- Using Numbers (arithmetic, integers, etc.),
- Measuring (metric system),
- Using Space/Time Relationships (geometry), and
- Communicating (graphs).
✓ Understanding does not produce mastery; practice does!
✓ Unchallenged students learn little!
✓ A major problem has been low expectations for all students.
✓ In my 1st-4th grade algebra lessons (TKA), new content was explained, linked to old content, practiced, and reviewed with feedback.
✓ Minimal Guidance = Minimal Learning
Chem Example: Explain how you would prepare 2.50 L of a 0.350 M solution of glucose. Students must solve three major mini-problems (steps) in a specific order, one step leading to the next, etc. Thus, the major steps all require knowledge of arithmetic and how to prepare solutions in chemistry, including using a volumetric flask. Clearly, the student would be lost without knowing the chem vocab. Moreover, "the ability to solve a wide range of problems comes only with extensive practice," writes Dr. Saundra Yancy McGuire, Louisiana State University chemistry professor. Students need to show the steps on an exam without notes or an open textbook. Students should master procedures. Incidentally, most of the math in chemistry is arithmetic and middle school pre-algebra, but not all (e.g., parts of algebra-2).
4th-Grade-Arithmetic Example (and a lot more): Find the area of a rectangle in centimeters squared with the width 10 cm and length 5 inches. It is not 10 x 5 or 15. Key Idea: The L x W units must match the A = LW equation to work correctly. (You must use the right units for length and width.) So either change 5 inches to centimeters or 10 centimeters to inches. Think! Right, I must change inches to centimeters so that the answer would be centimeters squared: cm x cm.) The area problem has at least two major steps (below):
(1) Apply the conversion factor (ratio) to change inches to centimeters (2.54 cm = 1 inch) so that the answer would be cm-squared (cm x cm). Therefore, 2.54 cm per inch x 5 inches = 12.70 cm. Inches cancel out, leaving centimeters (cm) in the numerator. Thus, 5 inches is equivalent to 12.70 cm. (Unit Analysis)
(2) Calculate the area (L x W) as both measurments are in centimeters (cm): 12.70 cm x 10 cm = 127 cm-squared (cm x cm) or squared centimeters.
Comment: Suddenly, simple arithmetic becomes decimals, fractions, conversion factors or equivalents (ratios), unit analysis, etc. The idea of Significant Figures or SigFigs is not discussed here. Students learn to set up equivalents, which are conversion ratios like 2.54 cm / 1 inch. Notice, I wrote 5 inches as "5 inches / 1", so students realize it is in the numerator. Conversion factors are written in ratio form. Identical units on top and bottom cancel (e.g., inches).
"In many K-8 classrooms, the instruction is diluted by the teachers' reliance on time-consuming games or hands-on activities that are inefficient as compared to direct instruction." R. Barker Bausell (Too Simple To Fail, 2011). Teachers are often told to seat students in small groups to implement minimal guidance instructional methods and discussion. In contrast, teachers are seldom told to use direct instruction. Also, frequently, teachers are placed in a classroom "comprised of students of widely diverse knowledge levels," which is an untenable situation, in my opinion. Bausell recommends that students be sorted into homogeneous classrooms for math as much as possible. "Heterogeneity in student knowledge has been a major drawback to classroom instruction since its birth."
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| Students cannot do higher-order thinking in math without knowing some math. |
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| Math Got You Down? You are not alone! |
✓ Elementary School Mathematics Priorities
by Dr. W. Stephen Wilson, a mathematician
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
The radical progressives have tossed traditional arithmetic and standard algorithms into the trash--a ruinous idea. Here is more bad news: The role of teachers was redefined as the guide-on-the-side rather than the sage-on-the-stage. The switch has been advocated by professors in many education schools, including Jo Boaler at Standford. Also, there were calls for teachers not to "uphold spelling and punctuation rules," writes Mark Bauerlein. In fact, students read fewer books and spend more time looking at screens. Is this what we want for children?
In reform math, memorizing math facts or standard procedures was not essential, and paper-pencil arithmetic was obsolete. In contrast, kids need strong calculating skills to solve problems, not reform math.
