Wednesday, December 23, 2020


Observations, Ideas, and Opinions on Math Education by a Contrarian in 2021, a divergent view 😎

Surely You're Joking, Mr. Feynman
Good education increases differences. 
You don't know anything until you have practiced.

Thought comes from knowledge, not thin air!

"To think math, kids must know math facts!" 

🙏 There were many inspiring events from 2020. Perhaps, one of the best was Andrea Bocelli and his 8-year-old daughter Victoria Bocelli singing Hallelujah as part of an online Christmas concert (12-12-20) to give hope and optimism to the world. Happy Holidays and Happy 2021. There are other father-daughter duets on YouTube: Adrian & Emma-Jean, Martin & Faye, Dad/Mom & Karolina, and others. (The duets are on YouTube.)

🍎 SAT Changes (See Reflections2)


Unions want billions to reopen schools! 
Remote has been costly ($$$$$) and plagued with tech problems and little learning!

Note: The teacher unions, many professors in schools of education, supporting educationists, and the media are Marxist-like in disguise, and anyone who disagrees with the "establishment" is racist and needs deprogramming. Kids, First, is empty rhetoric. Schools are shut down, which means "Kids, Last." Socialism has silently crept into education over the decades as every student gets the same such as Common Core. It's called equity, which often is a fallacy of fairness, explains Thomas Sowell. Thus, no student should get ahead.

🍎 Math & Science
Other nations are out-educating us in math and science. Many U.S. students have difficulty reading and grasping content-rich texts in science. Online learning puts students at a disadvantage (behind). Kids are not learning as much as they should online. The trend can be reversed. But will the progressives (aka liberals) and teacher unions support it? Why aren't students in the classroom with in-person teachers full time? The people blocking the full opening of schools are radicals. They may not look or sound like radicals, but they are because they have hijacked your child's education. 

✍️ First, teachers must return to the classroom in-person, full time. Second, they need to deflate grade inflation and get back to reading textbooks, not screens. Third, teachers need to rethink reform math and minimal guidance methods of instruction and bring back traditional arithmetic with standard algorithms through explicit teaching with explained examples. Teachers should teach, not videos. There have been continual tech problems with Zoom and the Internet. Tech maintenance is a costly nightmare ($$$$$).

Mom, it's not working again!!!! 
Tech Problems with Zoom, Internet speed, and hardware plague remote and hybrid learning. Many students and parents are frustrated. (Credit: GabbyB)

✍️ Educators should stick to standard arithmetic for mastery, reduce group work and inefficient discovery learning in math class, and teach arithmetic explicitly with worked examples and plenty of in-class retrieval practice (flashcards). But, I wonder if this is possible. "Elementary school mathematics must prepare students for algebra," explains W. Stephen Wilson, a mathematician. The preparation begins in 1st-grade. To prepare for algebra and higher-level math, students need to memorize math facts and become fluent using standard algorithms, says Wilson. But reform math doesn't prepare students for Algebra-1. Students stumble over simple arithmetic. Consequently, many students end up in remedial math courses (Pre-Algebra, Algebra 1, and Algebra 2) at colleges. (Hint: Students should review and practice algebra fundamentals before taking the placement test. But review won't help much if students are weak in arithmetic fundamentals.)

✍️ Without up-to-date student testing, educators won't know the content students have or have not mastered in the past two semesters. Remote or hybrid instruction is a lousy substitute for full-time, in-person classroom teaching. To measure student progress, some advocate engagement and attendance as alternatives to testing. Really? The teacher unions are lined up against testing and in-person instruction full time. They assert, "It's not safe for teachers to return to the classroom." Really? Parents should be furious, and many are! Many are pulling their children out of the public schools. Remote is a substandard substitute for in-person classroom teaching full time. Tech problems with Zoom and the Internet frustrate students and parents. Students are not set up for success.

✍️ Unions want billions and billions to reopen schools. 

