Friday, December 24, 2021


Welcome to MathPage2

March 14-15-16-17-18-19-20, 2022

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See the latest:  MathPage3

😇  Ideas:

✓ Competency in using numbers is essential for advanced math such as algebra and above, for science, especially chemistry and physics. Students must master the fundamentals, starting with arithmetic in 1st grade to get there. For example, in Science--A Process Approach (SAPA, K-61967), the "Using Numbers" process introduces 1st-graders to integers (-9 to 9) by "identifying and naming" their positions on the number line and counting how many "steps from one integer to another such as 2 to -7," etc. By the 2nd grade, SAPA students use a number line to combine integers, such as -5 + 8 = 3, which is what I am teaching 2nd graders in March 2022. In short, the math needed to do the science was taught as a key process in SAPA.  The experts who put together SAPA lessons knew what they were doing. In fact, 4 of the 6 processes taught in 1st-grade SAPA were math or math-related
  • Using Numbers (arithmetic, integers, etc.), 
  • Measuring (metric system), 
  • Using Space/Time Relationships (geometry), and 
  • Communicating (graphs).
SAPA 1st-grade math was a step up for the 1st-grade curriculum!

 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

✓  Bright students covet advanced math!
✓ Students are not equally creative or academically equal. 
✓ Correlation is not causation. 
✓ It is that simple! Students aren't taught! (National Math Advisory Panel, 2008)
✓ Try something. See how it works. Try again. (Joe Morgan, Zig Engelmann
✓ Teach children how to do things well.
Equity as equal outcomes by lowering standards is a "fallacy of fairness." (Thomas Sowell)
✓ Teaching less content is a recipe for mediocrity.
✓ "To learn something is to remember it." What have you learned if you can't retrieve the fundamentals from long-term memory (instant recall)?
Why should low-income Asians be punished for doing well academically? What has happened to merit?

Note: I use direct instruction, i.e., explain examples of key ideas. 
"The term direct instruction refers to (1) instructional approaches that are structured, sequenced, and led by teachers, and/or (2) the presentation of academic content to students by teachers, such as in a lecture or demonstration." (Google)

When we ask children to learn arithmetic, "we reinforce the idea that if something is not a blatantly employable skill, it's not valuable," writes Joe Morgan. Really?  But learning arithmetic is valuable and essential. Children should know it well, as it prepares them for higher-level math, starting with algebra, trig, precalculus, and later calculus, and preparing students to solve problems with knowledge and logic, including chemistry and physics problems. 

Students must learn to break a problem into smaller parts. In math and science, such as chemistry, there may be several mini-problems or steps you need to do (i.e., multi-step problems) before you get to a final solution. For example, to solve a complicated trig problem, I went through 4 major steps (i.e., calculations) in a specific order to arrive at the final solution--an answer that made sense. The bottom line is that I needed to know trig to solve trig problems. So I also devised a plan of attack.

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."

"Factual knowledge must precede skill [i.e., critical thinking skill]." 
--Daniel T. Willingham, a cognitive scientist.
Thinking in math requires knowing facts and efficient calculating procedures in long-term memory. 
Students cannot do higher-order thinking in math
without knowing some math. 

  Math Got You Down? 
You are not alone!

If a student struggles with Algebra-1 or higher math, the problem is usually linked to not mastering traditional arithmetic in elementary school. Sad to say, basic arithmetic has been changed to reform math. Or it is poor teaching of pre-algebra and prerequisites, even low expectations. In my opinion, calculators, class notes, or open textbooks should not be allowed on tests or exams. Kids must know the fundamentals in long-term memory, which requires ample practice and review

Traditional Arithmetic 

✓ 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

5. Problem-solving  

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. 

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.

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 fractionsx + 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 BauerleinThe Dumbest Generation Grows Up, 2022) So why don't teachers know this? Students need more than a "screen and a keyboard," argues BauerleinKids 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!

U.S. Kids Stumble Over Simple Math!

With Common-Core-based-state standards and the concurrent resurgence of reform math ideas and methods that had failed in the past, many K-8 students continue to stumble over simple arithmetic. The main reason for our educational problems is "the teaching" in the classroom (Zig Engelmann), but many teachers don't think that way. For decades, educators have been fixing the blame on parents, society, and money instead of fixing the problem, which is "the teaching" in the classroom. The reformers insist that students dive into problem-solving before the basics are taught and learned--a detrimental strategy. Students cannot calculate perimeters if they cannot add, areas if they can't multiply, etc. The excuse has been, "Well, the kids can use calculators."
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! 

Children are not little mathematicians. They are novices.  

Knowledge is the goal of learning and the basis of critical thinking. 

"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.