After nearly a decade of Common Core reform math, national and international math scores are stagnant. Only 25% of 12th graders are proficient in math. The SAT writing test is long gone. It's all part of the Cult of Wokeness, which attempts to build a utopia. Peter Wood writes that "young people will be unprepared for real life. They would face the world bereft of knowledge, faith, and sound judgment."
Teachers must boost the math curriculum that leads to Algebra-1 in middle school for most students and the literacy curriculum to include literature and novels, science, history, geography, art, etc. Kids should read books, not screens.
Learning the basic arithmetic of whole numbers in grades 1 to 3 is essential. All four operations for whole numbers should be learned as standard paper-pencil algorithms by the 3rd grade, including long-division. Look at the typical equations that 3rd-Grade Russian children learn to solve.
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| Grade 3 Equations (Russia) |
Note. American math programs are substantially behind many other nations. One way to catch up is to introduce basic algebra starting in 1st grade and make sure standard algorithms are given top priority, starting in 1st grade. Indeed, memorization is good for kids! Also, practice, lots of it, is good for kids.
The problem is that math is not taught for mastery! Instead, it is taught to score high on a state test.
The idea to connect algebra to arithmetic is not new. It's been around in America since 1957. It's what I do when I teach my Teach Kids Algebra program (TKA) to young elementary school students, starting with the 1st grade. TKA began as a reaction against Common Core reform math. Improving the curriculum with algebra requires students to know both factual and procedural arithmetic. Students should memorize single-digit math facts (e.g., 3 + 6 = 9) in 1st grade. The number line shows students that 3 + 6 is 9. At first, I begin with missing addend equations such as 5 + x = 12 and advance to y = x + x + 2: equation, table, and graph. Almost everything in TKA involves arithmetic.
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| TKA: 1st Grade, Spring 2011 Given the equation y = x + x +2, build a table of values, and graph the number pairs in Q-I. |
In 2011, I developed a program of supplementary work in algebra connected to arithmetic for young children, ages 6 to 10, approximately 1st grade to 4th grade. What's different? I lecture, that is, I explain carefully selected worked examples of ideas--the how not the why. I do not use an inquiry/discovery approach, which doesn't work well because I have only an hour a week for TKA instruction. Also, I don't always meet students every week because of school schedules. It might be every two weeks, sometimes just once a month. I meet 1st and 2nd graders for 6 or 7 sessions, and for older students, the sessions last most of the school year. But, a weekly session is only 1 hour.
Algebra should not be a problem for students because it figures out stuff using what we know. For example, a variable like x is an unknown number. Also, there are simple procedures (inverses) for solving the unknown number: x. I teach most of the inverses in 3rd and 4th grade. Often, precocious 2nd graders figure out inverses by themselves. The algebra part is easy for kids. Arithmetic is harder, including memorizing math facts and learning standard procedures (algorithms). Almost everything in my TKA algebra program involves basic arithmetic. Kids don't like to practice arithmetic.
Currently, I meet with 2nd graders for an hour once a week for 7 sessions. It is instructive to witness how little kids deal with abstraction and observe what they can learn (the how) given proper instruction. When I first piloted the program in a 3rd-grade class in the second semester of 2011, the hour often stretched to 90 minutes. Today, that's not possible. Also, in the 2011-2012 school year, I met three 4th-grade classes twice a week, all year. Meeting twice a week worked the best; unfortunately, tight schedules do not allow this.
✔︎ Here is a typical problem: 5 + x = 3
The problem doesn't make sense to young students. Even my 2nd graders who have experience with negative numbers on an integer number line had difficulty at first. It demonstrates the need for negative numbers. (x is -2)
✔︎ Here is a 3rd or 4th-grade problem that shows the need for fractions: x + x = 5, where x must be the same number to follow the algebraic rule for substitution. Again the problem doesn't make sense to students. (Hint: It literally means to chop 5 into two equal parts.) So, again, don't tell the students what number is x. (x is 2 1/2)
Critical thinking is the product of knowledge.
Today, woke teachers "emphasize critical thinking over subject matter knowledge, but critical thinking about a subject only happens on top of thorough knowledge of that subject, explains Mark Bauerlein." (Mark Bauerlein, The Dumbest Generation Grows Up, 2022) So why don't teachers know this? Students need more than a "screen and a keyboard," argues Bauerlein. Kids need content knowledge, lots of it! Bauerlein suggests that students read great books and envision "history as more than knowledge but moral truth." None of this happens today, of course.