Unions won't agree to reopening schools full time without a massive influx of federal money. The National Education Association or NEA, the largest teacher union, states, "Tax revenues are dropping sharply due to the widespread decline in economic activity and consumer spending." The NEA estimates 175 Billion more. Still, public schools' primary funding has been property taxes and state taxes, not federal dollars. Students leave the public schools in droves, causing a shortfall of revenue in large school districts. Some people in high-tax states like New York and California are moving to states with lower taxes, thus decreasing the revenue base. Schools closed because it was not safe for teachers to return, according to the influential teacher unions and the media. Now, it's more cash. In contrast, many private or independent schools stayed open. Going remote has strained school budgets as each student needed a working laptop or tablet and a good Internet connection, which requires huge amounts of money. Was education any better? I think not. (To Be Revised) 

✍️ Socialism has silently crept into education over the decades as every student gets the same. It's called equity that often is a fallacy of fairness, explains Thomas Sowell. Thus, no student should get ahead. But sameness does not work. It hurts most kids because kids are not the same genetically (DNA) and widely vary in learning what is taught in school. It is called genetic variability. Children get their smarts and abilities from their parents. Therefore, outputs can never be the same, regardless of the rhetoric from liberals. Some kids are better in math than others. I do not think that inputs can be the same either, because teachers vary widely in knowledge, talent, and ability.
✍️ Even with testing, parents still didn't know how much grade-level material had been mastered, especially arithmetic and algebra. Most kids are not proficient in math, according to government testing (NAEP). With substantial feel-good-for-no-good reason grade inflation, parents cannot trust report-card grades. U.S. students are behind in math compared to their peers in top-performing nations (TIMMS). One reason is that Common Core or Common-Core-based math standards are not internationally benchmarked.  

✍️ Common Core Cuts Content
In Common Core, the stress is not on solving equations to solve problems but on analyzing functions. The shift is ill-advised. The new science standards (Next Generation Science Standards or NGSS) cut back on stoichiometric calculations from high school chemistry. Of course, if equation solving has been de-valued in Common Core algebra classes, students won't have the mathematical background needed for college-level chemistry or physics calculations, much less STEM. 

When was the last time Daro, McCallum, and Zimba, the leading writers of the Common Core math standards, taught K-5 arithmetic? Kids don't think like adults, much less reason like mathematicians. Kids are novices, not experts. They are not little mathematicians. 

✍️ Some educators praise Eureka Math, saying it is the best curriculum ever. But, they are wrong. Why? It doesn't measure up to international benchmarks. The math curricula in many Asian nations are far superior. Eureka Math, which is Common Core reform math, is better than before, but that doesn't make it first-rate in my mind. For example, Singapore 1st-grade students learn multiplication, memorize math facts, and practice formal algorithms (i.e., standard algorithms) for calculating. They also write and solve equations from word problems (operations: +, -, x). In short, calculating skills are essential for problem-solving in mathematics. Moreover, factual knowledge (lower-level thinking) in long-term memory must precede higher-level thinking, writes Daniel T. Willingham.  

✍️ The number line is essential for 1st graders learning operations such as addition, subtraction, multiplication, and calculating simple combinations, yet it isn't found or marginalized in many U.S. 1st-grade textbooks. Kids don't need manipulatives or a lot of colorful graphics that distract in textbooks; however, they do need a simple number line for the first couple of school weeks. Then, the single-digit addition facts need to be systematically memorized to support the standard algorithms. The number line is understanding for novices. In short, students should learn numbers as numbers by how they relate to each other.

0-20 Number Line is Basic Arithmetic!

✍️ Kids need a baseline of knowledge. Jeff Litt is quoted as, "People talk about critical thinking. You cannot think critically if you don't have something to think about; knowledge matters." (Education Next, Fall 2016) 

✍️ Common-Core-based NCTM reform math not only de-emphasizes traditional computational skills but often substitutes calculators for basic skills. Moreover, the role of the teacher has changed from an academic leader to a facilitator. The idea is that children should invent their own solutions to problems even in the 1st and 2nd grades. Somehow, by magic, children can move on to doing math without knowing the basics first. Really? The bottom line is that students cannot "think" their way to the solution of a percentage problem without knowing the basic arithmetic, percentages, and equation solving techniques. They can't solve a trig problem without knowing trig. Likewise, they cannot translate Latin without knowing Latin. In math, critical thinking is called problem-solving. 

🍎 Fractions (See Reflections2)
Mathematician Morris Kline wrote, "The operations with fractions are formulated to fit experience."

