know how to teach advanced math or know what it looks like, not even in 1st grade. Most teachers do not know what a ring is in math. They also think Einstein was bad at math, which is absurd. 

In 2011, I found that ordinary 1st graders can use variables to build simple equations and learn the linear equation sequence: equation-table-graph, the three representations of a function. Yes, in 2011, after only 7 hours of instruction, the 40 1st graders (2 classes) were able to use a linear equation, such as y = x + x + 3, to build a table of values (x-y) given the x-values, then plot the number pairs (x, y) as points in Quadrant I. Performing math is a step toward understanding it. Math is abstract and requires a lot of practice-practice-practice. Numbers begin at the symbolic level: 3 + 4 = 7.    

The Johns Hopkins Center for Talented Youth (CTY) tests children to qualify for online courses or on-campus summer programs. CTY gives the School and College Ability Test (SCAT) two years above grade level in quantitative and verbal skills. Thus, a 2nd-grade student would take the 4th-grade SCAT. The CTY standards are much higher, seeking the best of the best. 

Unfortunately, most of the students in various talented and gifted intervention programs in many U.S. school districts would not make the CTY cut. The main reason for talented and gifted programs in school districts should be acceleration, especially math, but it is not the case. Most school programs are enrichment.

Tom Loveless writes, "All students take common, heterogeneously grouped math classes through 10th grade," according to the 2021 California Mathematics Framework, which asserts, "The lack of tracking or acceleration will allow all students to regard mathematics as a subject they can study and in which they belong." Really? It is a classic case of "equalizing downward by lowering those at the top" and a pathetic excuse for dumbing down content in the name of equity. In an essay, Thomas Sowell described it as a "fallacy of fairness." Consequently, all Talented/Gifted and accelerative programs would disappear should the framework pass. Thus, above-average math students, especially Asian Americans, would suffer the most. In my opinion, the framework is biased against Asians and other students who work hard and study more.

What has happened to math education?
Closing gaps should not be an educational goal, observes Sandra Stotsky, The Roots of Low Achievement, 2019. For decades, we have pumped billions and billions of dollars ($$$$$) into helping struggling students and have little to show for it. We still don't know how to change low-achievers into high achievers, explains Stotsky. We have been chasing after the wrong mark, she writes. More money hasn't worked!

According to Sandra Stotsky, upgrading the teaching and curriculum for all students--not just low achievers--should have been the goal. In contrast, what I hear from progressive educators and unions is more of the same that failed in the past (e.g., NCTM reform math; "dumbing down the curriculum so everyone can pass, but no one can excel" (Charles J. Sykes); group work; grade inflation; more money-money-money; etc.). 

K-8 students should not use a calculator as 
a substitute for arithmetic skills.

Starting in 1st grade, students should learn factual and procedural knowledge to automaticity to free space in working memory for problem-solving, let's say in arithmetic and algebra, etc. Also, a daily review should be a significant part of that practice. Students can test themselves (study-test-study-test, etc.) by using flashcards at school and home, says Stanislas Dehaene (How We Learn, 2020). Flashcards give students instant feedback. Since working memory is limited, it is vital to introduce new material slowly, linking it to previously learned material. I rely on carefully selected worked examples or models. Explaining the examples (i.e.,  models) is essential. In math, one idea builds on other ideas. You can't skip around. Indeed, the sequence must be correct. (Robert M. Gagne: Hierarchical Theory). 

I would give students problems as "supervised practice" to ensure they could work the problems on their own before assigning independent work (i.e., homework). During Supervised Practice, I would walk around the room to observe the students' work, answer questions, and re-explain ideas one-to-one as needed. Also, I liked Saxon Math because most of the 30 practice problems were review questions, and only a few were new material. Unfortunately, there are very few K-8 textbooks with good practice problems.   

When I taught science, I introduced basic concepts and the mathematics needed for the science topic. Repetition is vital in learning new material. Some of the significant ideas were reinforced in labs. In short, the experiments came after the instruction, not before. Also, I would explain complicated ideas step by step. Teachers must distill and explain the essential principles, which means they need an excellent background in biology, chemistry, physics, and the mathematics of science.