I have known many smart kids over the decades, but they don't know much. The reason is that many progressive educators emphasize critical thinking over knowledge. For example, many students can't do simple arithmetic without a calculator. Instead, students should learn content knowledge in long-term memory from which critical thinking and problem-solving arise.
Thus, students need to read science, literature, and history--from books, not screens. Also, they should study and review worked examples in math textbooks. Furthermore, on K-9 math tests or exams, students should not be allowed calculators, notes, open textbooks, etc. In short, they should practice factual and procedural knowledge for mastery. Moreover, teachers were exhorted to "switch from sage-on-the-stage to guide-on-the-side," writes Mark Bauerlein. But unfortunately, the shift has been a terrible idea in math teaching.
Missing the Mark, Again and Again...
It's not that Asian kids overachieve; it's that American kids underachieve! They miss the mark! We should focus on the mastery of fundamentals like Singapore, not state test proficiencies. But we don't. Students don't drill or practice enough to cement essential math skills in long-term memory. Unfortunately, it seems that repetition, drills, and memorization are considered Old School and bad pedagogy by the progressive far-left. Really? They work!
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| U.S. Kids Stumble Over Simple Math! |
Also, the so-called math educators from the schools of education claimed that young students would pick up the arithmetic along the way and invent their own math through the discussions of math problems in small groups, which is another silly idea. But, then, there is calculator use. The reformers assumed that, somehow, perhaps by magical intervention, children would invent their own math. Wrong!
"Lacking skills, knowledge, religion, and a cultural frame of reference, Millennials are anxiously looking for something to fill the void," observes Mark Bauerlein (The Dumbest Generation Grows Up, 2022), so they turn to politics. They want to "tear down the inherited civilization and replace it with their utopian aspirations," that everyone should be happy, explains Bauerlein.
✓ Elementary School Mathematics Priorities (Dr. 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
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| Thomas Sowell: "Clearly, we cannot all be equally capable of doing concrete things. Asian American students spend more hours studying than either white or black American students." Thus, their progress should not surprise us, explains Sowell. |
Teach Academics, Not CRT
Elementary and middle school teachers should stick to the fundamentals of reading and vocabulary, writing, arithmetic, and pre-algebra, and kick other stuff out, such as identity and critical race theory. Equity has come to mean equal outcomes. Really? "Equalizing downward, by lowering those at the top," is a "fallacy of fairness" and a "toxic social vision," exclaims Thomas Sowell, Discrimination and Disparities, 2019. So stop lowering standards and downgrading curriculum to level student outcomes. Toss out grade inflation practices.
Strong Calculating Skills are needed for problem-solving and advanced math. Since November 2021, I taught algebra to two 4th-grade classes and added a 2nd-grade class in February (2022) at a city, PreK-8, Title 1 school. I fused fundamental algebra ideas with traditional arithmetic. In my Teach Kids Algebra (TKA) algebra program (once a week for an hour), students use arithmetic in almost everything we do. The algebra part is easy for very young children but not the arithmetic needed for algebra. Also, in many progressive classrooms, the focus has been on Common Core reform math or a "state" variant, not standard arithmetic. Consequently, if middle or high school students struggle with Algebra-1, it can be traced to inadequate preparation in basic arithmetic and pre-algebra in elementary and middle school.
Unfortunately, some math teachers allow class notes, calculators, and open textbooks on math tests, which indicates, at least to me, that these students have not or be expected to master the fundamentals and, consequently, won't be ready for a top-notch Algebra-1 course. Furthermore, the expectation of mastering key content is not there either! In short, too many students lack the calculating skills needed to solve problems. According to Daniel T. Willingham, problem-solving can be as simple as adding fractions in working memory or paper-pencil based on factual and procedural knowledge flowing from long-term memory. (January 2022)
I figured out how to teach advanced ideas to very young children.
Experts are not always experts.