I wrote about math reforms in October 2006. Nothing has changed! We still have reform math, repackaged in various flavors such as Focal Points. Only what students discover for themselves is truly learned.” Really? It isn't true and never has been. The idea that children should "invent their own algorithms" to do arithmetic is nonsense. Students first need to master standard algorithms. Repetition to automaticity is the key to learning, explains Daniel Coyle (The Talent Code, 2009).

NCTM Reforms Have Fallen Short 

The reasons for lackluster improvement can range from bad instruction, ill-trained teachers, misnamed courses, poorly written curricula to an over-reliance on calculators, and ineffective math reforms that do not focus on mastery of content and fluency of skills. The National Council of Teachers of Mathematics (NCTM) reform movement, which has dominated math instruction for over 20 years, falls well short. Its main emphasis on understanding and problem solving, which sounds great, has not produced an upsurge in math achievement because content mastery and skill fluency are not the central nuclei of math programs. The consequence is that our kids are almost standing still while their peers in other nations are racing ahead. LT 

Note: The NCTM reforms started in 1989, over 30 years ago. Long division and other standard algorithms were major casualties. "Long division is a pre-skill that all students must master to automaticity for algebra (polynomial long division), pre-calculus (finding roots and asymptotes), and calculus (e.g., integration of rational functions and Laplace transforms.). Its demand for estimation and computation skills during the procedure develops number sense and facility with the decimal system of notation as no other single arithmetic operation affords." (The Washington Post, May 31, 2005, 10 Myths About Learning Math)

The content taught in 1st and 2nd grade has a tremendous impact in later grades. There is a problem if students don't memorize the multiplication table and practice the standard algorithms for long multiplication and long division by 3rd grade. Singapore starts multiplication in 1st grade. Many reform-minded teachers in progressive classrooms don't realize this. Third/Fourth graders should be taught that a regular fraction is a quotient of two integers, such as 8/2 or 1/3. What is 8 divided by2? {4} What is 1 divided by 3? {.33333...} It's a repeating decimal, but students won't be able to calculate it because long-division isn't taught. Most K-5 elementary school teachers don't know this. Note: Decimals and Long Division should be taught no later than 3rd grade. Why is 4/5 = 0.8? Division links fractions to decimals. 


End Insert


🍎 In 2021, we need to return to a bold pursuit of educational excellence. Progressive ideology and policies, fads, computers/tablets, and Common Core reform math, and minimal guidance methods have sidetracked us. Educators should teach standard arithmetic for mastery but don't. Instead, they substitute reform math for standard arithmetic. Children need explicit teaching via worked examples--not group, discovery, or project work, much less six different ways to multiply, and eight so-called mathematical practices. 

1. Grade Inflation falsifies the level of performance or achievement. Still, it dominates K-12 education.
2. Discovery Learning and other minimal guidance, constructivist methods are substandard approaches to teaching arithmetic and math in general. The math taught is dumbed down.
3. Charter Schools are under assault. Thomas Sowell (Charter Schools and Their Enemies, 2020) "defends charter schools against the teachers' unions, politicians, and liberal educators who threaten to dismantle their success."
4. Science, Where's the Math? To learn science well, kids need to know some math. Richard Feynman complained about the lack of math in K-6 science textbooks. That was then. Now, it is the same thing. Many students have difficulty reading chemistry and physics textbooks, which are complex, content-rich texts. Math is barely visible in K-8 science texts. 

Radical, liberal educators want to kick out grades. They say grades are racist. To many liberals, everything is racist, so I don't give them much credence. "Some school districts are embracing trendy but dubious ideas about how to fight racism in the classroom," writes Robbie Soave ( "District officials evidently believe that the practice of grading students based on average scores is racist and that "anti-racism" demands a learning environment free of the pressure to turn in assignments on time" or showing up for class, completing assignments, or practice/review work, etc. Really?

Soave explains, "Under the new system, pupils will not be penalized for failing to complete assignments or even show up for class, and teachers will give them extra opportunities to demonstrate their "mastery" of subjects. What constitutes mastery is not quite clear, but grades shall not be influenced by behavior or "nonacademic measures" such as quantity of work completed, according to guidance from the district."

Note: The "work completed" is not a "nonacademic measure." It is an academic necessity that helps students transfer important content to long-term memory for problem-solving, i.e., critical thinking in mathematics. Learning arithmetic and math requires retrieval practice-practice-practice and review-review-review. (Incidentally, Einstein was a stellar math student in grade school. Curiosity does not produce innovation. Knowledge fosters high-level thinking and creates innovation.)