It's the teaching!
Many students can't read, write, or calculate proficiently. And, the pandemic didn't help, putting many kids almost a year behind. However, after nearly a decade of Common Core reform math, only 25% of pre-pandemic 12th graders were proficient at mathematics (NAEP). What's wrong? It's the teaching--a combination of an inadequate curriculum, ineffective instructional methods, fads, early calculator use, false assumptions, and concocted narratives.  

Also, we should train children in arithmetic when they are very young. For example, there was a time when teachers introduced negative numbers to 1st graders (Science--A Process Approach, 1967). I still have the 1st-grade SAPA lessons. In my algebra program, I start negative number operations in 2nd grade to calculate 6 + -8 on an integer number line (-10 to 10) and use debt as a negative number. Singapore students start multiplication as repeated addition in the 1st grade (e.g., 3 x 5 = 5 + 5 + 5 = 15) and use a formal (i.e., standard) addition algorithm. 

The standard algorithm for addition is the best model for place value, but it isn't taught in many 1st grades and, sometimes, not even in 2nd grade. The U.S. curriculum is weak. It delays the memorizing of addition facts, such as 5 + 7 = 12, that support the standard algorithm, which is delayed as well.

I can only conclude that American teachers have low expectations for children, even though they often say the opposite. (Or, they just don't know.) Why else would they teach the lattice method or four other complicated, confusing multiplication algorithms, avoiding the standard multiplication algorithm, which is a major priority and requires memorizing math facts? 

The multiplication facts should be memorized to accommodate the standard multiplication algorithm in the 3rd grade, 1st semester. The focus in the 2nd semester of 3rd grade should be a review of multiplication and the standard long-division algorithm. 

In the 4th grade, the standard fraction operations, decimals, and percentages should be taught. Also, 3rd and 4th graders should solve more complex equations, such as those shown below from a 4th-grade Russian textbook. 

Note. Kids can't solve math problems efficiently without knowing the standard algorithms. So let's get priorities straight! Calculating fluently with standard algorithms is required for problem-solving, and higher mathematics observes mathematician W. Stephen Wilson.

✓ 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

5. Problem-solving  

5th-Grade Math in Russia
The 5th-grade math textbook at the government-supported Bolshoi Ballet Academy may be falling apart. Still, it contains Old School basic arithmetic and algebra (solving equations), not Common Core reform math or fads. In short, basic arithmetic doesn't change. Still, the U.S. progressive reformers have replaced Old School standard arithmetic with reform math, which is, I think, a major flaw in our math instructional programs, starting in 1st grade.

FYI: The future of Russian ballet lives at the Vaganova and Bolshoi ballet schools. At Vaganova, for example, the ballet entrance exams begin at age 10 or 4th grade. They accept around 60 girls for the first-year class, turning away almost all the girls. "Every year, over 3,000 children audition [for Vaganova BalletSt. Petersburg, Russia], and only approximately 70 children are accepted." 

Below are some of the sixty ten-year-olds who were accepted for the first-year class, lining the perimeter on the first day of school for the Day of Knowledge at Vaganova. (Some students are already 11 by September.) In Russia, education is stressed even at elite ballet schools. I wish American schools would have a Day of Knowledge for a pep talk. Students bring their teachers flowers.

4th Grade Russian Mathematics Textbook
No calculators!
Note: It is clear that we need to push more key content to the lower grades, but we may not have enough content-capable teachers. In my mind, the rigor of current math content and instruction is deflated compared to other nations. I taught algebra bounced off standard arithmetic to elementary school kids to beef up the curriculum. I still do. I have a 2nd-grade class and two 4th-grade classes. Also, 3rd graders comprise about half of one of the 4th-grade classes. The 3rd graders are expected to learn the same algebra. 

For decades, the far-left progressive agenda (a ruse, in my opinion) has been to dumb down our kids and equalize outcomes (redefined as equity) by lowering expectations, cutting or delaying content, using substandard instructional methods, inflating grades, etc. Of course, not all teachers bought into this ploy, but many have. Also, high schools resort to credit recovery as a workaround to artificially boost the graduation rate. Therefore, a high school diploma doesn't mean much anymore.  

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.

Notes: The far-left has redefined equity to mean equal outcomes, which is unattainable nonsense, in my opinion. Manipulating and redefining words is pure woke. "Critical theory should be treated more like creationism in public schools than scholarship: an unfalsifiable form of religion," writes Andrew Sullivan

What irks me is that many so-called ethnic studies are filled with Critical Race Theory ideas. The idea that race or identity/gender determines your future is wrong. In my opinion, teachers should not teach racism and identity stuff (CRT) to little kids when they should be teaching reading, writing, and arithmetic, along with parts of history, geography, science, algebra, measurement, and literature.  

Reading Comprehension and the Role of Vocabulary 
Balanced literacy for "black and brown students" has not worked, observes Robert Pondiscio. Phonics isn't enough. For comprehension, students need to study vocabulary early on by spending more time on science, history, math, literature, the arts, and more, and less time on so-called "reading skills" instruction. For comprehension, kids need to learn a wide range of vocabulary from different academic disciplines!

