Musing

Welcome to Musing

10-13-21

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. 

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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%

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"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 goal. Sandra 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. 

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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. 

© 2021 ThinkAlgebra/LT

 

 

Back To School-2

Back To School - 2
10-1-21

Some content has been placed on Musing.

Sandra Stotsky (The Roots of Low Achievement, 2019) points out that to close achievement gaps, a stronger curriculum is needed for all students, not lower standards. And a giant leap in expectations and better teaching.

 

Students are confused and behind!
Discovery lessons in groups have not been successful in math instruction, but explaining worked examples with lots of practice is highly effective, points out researcher John Sweller. The caveat is that K-8 teachers must know math well to teach it well, but many don't.

Student test scores have stagnated for at least ten years. It's the teaching, as the late Zig Engelmann would retort. John Sweller ("Why Inquiry-based Approaches Harms Students' Learning" 8-21) concludes that discovery/inquiry learning (group work) is why test scores are not getting better while peers in other nations leap ahead. Sweller writes, "Inquiry learning places an increased emphasis on learners discovering information for themselves rather than having the information explicitly presented to them. This paper suggests a causal relationship between the emphasis on inquiry learning and reduced academic performance." Schools should teach "domain-specific skills such as how to solve particular types of problems that we learn with conscious mental effort." It is particularly true for arithmetic and algebra, where one idea builds on other ideas. Sweller also explains that worked examples are critical for students to study because of Cognitive Load Theory. 

Learning doesn't matter much anymore. 

In the name of diversity, equity, and inclusion (DEI), lowering standards does not fix anything. It gives students a false impression of being better than they are. Grade inflation is another simple-minded idea that has pervaded education since the 1970s. How will students handle college-level mathematics when educators have downgraded standards and inflated grades? It is counterintuitive. 9-23-21

Students are not ready for college math when they have not covered a solid Algebra-2 course or pre-calculus, a combination of trig and college algebra.  Equity as equal outcomes by lowering those at the top is an illusion of fairness, explains Thomas Sowell. Many students are unprepared for college mathematics, so they end up in remedial math courses, which are middle school and high school, at community colleges. 

Note: To narrow racial achievement gaps, erase the tests that point out the gaps because black kids do not test well. Really? It's another stupid idea promoted by liberals! In contrast, I expect black kids in my elementary school TKA classes to learn the basics of algebra. As a result, many black students are my best algebra students in grades 1 to 5. 

Sandra Stotsky (The Roots of Low Achievement, 2019) points out that to close achievement gaps, a stronger curriculum is needed for all students, not lower standards. And a giant leap in expectations and better teaching. 

Stotsky explains that closing gaps should not be an "educational commandment." Instead, a more rigorous curriculum for all students should be the goal. Teachers try to close gaps by teaching less content, which is a mistake. Stotsky observes, "Gap closing as an education goal has had damaging effects on teachers as much as on their students." Lowering the ceiling is not the path to closing gaps. Rather, educators should focus on upgrading the curriculum for all students, but that has not been the case.

Note: One of the best Algebra-1 textbooks is written by Dr. Mary Dolciani,  a member of the School Mathematics Study Group or SMSG. "To study algebra successfully, you need to learn its language, starting with variables and mathematical expressions." (It is difficult to find suitable textbooks or math programs with plenty of word problems and without the fluff of reform math or alternative algorithms that seem to have replaced standard algorithms.) For example, my copy of Dolciani's revised edition (Dolciani & Wooton) was copyrighted in 1970, but there are earlier editions. 

What? Do I have to take remedial math to start college credit courses? The math I took was college prep. (Sadly, college prep math in high school was watered down. It was college prep in name only, not in content.)

Not ready for college-level math...

Comment: Educators don't take academic achievement seriously enough. They inflate grades and use a curriculum that is not world-class. Students are passed on regardless of learning. Educators seem more concerned about diversity, equity as equal outcomes, social justice, self-esteem/social-emotional learning, universal pre-K, or free admission to college than academic achievement in reading, writing, history, or mathematics. For at least a decade, mathematics and reading scores have stalled. Algebra has been thrashed. Metrics, merit, and tests are attacked and labeled as racist by the equity hawks. Credit recovery is rampant in high school to boost graduation rates. 9-20-21

Paul Hill of the Center on Reinventing Public Education explains, "As schools reopen, it is less clear than ever what public education is for. States and districts resist testing students to see what they know now, and many won't judge schools on whether anyone learns. National assessments are on hold, and traditional elements of the curriculum—and even grading—are under assault as supporting white supremacy." Really? I guess learning is not important! Let's not measure it! (https://www.crpe.org/thelens/closing-void-core-public-education)

Hill guesses, "It might all blow over. In a few years, Americans might experience another Sputnik-era panic about how far our kids have fallen behind kids in Europe and Asia. Fear about our national competitiveness might rehabilitate hard subjects, rigor, clarity about results, and educator accountability. But the pendulum could swing too far, back to the narrow measurement-based pedagogy that was rightly abandoned a few years ago." 