"Experts say that teacher and student emotional and mental health needs must be addressed before academic gaps." I'm afraid I have to disagree. How is this approach going to close achievement gaps? Our kids lag behind kids in many other nations because of backward thinking like this. In short, American students are not taught content that is routinely taught in other countries. Why is that? We also know that combining in-person with remote has been a disaster.
It's anecdotal, not scientific.
After spending part of a day at the 1st-grade class in Austin, Texas, an observer concluded, "Many students were struggling with things like being able to use scissors, work independently and resolve conflicts." Really? Anecdotal observations are not scientific evidence. Furthermore, when is learning to use scissors more important than learning arithmetic.
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| 0-20 Number Line |
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| 5 = 5 |
Understanding does not produce mastery; practice does! 😇
- Statistics are not facts.
- Correlation is not causation.
- Equity has spread to mean equal outcomes by lowering those at the top, which is a "fallacy of fairness," explains Thomas Sowell.
- Be cautious of claims such as "Research shows..."
- "You learn math only through mastery!" (Zig Engelmann) And mastery requires practice-practice-practice.
Today, there are many issues in education, but my main focus is on learning mathematics. In 2011, I started ThinkAlgebra.org to teach algebra fused with traditional arithmetic to elementary school students, grades 1 to 4. It was a reaction against Common Core. Many students have weak calculating skills because they have been taught reform math, not traditional arithmetic, and weaned on calculators. Yet, good calculating skills are required for problem-solving in mathematics. In short, students must know arithmetic and algebra to advance to higher-level mathematics. Many don't.
Also, kids don't learn math well in groups. Indeed, popular minimal-guidance teaching methods, which have dominated math instruction for decades, do not work well either. Over the years, the problem with math instruction has always been the "teaching" of content in the classroom. Math is a bunch of related skills that must be mastered, starting with number facts, place value, and efficient algorithms for operations, i.e., the standard algorithms, according to the mathematician, Dr. W. Stephen Wilson.
However, mastery is impossible without a combination of explicit instruction, memorization, extensive practice, and continual review. It starts in 1st grade, even earlier. Therefore, academic standards must be set high. Don't lower the bar.
Note. Many 3rd-grade kids can't read or do multiplication well, but this was true before the pandemic, which made academic achievement drop even more. When kids reach the 12th grade, less than 25% are proficient in mathematics (NAEP). In my opinion, math education in the U.S. is at a catastrophic stage.
Reading words based on phonics is not the same as reading for comprehension, which requires vocabulary study. Students should know what words mean, but many don't.
✔︎ War on Merit & Asians
"Critical Race Theory (CRT) holds that racism is the ordinary state of affairs in American life." Really? This view is biased and radicalized. "CRT openly denigrates a key American virtue--merit, that combination of talent and hard work that makes for genuine, well-earned success. The denigration of merit (as well as individualism) leads it [CRT] not only to repudiate the foundational premises of America but also that has a disproportionate impact on one racial minority group in the United States more than any there: Asian Americans." (James Lindsay, Forward, An Inconvenient Minority by Kenny Xu)
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| Asian student in my 3rd-grade algebra class, 2011 |
Xu explains, "The Asian American parents were simply investing more in their kids' math education from an early age." Amy Chua points out, "Chinese parents spend approximately ten times as long every day drilling academic activities with their children. By contrast, Western kids are more likely to participate in sports teams." Chua also observes that rote repetition is undervalued in America.
The Leftist agenda asserts that Asians are not part of diversity, even though, by race, they are minorities. Asians have been "historically oppressed," such as the internment of Japanese Americans during World War II. Hostility toward Asians continues today with CRT, watered-down curricula, war on meritocracy, quotas to keep Asians out, and even physical attacks. Still, Asian students and businesses flourish because they believe in the importance of education, especially mathematics and hard work.
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| Thomas Sowell: "Clearly, we cannot all be equally capable of doing concrete things. Asian American students spend more hours studying than either white or black American students." Nevertheless, their progress should not surprise us, says Sowell. |
✔︎ Thomas Sowell, a black scholar, points out that the soft discipline in some of our public schools can lead to lawbreaking on the streets, which connects to prison. Consequently, our leaders have disregarded these well-defined connections. Instead, billions and billions are spent on public schooling while the academic outcomes don't improve. It's not cost-effective. Something is wrong! National and international tests (NAEP, TIMSS) show that most K-8 students are not proficient in reading, writing, or arithmetic to solve problems. Why not? Clearly, more money has not helped much. Also, test scores are dire for 12th graders--less than 25% proficient in math (NAEP). Over the past decade or so, math and reading scores have stagnated. What's happening in the classroom has not worked (e.g., reform math & minimal guidance methods). "It's the teaching," as the late Zig Engelmann would shout.