(Note: None of these ideas are new. They were commonplace in the 60s and early 70s, even earlier.) 

The problem I often see is the lack of specific, measurable, achievable objectives for lessons. Objectives are often too vague.

Robert Mager

Notice the capitalized action verbs in the objectives below. The objectives identify the performance needed to reach the objective. 

At the end of this exercise, the student should be able to 

1. IDENTIFY and NAME the numbers 0 1,-1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, -9. 

2. DISTINGUISH between any two positions on the number line and NAME positions by using the number line names. 

The objectives above were written for 1st-grade students. (Science--A Process Approach, Using Numbers 5: "Numbers And The Number Line," © 1967 by American Association for the Advancement of Science or AAAS.) 

When composing objectives, it is essential to state specifically an observable, measurable behavior that demonstrates the student has reached the objective. In short, the objective "identifies the kind of performance that will be accepted as evidence that the learner has achieved the objective." (Reference: Preparing Instructional Objectives by Robert Frank Mager, 1962) 

A major stumbling block to learning chemistry and physics is a weakness in prealgebra and algebra. "The ability to select a proper formula for a given situation is critical" in physics and math. I dislike the idea that students can refer to their notes when taking a test. Important formulas should be used and memorized. Even 1st graders should learn perimeter formulas for squares and rectangles. 

Important ideas include the recognition of problem types and the ability to calculate. Efficient calculating skills are necessary. Also, in my opinion, all high school students should take algebra-based physics. Note: Using a calculator won't help you answer most questions on the AP Physics 1, Algebra-Based Exam. Learn math, formulas, science, etc., through use


Note: Your high school grades, homework, and math courses matter if you want to complete a bachelor's degree in college. 

"Getting a four-year college degree depends a lot on how far you go in high school math." It is the reason that many excellent private schools require high school students to take PreCalculus for graduation. PreCalculus is a combination of college algebra and trigonometry, but it is for average high school students who are prepared.

Algebra I: 7.8%

Geometry: 23.1%

Algebra II: 39.5%

Trigonometry: 62.3%

PreCalculus: 74.3%

Calculus: 79.8%


"You cannot think your way to the solution of an algebra problem without knowing algebra," points out Charles J. Sykes. Likewise, you will struggle with algebra if you don't know basic arithmetic in long-term memory, including the standard algorithms, fractions, proportions, and percentages. Competency requires memorization and practice-practice-practice, which, unfortunately, have fallen out of favor in liberal classrooms since the 1970s. The early use of calculators was promoted by NCTM, even for Kindergarten students. Calculators are often used as a substitute for basic math skills. Furthermore, Common Core and state standards often delay the proficiency of standard algorithms. In short, Common Core (and state standards largely based on CC) did not adopt world-class benchmarks.  

Common Core: Delay Delay Delay

Sandra Stotsky (The Roots of Low Achievement, 2019) cites the "reduction in academic demands" and the "decline of the academic quality of teachers" as reasons for low achievement. Included are popular practices, such as block scheduling, group projects, etc. She writes, "Process-orientated activities came to dominate mathematics, science, and language classes." Consequently, academic content diminished over the decades. Add grade inflation to the mix, and you get the picture. American students start behind and stay behind their international peers.    

Gap closing has been the wrong educational goalSandra Stotsky observes, "Educators still don't know how to turn massive numbers of low-achievers into high-achievers." We have spent billions and billions on low achievers, and it hasn't worked. In contrast, improving the academic content, let's say in arithmetic and algebra, for all students should be a fundamental goal. It means substantially upgrading the school curriculum, beginning in 1st grade, and upgrading teacher education requirements. Future teachers should major in an academic subject, not education, and all teachers should take college-level math such as precalculus. They should also take biology, chemistry, and algebra-based physics courses and be able to read and digest the textbooks. Many K-8 teachers don't know enough math or science to teach these subjects well. They are told they are doing a good job, but national and international tests show otherwise. 