Okay, that's possible. Even so, it would be much better than now: American students have fallen significantly behind their peers in many Asia and European nations. For at least a decade, learning in math and reading has stagnated. We cannot afford to wait another few years to forcefully address achievement gaps internationally and at home.

A math program or curriculum that does not support memorizing math facts and standard algorithms of whole numbers from the get-go (1st grade) is substandard. The addition, subtraction, multiplication, and long-division standard algorithms should be learned and used by the 3rd grade. It requires practice-practice-practice. Good calculating skills are the heart and soul of problem-solving.

Comment: A Zoom-year has put our kids behind in math. Math is sequential: "One mathematical idea builds on old ones," explains Ian Stewart, a mathematician. Knowing bits and pieces, here and there, does not cut it. One cannot skip around. Math is not social science or science. 

Furthermore, in my view, students have not learned arithmetic that they need to know to advance, especially fractions, ratio/proportions, percentages, etc. In short, students are weak in calculating skills and techniques needed to solve problems, including whole numbers. Sadly, many 4th and 5th grade students did not master the multiplication table in the 2nd and 3rd grades. Also, the fluency of standard algorithms of whole numbers is often delayed or overlooked. Learning arithmetic and algebra requires memorization and practice-practice-practice. 9-16-21

Note: For many, math is torture. A mathematician at U.C. Berkeley, Edward Frenkel, explains, "First, mathematics is more abstract than other subjects, hence not as accessible. Second, what we study in school is only a tiny part of math, much of it established more than a millennium ago." Math relies on facts. "Everybody can grasp key mathematical concepts and ideas if they are explained in the right way." Israel Gelfand used to say, "People think they don't understand math, but it's all about how you explain it to them." That's my point. Many K-8 teachers are weak in math and don't know how to explain it to children. Children can learn much more math content than is taught in most U.S. schools.

Math must be practiced at school and at home! 

"You don't know anything until you have practiced." (Feynman)

Note: School achievement (i.e., performance on tests) is 60% inheritable on average, which shows what could be. The rest is nurture (40%), explains Robert Plomin (blueprint, how DNA makes us who we are, 2018 MIT). Educators have not come to grips with the importance of genetics. The 40% non-genetic is significant! It has always been known that children are born with different abilities, but 60% on average affords more opportunities for more students. Thus, what is done to develop academic abilities in the classroom is critical. Children need to be challenged! Schooling improves IQ, writes Sanjay Sarma (Grasp, 2020). But, in my opinion, the 40% is not done well. For children to develop logical-mathematical ability, they need to memorize math facts and use standard algorithms for efficient calculating and recognize problem types to solve problems in the earliest grades. 9-13-21

Most 1st and 2nd graders in my algebra class (Teach Kids Algebra or TKA), for example, were able to learn many fundamentals of algebra in a short period of time (7 hours). TKA is an opportunity that most kids don't get and a testament that very young children can learn much more than the curriculum currently teaches.   

Teach Kids Algebra (TKA)

I found that algebra is accessible to very young children through standard arithmetic, starting with true and false numerical equations, an idea I borrowed from Book 1 Modern Algebra (1970) by Mary Dolciani, when I started Teach Kids Algebra, a STEM math program for elementary school students. "A large part of mathematics is in fact about solving equations [technique]," points out Frenkel. First-grade children should start solving equations using guess-and-check method (x + 12 = 45). By 3rd grade, students should use inverse ideas to undo (solve) linear equations to isolate x, but they must know the multiplication table. Even in first grade, students build x-y tables based on linear equations (y = mx + b). y = x + x + 2 is a typical first-grade equation.  

Comment: Many progressive educators don't get that flashcards are integral to the mastery of arithmetic and algebra, or that success is built on doing lots of problem sets, i.e., practice-practice-practice. Many K-8 students have weak calculating skills because memorizing and practicing have fallen out of favor for decades in modern classrooms. 

Algebra

For a 2nd-Grade Teach Kids Algebra (TKA) Lesson Summary, Click TKA, which links equations to tables to graphs--the three representations of a function. TKA is STEM math for elementary school students. Algebra is accessible to very young children using standard arithmetic. However, we grossly underestimate the math content that children can learn given proper instruction. Being able to do arithmetic is the giant step to understanding it. Algebra, the same. If you can't calculate it, then you don't know it. Learning concepts is the easy part.