Note: Grading content knowledge is not racist! It is essential. In my opinion, averaging math test scores are objective determiners of mastery of content beginning in 1st grade. How else are teachers going to determine mastery of anything? Gee, I thought learning was one of the main purposes of schooling. 

Note: Engagement, the rage in today's progressive classrooms, is not the same as learning something. This idea is so important that it demands repetition: Engagement is not the same as learning stuff. Got it? Educators spend too much time on engagement, social-emotional activities, mathematical practices, and too little time explicitly teaching key content skills, ideas, and uses. Group work or group grades are a fallacy of fairness. "You don't understand anything until you have practiced," explains Richard Feynman. 1-5-2021

Theme: Other nations are "out-educating us in math and science." 
(1) The international math gap begins in the 1st grade. 
(2) Math needs to be taught, practiced, and reviewed daily. 
(3) Block scheduling and remote/hybrid learning are a bust for learning mathematics well. 
(4) In my opinion, reform math reduces content to level outcomes, a toxic idea
(5) "Elementary school mathematics must prepare students for algebra," explains W. Stephen Wilson, a mathematician at Johns Hopkins University. But, students are underprepared when taught reform math with minimal guidance methods (group work).  
(6) "Children differ in their ability to learn the things that schools teach," observes Charles Murray (Real Education, 2008).  
(7) "Equal opportunities do not create equal outcomes," points out Robert Plomin (Blueprint, 2018). 
(8) Standard arithmetic is simple but not easy to learn. It must be taught well by competent teachers, starting in the earliest grades of elementary school.
(9) Some kids are better at math than others because of genetic variation (Robert Plomin). Another reason is variability in instruction and expectation (LT, 
(10) Average kids who are adequately prepared in arithmetic and algebra can learn basic calculus in high school, suggests Richard Rusczyk, the Art of Problem Solving. 

Note: Typical urban Title-1 1st-grade students can learn basic algebra ideas when fused to traditional arithmetic. It's STEM math.
I know. I taught it. And, kids learned it. (LT, 2011)

✔️ There has been an attack on academic achievement since the 1930s. It's most recent form has been Common Core reform math, promoted by progressive professors like Jo Boaler in education schools. The result has been that many students stumble over simple arithmetic

Note: It is impossible to equalize outcomes (the progressive idea of sameness) unless the math content is dumbed down, down, down. The curriculum based on Common Core math standards, for example, does not meet international benchmarks, so students start behind and stay behind. What's worse is that the progressives interpret CC-math through the reform math prism and so-called "mathematical practices," making it confusing and more complicated for students. Under Common Core reform math, sameness for all seems more important than basics for all. Memorizing math facts, mastering standard algorithms, and learning the trio of skills-ideas-uses are essential arithmetic. Unfortunately, they have been cut back in many progressive classrooms. Knowledge is the basis of critical thinking (i.e., problem-solving) in mathematics, but reformers often skip to critical thinking and downplay core content, a colossal error.  

Common Core reform math confuses and frustrates students and parents. Note: Memorizing math facts, practicing standard algorithms, and drills-to-develop-skills are downgraded in today's reform math classrooms.  

Let's Get Back to Basics. Basics Are for Everyone.

I dedicate this page to educators and parents who want to improve the rigor of mathematics education in schools at all levels. We must strengthen the depth and quality of instruction and refocus on basic math skills, ideas, and uses. There is no free lunch in learning mathematics. It takes time, continual practice, review, and effort. Students need to memorize math facts and become fluent using standard algorithms, explains W. Stephen Wilson (Elementary School Mathematics Priorities, 2006). Math is "hierarchical," that is, "certain topics must be taught before others. The core content [basics] is actually quite small." The problem is that core content is not taught for mastery in the early grades of elementary school. (Quotes: W. Stephen Wilson, Mathematician, "Elementary School Mathematics Priorities," 2006)