I agree with Sowell and Murray: Disruptive students and thugs should be kicked out. ("Students who in any way threaten a teacher verbally or physically are expelled," writes Charles Murray in his plan to rid our schools of troublemakers (Real Education, 2008.) Indeed, parents are ultimately responsible for their children's conduct and education, but some parents have dumped ill-mannered children in schools and expected the schools to raise them. And, that's not right! Neither is grade inflation. A child's self-esteem has been the focal point in American education since the 70s, while Asian parents and educators could care less.
Praising kids for no good reason is counterproductive. It's a terrible way to raise children. Most Asian kids excel in math because they practice tenaciously. Most American kids do not.
The radical social vision of the progressive left to equalize opportunities and results included the dumping of money into the educational system and integrating black children. It didn't work, observes Sowell. It is foolish to think that all individuals or groups value education the same way, says Sowell. (Source: Discrimination and Disparities by Thomas Sowell, 2019)
Background: Teach Kids Algebra (TKA)
✔︎ Children can learn more content than the current curriculum allows. For decades, children were taught reform math instead of traditional arithmetic. It held kids back. So, in 2011, my challenge was to reformulate math, as I understood it, for very young students in grades 1 to 4, which gave birth to my algebra hybrid approach called Teach Kids Algebra (TKA). TKA fuses algebra basics with traditional arithmetic.
My Premise
Algebra is accessible to very young children via standard arithmetic.
Indeed, memorizing math facts that support the standard algorithms for calculating and problem-solving should be a top priority. In TKA, even 1st-grade students work with linear functions, build (x - y) tables, and plot (x,y) points. In addition to basic algebra ideas, TKA often includes parts of measurement, geometry, and fractions. Feedback, practice, and review are essential components, too. In short, my algebra program beefs up the elementary school math curriculum.
✔︎ Elementary School Mathematics Priorities
by Dr. W. Stephen Wilson, Mathematician
*****The five building blocks for higher mathematics:
1. Numbers (Memorize Number Facts)
2. Place value system
3. Whole number operations (i.e., Standard Algorithms)
4. Fractions and decimals
5. Problem-solving
✔︎ I simplify complex stuff for young children by thinking about content and performance, but I don't know how my unconscious mind does that. So, I ask myself, What do I want students to do? It's performance over knowledge, but knowledge must tag along. Kids must know stuff. I break down a math topic into smaller, step-by-step procedures. It seems to work most of the time.
Perhaps, that is a strength of my 1st- to 4th-grade algebra program, starting with fundamentals often overlooked in elementary schools, such as true/false, equal, properties or rules, variables, linear functions, graphing, order of operations, the algebraic rule for substitution, and equation writing and solving. Moreover, feedback to students is critically important, too.
Students must memorize math facts and practice standard algorithms for fluency starting in 1st grade. These should not be delayed, such as in Common Core reform math and many state standards. Students can get instant feedback on math facts using flashcards beginning in 1st grade. "Determining what a student should be able to do is far more effective than determining what that student should know. It then turns out that the knowing part comes along for the ride." (Ericsson and Pool, Peak, 2016)
Flashcards work, explains Stanislad Dehaene (How We Learn, 2020). They also give instant feedback. "To get information into long-term memory, it is essential to study the material, then test yourself, rather than spend all your time studying." To do this is easy with flashcards: study-test-study-test... We need to teach children how to learn essentials in long-term memory, so the limited space in working memory is free to solve problems.