2nd-Grade Student in TKA (L-7, Last Lesson: Review)

True/False, Integers, Perimeters, Order of Operations, & Equation to Table to Graph (Note: The student below was one of my best students.)

y = x + x + -4, for x = 2, 3, 4, 5 (build x-y table), then plot points (x,y) in Q-I. This was the 7th or last one-hour lesson for 2nd graders. In earlier lessons, by using the number line, they established that adding (-4) gives the same result as subtracting 4, an important pre-algebra idea: to subtract a number, add its opposite (a - b = a + -b). It is also called the "add-opp" property or subtraction rule. (Note: Addition is commutative, so you can add terms or numbers in any order.) Thus, for example, if x is 3, then, by the algebraic substitution rule, y =  3 + 3 + -4 or the sum 6 + - 4, which is the same as 6 - 4 = 2. The (x, y) number pair is (3, 2), plotted in Q-1. (My practice sheets are not fancy.)  

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Some education schools are teaching wannabe teachers that it is a matter of social justice when students do not grasp basic fraction ideas. Really? It's poor teaching, not racism. 

Some education schools are teaching wannabe teachers that it is a matter of social justice when students do not grasp basic fraction ideas. The social justice baloney comes from education schools (e.g., Deborah Loewenberg Ball, former dean of the University of Michigan's ed school, citing "patterns of racism and marginalization.") Of course, students often struggle with fractions, but this is not new. The problem has always been the teaching. Put simply, fractions are poorly taught or not taught at all.

If 5th graders do not know that the mark is 1/3, how did they get to 5th grade? One student said it was 1/7, and some students giggled. I taught this idea to 1st graders in a self-contained, desegregated, Title-1 urban classroom in the early 1980s. 

Naming 1/3 on a number line (5th Grade) is not a social justice issue, as Deborah Loewenberg Ball claims. But, she created one. She arranged a situation to videotape and justify her assumptions presented at conventions.

Ball asked a black student to the board to write her answer and to explain her reasoning. She wrote 1/7 and showed how she got seven equal parts, but from 0 to 1, there were only 3 equal parts. Some kids giggled, and a discussion followed. There were lots of black kids in the classroom. Unfortunately, this 5th-grade student is the product of poor teaching, not racism. Ball displays confirmation bias. 

The video of the student proves nothing other than poor teaching. First, naming 1/3 on the number line is not rigorous content. Second, it has nothing to do with social justice. Third, it is a matter of poor teaching. Teachers should not withhold or delay key content. Unfortunately, Common Core delays the standard algorithms, which are needed to solve problems. (Note: Core Knowledge should not be confused with Common Core.)

Ball also implies that practicing (drilling) math facts often leads to "marginalization, racism, and oppression." Really? No, it is basic arithmetic that all kids must know. What irks me is that kids in the 5th grade don't know basic arithmetic facts like 7 x 6 = 42 in long-term memory. Today, memorization and drill are considered bad pedagogy, but memorization is good for kids. Kids need to practice-practice-practice fundamentals to cement them in long-term memory for retrieval to solve problems.  

But, apparently, according to Deborah Loewenberg Ball, it is racist to expect black kids to drill math facts or master basic arithmetic. No, it is discrimination when black kids don't know fundamentals in long-term memory, which requires memorization and drill. (Comment: Even Einstein participated in math drills when he attended a Catholic elementary school. He was the only Jew in the class.)

Ball argues, "Many taken-for-granted practices in classrooms reflect and reproduce patterns of marginalization and oppression." Ball cites a drill sheet of basic facts. Really? Memorizing math facts is essential arithmetic. Arithmetic is not racist. She seems to argue against kids learning arithmetic because timed practices like the one below can result in "marginalization, racism, and oppression." Ball cites this video as proof of her theories. Is it any wonder that kids don't learn enough math? 

TKA

The algebra program I designed and taught off and on in grades 1 to 5 since 2011 is called Teach Kids Algebra (TKA). Algebra is accessible to very young children using standard arithmetic. This was illustrated by Dr. Robert B. Davis via The Madison Project (1957) and my TKA program in 2011. I noted that students who did exceptionally well with abstract ideas in TKA could be identified as capable in mathematics as early as the 1st and 2nd grades. I wish I would have been able to follow the 42+ 1st graders into high school; however, one of the TKA 1st graders (Kailey) is now in 12th-grade taking calculus and has already received a merit scholarship. 9-12-21

In 2011, Kailey was a 1st-grade TKA student. 
Today, she is in 12th-grade taking calculus. She already has a merit scholarship. 

In the name of what?