For example, when I teach algebra to 1st graders, I introduce a lot of new math vocabulary such as variable, equation, x-y table of values, number pairs (x, y), coordinate geometry/graphs, point plotting, solve, solution, expand, substitute, place value, guess-and-check, etc. Students perform algebra fused to basic arithmetic.

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 fairnessand 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

Unfortunatelysome 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.

0-20 Number Line
I recommend that teachers teach 1st-grade kids to use a number line (0 to 20) and determine whether an equation is true or false in the first week of school, including the commutative rule. In short, teach academic content. The number line is important mathematics. Also, if you want kids to pay better attention and work independently, avoid seating them in groups.

The basic idea of an equation is "balance."
Equations can be taught in 1st grade, even earlier. For example, an equal-arm balance is a metaphor for an equation, where the value on the left side of the equal symbol (=) balances the value on the right side of the equal symbol. 

Thus, 2 + 3 = 12 - 7  is True because 5 = 5. 

The idea of balance, signified by the equal sign (=), should be taught early in the 1st grade, but it isn't. For example, does 3 + 4 = 6 - 1? No. It is a false equation because 7  5. (The  symbol is an inequality symbol that means "not equal to.") The left side of an equation must balance the right side in value. There are three types of equations in lower elementary school: true, false, and open. An open equation has a variable, such as x - 62 = 140 (3rd-grade equation). To solve (find x), "add 62," the inverse of "subtract 62," to both sides. At this level, all equations have an equal sign. There are also formulas. 

5 = 5

Note1. My Teach Kids Algebra (TKA) students use guess and check in the 1st and 2nd grades to solve equations; however, students use inverse operations (undo) by the 3rd and 4th grades, a step they must show when solving equations. 

Special Note2. I love photography and shooting photos for fun and practice, helping models from Barbizon, and illustrating my math websites (above). Special thanks to the children and teens who posed for photos and their parents. Also, parents of wannabe models can contact me at or visit my photography website. 

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. 
Math education in the U.S. is at a catastrophic level. 
0. "Asian American students spend more hours studying than either white or black American students." (Thomas Sowell)
1. Less than 25% of 12th graders were proficient in mathematics before the pandemic (according to NAEP test scores). Low test scores indicate that Common Core taught as reform math didn't work. The students had nearly 12 years of Common Core.  
2. Algebra is accessible to very young children via standard arithmetic, but, unfortunately, standard arithmetic was replaced by reform math in progressive classrooms, a wrong turn. 
3. E-mail to make comments:
4. If you want to know what musical ability sounds like, click here (Himari in violin competition. She is one of a kind.)
5. Good calculating skills are required for problem-solving in math. Learning number facts that support standard algorithms requires memorization, extensive practice, feedback, and review. 
6. Starting in the 1st grade, students should practice math facts with flashcards for immediate feedback (Stanislas Dehaene, How We Learn, 2020). 
7. Memorization or rote repetition is underrated in America, writes Amy Chua. Instead, Chua points out, "Tenacious practice, practice, practice is crucial for excellence ... Nothing is fun until you are good at it." 
8. Knowing something is correct doesn't mean you understand it. (Richard Feynman)
9. "Asian parents taught their children to add before they could read." (Amanda Ripley, the smartest kids in the world, 2013)
10. About 75% of students who take a solid precalculus course in high school are more likely to earn a college degree. Math unlocks the future! Amanda Ripley points out, "Math is a language of logic. It is a disciplined, organized way of thinking. There is a right answer; there are rules that must be followed." 
11. Paying attention in class is the key to learning! But, U.S. students today have difficulty staying on task. 
12. Other nations, such as Korea, are substantially ahead of American kids in math because Korean kids attend after-school tutoring programs. In many Asian nations, excelling in math is emphasized over sports. Compared to students in other nations, U.S. students don't study much because they don't have to. "Not much is demanded of U.S. students."
13. Unfortunately, the goal of teaching has been to improve test scores, but schooling should be much more than teaching children how to pass state tests! 
14. Some schools resort to credit recovery as a workaround to artificially raise the graduation rate.  

🐸 Good math teachers require that students show steps to get to a solution. Students must know rules and factual and procedural knowledge in long-term memory. Math not only embodies rigor, but it is also a method for "mastering higher-order habits of mind," such as reasoning and finding patterns, etc. In short, students must have efficient calculating skills to solve problems. Kids must be fluent in math skills, not "calculator skills," starting with standard arithmetic in grades 1 - 5, but many are not, according to national (NAEP) and international (TIMSS) tests. 