It's Crazy! So Is Grade Inflation!
"Equalizing downward by lowering those at the top" is a "crazy idea" taught in education schools. It has impeded math and science achievement. Today's math is a dilution to sameness, observes Thomas Sowell, who explains that equalizing downward is dumbing down math and a fallacy of fairness. Reform math is equalized-down math, in my opinion. It's from a progressive ideology that attempts to artificially "arrange" the sameness of outcomes, a toxic social vision. Thus, "the curriculum has been dumbed down so everyone can pass--but no one can excel," explains Charles Sykes. The self-esteem movement has undermined grades and achievements via grade inflation. Most everyone passes with good grades whether or not they have learned the grade-level curriculum. Students graded by letters like Ds or Fs can now receive a "Pass" grade in some districts. Likewise, A and B students get the same "PASS" grade as a D or F student. What foolishness! Also, the grade-level curriculum, especially in math, is not world-class. But, educators and reformers pretend not to know this. Good intentions are not good enough for failed programs. The excuse, "We meant well,"  is nonsense! I have heard that for over 50 years in education. Here's another: "If we only had more funding, it would have worked." What irks me is that Congress keeps funding programs that don't work. I wonder why that is?     

Content should drive the curriculum, not state tests. Why? If you change the test, you adjust what happens in the classroom, i.e., the teaching of content and methods used.  "Watch a video at home then do the homework in class" (flipped classroom), in my view, shows that some teachers don't know how to teach mathematics well. Also, it doesn't work well. 

Note: Often, teachers don't teach important topics or ideas because they are not on the test. I frequently would ask a teacher why are they teaching XYZ for two weeks, even though it is not consequential? "It is on the test." Why are you not teaching the standard algorithm? "It's not on the test." I asked a teacher why she is not teaching standard long-division. "Kids don't understand it." Do you think kids understand the lattice method? Kids don't understand anything until they use it repeatedly. There is no such thing as instant understanding. Understanding develops slowly over time.

Richard Feynman, Nobel-Prize Physicist, a curious character, concluded that the elementary school math and science textbooks were "UNIVERSALLY LOUSY!" Richard Feynman wrote in his book, Surely You're Joking, Mr. Feynman, "I chose to read all the books myself." He was a great inspiration. Not much has changed over 65 years! 

Richard Feynman at Cal-Tech
Channeling Feynman: "You don't know anything until you have practiced!

The math curriculum today, for example, is below world-class by at least two years. So why did almost every state adopt Common Core reform math? Reform math is dumbed-down math designed to level outcomes, which is a radical idea.   

If math is dumbed down, then science is dumbed down, too. 7th Graders should get chemistry and physics, but this rarely happens in progressive schools. Kids need a math background in ratios/proportions, fractions, percentages, square roots, trig, solving for variables in equations, etc. U.S. kids don't have the math background they need to learn, do, or understand science. Also, students lack the reading and vocabulary skills needed for reading science texts. When I taught and later tutored 7th-grade prealgebra, the best textbooks included the trigonometric ratios and excellent worked examples. It's not advanced math. 
MariaB 7th Grade

Some of the math skills and ideas needed for science are whole number and fraction operations; measurements and significant figures (sig figs); scientific notation; unit analysis and conversion factors; SI units; ratios; exponent laws; area, volume, speed = d/t, d=rt, and density; trigonometric ratios; rearranging equations (algebraic manipulations) and solving equations for a variable. (F=ma, solve the equation for mass (m); 12 feet is how many centimeters, etc.) Notes: Common-Core-based reform math and Next Generation Science in K-6 won't get kids to this level. Seventh graders need a strong prealgebra course, or they won't be ready for Algebra-1 in 8th grade.

Reform math in elementary school over-stresses manipulatives and downplays symbolics. Reformists claim that using symbols would be too abstract for kids to grasp. Really? (I guess symbolics like 2 + 3 = 5, x + 3 = 5, or y = x + x = 3 are too difficult for 6-year-olds to grasp. Not!) The reformists are wrong. Piaget's developmental stages don't work. Our kids are underprepared. "Other nations are out-educating us in math and science." We don't seem to care. 