Cutting the math curriculum in the name of equity is no way to improve achievement. In my opinion, equity should not dismiss merit. Without merit or high-quality standards, there is little incentive for students to excel. Better teaching by applying a world-class curriculum and efficient instructional methods will boost achievement. The problem is that many K-8 teachers are weak in math, according to H. H. Wu, a mathematician at UC Berkeley, who conducts PD courses and summer classes for current and future teachers.

In education, equity now means equal outcomes for all students by lowering the content and inflating the grades so almost all students can pass a dumbed-down curriculum. But kids are not the same. Academic ability widely varies, points out Charles Murray (Real Education, 2008). Murray wrote that disruptive students should not be permitted to remain in class, and the Core Knowledge Curriculum should be taught to almost every student in grades 1 to 8, which leads to Algebra-1 in 8th grade. (Don't confuse Core Knowledge with Common Core.)

In the U.S., K-12 math is dumbed down in the name of equity, diversity, and inclusion. Likewise, academics and merit are devaluated in many schools, but not in China or other Asian nations. 

American mathematicians Deift, Jitomirskaya, and Klainerman wrote in Quillette, "Needless to say, China pursues none of the equity programs that are sweeping the United States. Quite the contrary: It is building on the kind of accelerated, explicitly merit-based programs, centered on gifted students, that are being repudiated by American educators." Sergiu Klainerman, Princeton math professor, observes, "It's very clear that there is talent in math, just like there is talent in music." 

Many U.S. students lack good calculating skills--not only whole numbers but also fractions, percentages, equations, etc. 

We need to teach more basic arithmetic and algebra, not less. 

But, unfortunately, many of our K-8 teachers are ill-equipped to do that, explains H. H. Wu, a mathematician from U.C. Berkeley, who works with teachers and wannabe teachers. They are not appropriately prepared in education schools or required to take advanced math courses, such as precalculus or calculus. As a result, they lack factual and procedural knowledge and cannot explain math to kids using worked examples. 

Automaticity! Automaticity! Automaticity!

"Learn skills all the way to automaticity!"

Doug Lemov (Practice Perfect, 2012) points out, "The power of learning things by rote is that it allows you to do them with unconscious efficiency. ... It's all but impossible to have higher-order thinking without strongly established skills and lots of knowledge of facts." Rote learning does not get in the way of higher-order thinking, as some claim. In arithmetic, students should practice getting it right. "While failure may build character and tenacity, it's not good at building skills. ... Many types of higher-order thinking are in fact founded on and require rote learning," explains Lemov. Thus, practice getting things right!

We need better teaching and methods that work, not Critical Race Theory (CRT) or other distractions. Putting students into groups (i.e., tracking) to learn math is efficient teaching. Memorization of basic facts is necessary to support the standard algorithms and problem-solving. Teachers should explain arithmetic by carefully selected worked examples. Success in learning arithmetic means practice-practice-practice. Also, Stanislas Dehaene (How We Learn, 2020) advocates that teachers should bring back flashcards. Flashcards work! Children need to overlearn and automate certain fundamentals in arithmetic and algebra. If your child has not automated the multiplication table by the 2nd or 3rd grade, then your child can't move forward with long division, fractions/decimals, ratios/percentages, etc. In short, they hit a brick wall.  

Tom Loveless points out, "A decade later, scant evidence exists that Common Core produced any significant benefit." (Between State and the Schoolhouse: Understanding the Failure of Common Core, 2021) 

Divergent Schooling

Himari Yoshirmura, age 9, performs Paganini Violin concerto with New Japan Philharmonic.

Himari, 9, plays Paganini with zest and the New Japan Philharmonic.

Himari is intense when she goes into her zone. I do not know what happens in her brain, but it's on fire. She is remarkable, nailing 40 minutes of Nicolò Paganini: Violin Concerto No. 1 in D major, op. 6. Incidentally, Himari, age 7, was the Grand Prize Winner at the International Grumiaux Competition in Brussels. 

See Himari when she is not in her zone, click here. She is a happy child who enjoys practicing and doesn't know how good she really is!)

Question: Are we equipped to identify future violinists like Himari or very young children who are advanced in math in our typical gifted programs? In my opinion, the only justification for talented and gifted programs is for acceleration. Instead, we have project-based enrichment in many schools, so I wonder if we would be able to identify, much less train, the next Newton or Mozart? Are we wasting a lot of talent? We need highly specialized education for kids who actually display talent early. In Russia, for example, there are two major ballet schools supported by the government: Vaganova and the Bolshoi. These schools continue to produce the best ballet students in the world. At Vaganova, students must pass screening exams. The first-year class consists of about 60 4th graders (about 10 years old). If you get in, the children face rigorous dance lessons and academic studies, including mathematics and English. While thousands of would-be ballerinas audition, the school selects only 60 kids as first-year students.   

To Be Continued

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