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| Feynman: "It's crazy but correct!" |
Even if you know something is correct doesn't mean you understand it. Even if you think you understand something, think again! If you can't calculate it, then you don't know it. Feynman's point was that his students may never understand quantum mechanics, but they need to accept it, which is okay. For example, while the long-division algorithm isn't quantum mechanics, it's okay to accept it without a solid understanding of why the algorithm works. Just be able to do it, apply it, and move on. If anything, "understandings grow slowly" over the years, points out Dr. Robert B. Davis (The Madison Project, 1957), who influenced me to fuse algebra to arithmetic for very young children, especially 1st through 3rd grade. "The truth is that the child's understanding must develop gradually," writes Davis. But, it is no excuse for holding content back. Kids need content.
In K-8 math, we spend too much time on understanding and not enough time performing key content--the how. In mathematics, content and performance are sine qua non (i.e., essential conditions).
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Paganini has come home to live in a 9-year-old girl.
How does Himari Yoshimura (Japan) nail Paganini at age 9? I don't know, but I would guess her instruction centered on content and performance--the fundamentals--with extraordinary perseverance for practice-practice-practice. If she didn't enjoy practicing, then she wouldn't do it. It isn't easy!
[Note: Start at 3:30. Watch Himari dance to Paganini, etc.] She has a happy childhood.
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| Himari, 9, has extraordinary musical ability. |
| SAPA Equal Arm Balance Science--A Process Approach (SAPA 1963) |
The three types of equations in TKA are all easily demonstrated. Note: An open equation contains a variable or symbol for a missing number.
- true equation: 2 + 3 = 3 + 2 [5 = 5]
- false equation: 2 + 3 = 2 + 4 [5 ≠ 6]
- open equation: x + 2 = 12 [x = 10 True, because 12 = 12]
We don't know if an open equation with a variable or symbol for a number is true or false until we substitute (replace) a number for the variable and test it.
A variable is a number!
For example, in the equation x + 7 = 12, if x is 3, then the equation is false because 10 ≠ 12. Also, if the right side is 12, then the left side must also be 12. What number must x be? It must be 5 because 12 = 12.
The equation 4 + 5 = 10 - 2 is false, also, because 9 ≠ 8. True and false are essential in mathematics and can be easily taught using the balance idea (metaphor). We work with only true statements in math and derive new true statements from old ones. In short, students learn to "Think, As a Balance."
When I saw similar true/false exercises in Mary Dolciani's Algebra-1 textbook (1973), I knew I was on the right track to teach fundamentals of equality (=), first. Mary had been a member of the School Mathematics Study Group (SMSG, 1958-1977), a math reform think tank.
FYI: The symbol ≠ means not equal. It is an inequality symbol. Kids in the 1st grade easily catch on to = and ≠.
✔︎ Excellent calculating skills are essential for problem-solving.
Good calculators "recalled basic combinations readily and followed orthodox strategies closely. Poor computers [students who are not good at calculating] often derived basic combinations they could not recall and devised quite unorthodox strategies to do this." (1972)
Many students are taught reform math strategies instead of memorizing basic facts or applying standard algorithms proficiently. Thus, without instant recall of math facts or using complex, nonstandard algorithms, calculating 23x6434 or 3452÷84 becomes a difficult chore that very few children can do.
Much has been said of the 15-year-old Russian figure skater Kamila Valieva who did two quads in the free skating team program, and rightly so. Skaters at the Olympic level have never done anything like this before. Even the triple axel was considered difficult.
Nevertheless, Kamila is the best skater globally, ever. At 15, Kamila was too good, so malicious attempts to discredit her were made. Who supported this child? Many jumped to conclusions without knowing the facts. Such is the vindictive nature of social media and poor journalism. We should all be disturbed by the bitter treatment of a 15-year-old skater. The adults didn't protect her. Have the Olympics become that corrupt?
I hope Kamila comes back, facing the new kids on the ice. Russian skaters are her best competition.
Look out for new Russian skaters coming up the Junior ranks, such as Zhilina.
In the wings are a bunch of eager Russian Junior skaters--the quad squad--like 13-year-old Veronika Zhilina, who did three quads and a quad-triple combination at a Junior international meet (9-4-21). Some Russian figure skaters start quads as early as age 10-11 in practice but not competition. In other words, quads are built into the skating lessons. 2-6-2022,
American skaters are far behind the Russian skaters starting at the Junior division.
Comment: If you strive for a medal (Gold, Siver, Bronze) in international competitions, learn quads, even at the Junior Level.
©2022 ThinkAlgebra/LT