One can also think of 2 + 3 and 5 as equivalent numerical expressions because they name the same point on the number line (5 = 5). It is the start of algebra when an unknown x is introduced: x + 3 = 5. It is 1st grade STEM math. In an equation, the left and right sides must be the same value (equal). Because the right side is given as 5, then the left side must be 5. Therefore, the unknown number x is 3 to make the equation true. Equation structure is: expression = expression. While 3 + 2 = 5 is a simple equation using a memorized math fact, more complicated equations can be solved using guess and check and the idea of convergence.

x = 22
22 - 3 = 19
19 = 19 True

Over 65 years ago, Feynman, who loved teaching and education, blasted the lack of arithmetic in K-6 science and dismissed content such as strange vocabulary (e.g., "renaming" instead of carrying and borrowing), set theory, and different bases as irrelevant and a waste of valuable instruction time. Let's say in 1st grade, young kids should initially use a 0 - 20 number line to calculate easy combinations and then memorize the facts used repeatedly to prepare for the standard algorithms. The students can use guess and check (convergence idea) and logic to solve equations like 17 + x = 32. Thus, they need to know math facts, place value, rules, "carry," and the concept of an equation, that the left side must equal the right side in value. For example, 2 + 3 = 6 - 1 is a true statement because 5 = 5. This is content-rich arithmetic. This is the same equation in terms of an unknown number x

2 + 3 = x - 1, 

5 = 5 iff x = 6

iff means "if and only if."

Note: If 2 + 3 and 6 - 1 name the same point on the number line (5), then they are equivalent, which is the Transitive Law: Things equal to the same thing are equal. Also, if 5 - 3 = 2 and 5 + -3 = 2, then 5 - 3 = 5 + -3Thus, to subtract an integer, add its opposite (3 and -3 are opposites and sum to zero). Why change subtraction to addition? Simple! Subtraction is not commutative, but addition is. Also, to divide by a fraction, multiply by its reciprocal: ÷ 1/2 = 5 x 2 or 10. (1/2 and 2 are reciprocals whose product is 1). Working with integers on a number line should start in 1st grade.

Convergence, Instant Recall, Standard Algorithms, Critical Thinking, Abstracting (Math is abstract)

With guess and check, students learn the "idea of convergence, a basic step towards calculus," writes W. Stephen Wilson, a mathematician who points out that students must memorize single-digit addition and multiplication facts for instant recall and learn the standard algorithms. "Students must be fluent using standard algorithms." He also states that "mathematics is something you do. For example, multiplication is not understood if you cannot do it." Wilson writes, "Problem-solving at the elementary school level is a well-understood process that can be taught. Going from one step to two steps to multi-step problems gradually increases the level of critical thinking. New skills allow students to solve problems that old skills did not suffice for." Abstracting mathematics from a word problem is critical thinking.  If numbers are abstract, then why not teach numbers as numbers

More Clutter!
Elementary students K-6 don't need to learn probability, four-color theorem, networks/nodes, statistics, except for three averages, or the so-called Mathematical Practices (NCTM) that clutter the curriculum and waste time; however, in addition to basic arithmetic, they do need to learn selected ideas in algebra, measurement, and geometry beginning in 1st grade. Children are novices, not little mathematicians. Mathematical content should drive pedagogy, not mathematical practices. We should not be training kids to be little mathematicians. Moreover, knowledge drives innovation and problem-solving (critical thinking). A child engages in critical thought when solving an equation in one variable, such as 5 + x = 12 or x + 34 = 112, explains mathematician W. Stephen Wilson. Students first recognize the equation with a "missing addend" and use a process, perhaps guess and check for 1st and 2nd graders, to figure out the correct answer. Students should also use arithmetic rules and memorized factsAlso, 3 + 5 = 8 is a "number line" understanding, which is okay for novices. 

H. Wu, a mathematician at UC-Berkeley, writes, "The most difficult step in becoming a good teacher is to achieve a firm mastery of the mathematical content knowledge. Without such a mastery, good pedagogy is impossible." It is mathematical content that drives pedagogy, says H. Wu.  

First Grade Teachers who grasp the underpinnings of algebra know that the arithmetic content they teach builds the groundwork for higher mathematics. Math facts such as  3 + 5 = 8 are simple equations. Expanded notation, e.g., 362 = 300 + 60 + 2, which are equivalent numerical expressions, is analogous to polynomials. The distributive rule or property used extensively in algebra is the fundamental building block of multiplication. Lastly, there is a high correlation between how much math an elementary teacher knows and the quality of math the student learns.