Unfortunately, many standardized tests like the SAT are online only and require online calculator use. Other hour-long tests associated with the SAT have been eliminated, such as math, chemistry, history, etc. Also, the writing test is gone, too. The new SAT is adaptive. Students answer some questions at the beginning of the SAT, which then "adjusts" the questions to the student. In short, students do not have the same test questions.

"We are inflating grades and scores to make things look better than they are," writes Chester E. Finn, Jr (, "Blinding Ourselves to America's Achievement Woes"). Grade inflation is commonplace in K-12 and college. "Even AP students now seek out 'easy' subjects, such as Human Geography," or Psychology, Environmental Science, etc., rather than harder AP courses such as Physics, English Literature, Chemistry, Calculus, etc. 
✔︎ Ed Issues

Today, there are many issues in education, but my main focus is on learning mathematics. In 2011, I started 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

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.

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.  

Feynman: "It's crazy but correct!" 
Even if you know something is correct doesn't mean you understand it.

The late Richard Feynman, a brilliant physicist, much better than Stephen Hawking, always starts with fundamentals to figure out physics. He said he could not function without teaching, which forces him to focus on content basics and performance. Sometimes, he would stop in the middle of a lecture after writing lots of equations on the blackboard and suddenly exclaim, "It's crazy but correct!" Nobel-Prize winner in physicsRichard Feynman, admits, "I don't understand it (i.e., quantum mechanics). I'm not going to simplify it. That's the way nature is--absurd. If you don't like it, find another universe." 

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 howIn mathematics, content and performance are sine qua non (i.e., essential conditions). 


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. 

Himari, 9, has extraordinary musical ability.
But it is meaningless without a super-sized combination of expert instruction and the joy, eagerness, inclination, and perseverance to practice hours on end, not just now and then, but daily. 

It's DNA!
No matter how much I practice violin, I will never get close to the level of 9-year-old Himari. Clearly, practice can improve a person's violin performance up to a point, but it doesn't cause or create musical ability. There has to be something there, to begin with, which is DNA. Likewise, practice can improve a child's math performance to an acceptable level in arithmetic and algebra, but practice doesn't cause or create mathematical talent. Instead, it develops the ability the child was born with. Even calculus is for average students who are prepared and who practice-practice-practice, but practicing doesn't cause or create talent. It exposes hidden abilities. Thus, mathematical or musical abilities are not developed without extensive practice.

You don't get better at something by not practicing! 

Robert Plomin (blueprint, 2018) clarifies that "performance on academic achievement tests is 60% heritable on average." Educators have always known this but have chosen to ignore it. Instead, educators hide behind the idea that "All Children Can Learn." Learn what? However, in the real world, some students have more mathematical ability than others, or more verbal ability than others, or more artistic ability than others, or more musical ability than others, or more athletic ability than others, etc. And some students have more opportunities than other students. Even if it were possible to equalize the opportunities, the results would differ because kids are not the same. Like it or not, "DNA makes us who we are." Plomin clarifies that what is is different from what could be via the nongenetic 40%. Therefore, the 40% is critical.   


SAPA Equal Arm Balance
Science--A Process Approach (SAPA 1963)

"Think, Balance"

✔︎ An equation is a balance is an example of simplicity, so, in the first lesson, I pulled out an equal arm balance, which is good, but not as good as the old SAPA balance with a fulcrum and riders shown above. If there are 5 cubes in the left pan and 5 cubes in the right pan, then 5 = 5 is an equation that shows the balance. The left and right sides must balance in value for an equation to be true. Thus, in the equation, 3 + 4 = 10 - 3, both sides are 7 (7 = 7) and  the equation is true.

The three types of equations in TKA are all easily demonstrated. Note: An open equation contains a variable or symbol for a missing number.

  1. true equation: 2 + 3 = 3 + 2 [5 = 5]
  2. false equation: 2 + 3 = 2 + 4 [5 ≠ 6]
  3. 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 ≠ 12Also, 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 ≠ 8True 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. 

Tip: “Kids should do their homework on their own—after all, it’s their assignment.” But, I found just the opposite. Students are more likely to excel when parents support their children's education, such as checking homework daily, making sure all the assignments are done, making sure the study place is quiet, etc.

How, not Why
When I tutored high school mathematics, mostly Algebra-2 and Precalculus, students often said they understood the concept but couldn't work the problems. I told them the opposite. If you can work on the problems, your understanding of the concepts is enhanced. Thus, I showed students how to solve the problems. Students can function by recognizing problem types and knowing how to solve those problems. I help students with homework problems and follow each step. If the student errs, I immediately stop the student from making a correction. Soon, the student gains more confidence, making fewer and fewer mistakes. I cover most of the homework, which the student finishes at home. 

I use the same "how-to" approach in classroom teaching. I know what students should be able to do and show them how it looks. In short, I focus on the how with some logic built-in. Explaining examples, the how, on the board is a significant part of my instruction. Giving feedback is another.   


Zhilina age 13: 3 Quads

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