We should place fast students with other fast students, not with struggling students. Thomas Sowell wrote in the 90s, "Equalizing downward by lowering those at the top" is a "fallacy of fairness." That's what we have, but it goes deeper. The radical progressives have redefined equity as equivalent outcomes, which is not possible. Children are not the same. They differ in IQ, abilities, and attitudes. 

We should have a rigorous, knowledge-based curriculum equivalent to top-performing nations, but we don't. Instead, we have reform math, which is not based on world-class benchmarks. As a result, our students start behind and stay behind their peers in top-performing nations. 

There is a lot of misinformation in education, such as the claim that teachers are more influential than parents at motivating students (Hechinger Report cites 150 studies, 9/21); however, since the comprehensive Coleman Report nearly 60 years ago, we have known that "family background carries more weight than schools and teachers in explaining academic achievement." (Robert Plomin confirms it.) If you never heard of the Coleman Report, then what did you study in ed school? "All factors considered, the most important variable--in or out of school--in a child's performance remains his family's education background." Indeed, family background is far more critical than people realize. Even so, "equal opportunity does not produce equal outcomes," as Thomas Sowell often explained. Also, equal does not mean identical. (Note: There are exceptions, of course. Some parents who missed educational opportunities often pushed their children onto a pathway to college.)

Robert Plomin (blueprint, how DNA makes us who we are, 2018) explains, "Socioeconomic status of parents is a measure of their educational and occupational outcomes, which are both substantially heritable. Finding that heritability of school achievement is higher than for most traits, about 60 percent, suggests that there is substantial equality of opportunity. ... Environmental differences account for the remaining 40 percent of the variance." So the 40% is critical. For example, AP Calculus is for average students who are prepared.  

Larry Cuban writes, "Classroom teachers ultimately decide which of the goals, policies, and curricular content and skills assigned to be taught in fourth grade or high school physics turn up in actual lessons." Thus, K-8 teachers weak in math can teach the content they select and use popular methods, such as discovery/inquiry lessons in group work. In my opinion, the content taught is spotty at best, and the teaching inefficient. 


Concepts & Doing

Learning concepts of arithmetic and algebra is essential; however, it's not nearly enough. One can't live on ideas alone. Equally important is for students early on to do or perform arithmetic to solve problems. Doing arithmetic often helps students grasp concepts better, such as perimeter (1st Grade). In addition, students must recognize problem types. The early development of efficient calculating skills through models (i.e., worked examples) should be paramount. To learn arithmetic means to be able to do it. In short, solving problems in arithmetic or algebra requires good calculating skills. Also, students must be able to manipulate equations and use equivalents. For example, one critical idea is to turn subtraction, which is not commutative, into addition, which is commutative. In a "ring," only addition and multiplication are allowed. Subtraction and division are changed to addition and multiplication, respectively. 

In first grade, 5 - 3 makes sense, but 3 - 5 is not the same and does not make much sense. (How can you subtract a larger number from a smaller number?) Subtraction is not commutative; however, its equivalent, 3 + -5 (or -5 + 3), is easily calculated on the number line or using common sense. Thus, 3 - 5 = 3 + -5 or -2. I introduced 1st-grade students to negative numbers using debt (a negative number), which kids can grasp via an integer number line. 

In one of his lectures in teaching, G. Polya (How to Solve It, 1945) stated: "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean to be doing mathematics? In the first place, it means to be able to solve mathematical problems." Arithmetic and algebra are mathematical tools that solve problems. If you can't calculate it, then you don't know it. Also, Polya pointed out that students should first be exposed to routine problems and then more complicated procedures and problems

Aside: When I taught at a private school in the early 90s, I had a 7th-grade honors pre-algebra class. (I had these students as 6th graders the year before.) The students were invited to take the College Board SAT test. Their SAT scores, especially in math, were high enough to qualify for CTY. The core of these kids placed 2nd in the state Math League. Still, the other teachers on the team wanted to eliminate the honors section. I said "No," walked out of the meeting, went to the teacher lounge to inform the high school math teachers what was being planned in the middle school. To make a long story short, the administrators did not approve it. The honors classes were not eliminated. 

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