Teach Kids Algebra (TKA) has been a response to the Common Core reform math. Algebra ideas are accessible to very young students, including 1st graders, via traditional arithmetic. Teaching algebra to little kids in grades 1 to 5 has been a blast. Below is my first 3rd-grade class: 2011 January to May (one semester), twice a week up to 1.5 hours per session. My Teach Kids Algebra program was born in 2011 with two first-grade classes, two second-grade classes, and one third-grade class. I volunteered as a guest algebra teacher and had administrative and teacher support. All classes learned guess and check and convergence. Also, in 1st grade, adding 3 to 6 by counting on 7, 8, 9 and adding 3 to 29 (30, 31, 32) are not different, writes Richard Feynman. Both should be taught in 1st grade. In 1st grade, the place value system (e.g., 12 = 10 + 2 or t + 2) and the standard algorithm of addition should be introduced in the first few weeks of school because "counting on" is inefficient with large numbers. Kids need to memorize important single-digit math facts and learn large numbers. 

My 2011 Algebra Class (Teach Kids Algebra) Grade 3
Novices need guidance, encouragement, explained examples, 
and immediate feedback.

In 3rd grade TKA, students solved equations like 92 - (43 - x) = 68 using guess and check (April 19, 2011). Students did not work in groups or share their answers with other students. They had to show me the numbers they tried and the calculations. A student had to rely on themselves, not the group. In addition to working with linear equations, table building, plotting points, and interpreting graphs, 3rd graders worked on more challenging word problems such as this one: A machine uses 4 gallons of gas to operate for 1 day. How many gallons of gas are needed for 16 machines to operate for 1/2 day? (Students kept their answers secret. But, they had to explain to me their reasoning and demonstrate correct calculations.) 

Sometimes, I would say, "Your reasoning is okay, but your calculations are not. You have to show me your calculations to get the right answer. Here's a hint." Getting the right answer through convergence (guess-and-check estimation process) is the goal. 

In short calculating skills are an intrinsic part of math programs and problem-solving in math. Some students got the correct answer but couldn't show me how they got the answer. "I guessed it" was not acceptable. "I cannot accept your answer if you can't show me how you got it. Try again." Kids need encouragement and guidance. 

Perseverance and confidence do not grow on trees. 

In an equation, Mari, a third-grade student, tried 8 different numbers in guess and check. She was frustrated. "Is it a fraction?" No, it is an integer (whole number). I inspected her numbers and noted that 20 almost worked (69). I asked her, "What adjustment would you make to 20 to get to 68?" Convergence. She did, and her adjustment (to 19) worked. She was overjoyed and yelled, "I got it!" 😊😊😊 I shall always remember Mari and students like her who put forth the effort needed to learn. Sometimes, kids miscalculate. But, kids who are weak in calculating should not use calculators as suggested in Everyday Mathematics, a popular K-5 reform math program. Instead, they need extra practice to cement math facts into their long-term memory--drills with flashcards work. 

Students must get the process right and the calculation right to get the right answer. I hate it when teachers tell kids that explaining how to get the answer is much more important than getting the correct answer. How silly.

Kids need to be persistent in solving problems: "You can't give up! "It's not always easy," and I explain this to students. Mistakes happen and help you learn. Also, I avoid praising students for no good reason.

Thus, in first grade, students use a number line to calculate and then memorize the combinations used repeatedly. Use flashcards for instant feedback. To solve equations such as 11 + x = 72, 1st-grade students should use guess and check. Try numbers to find out which number worked. Counting on is not efficient in many equations 1st graders can solve, but guess and check is very important. There are equations in engineering and physics that can only be solved via guess and check, says Feynman. (Some smart students might "see" a pattern. Subtract 11 from 72 to get the number x, which leads to an inverse algorithm. How are 11 and 72 related to getting x = 61?) In 1st grade, students should learn addition with carrying and simple subtractions with borrowing. At the least, it requires extensive practice and knowledge of single-digit addition facts.

Teach Kids Algebra (TKA)

STEM Math for Grades 1-5
TKA started in 2011 as a reaction against Common Core reform math. I fused basic algebra ideas with standard arithmetic, not reform math. The importance of traditional arithmetic was stressed, starting with the automation of single-digit math facts that supported standard algorithms. It begins in 1st grade with counting and the number line, and a place value system. 

Note: Giving immediate feedback is important when children are learning algebra fused to traditional arithmetic. It's STEM mathChildren need guidance and encouragement. For grades 3 to 5, hour-long lessons were once a week for the school year. For 1st and 2nd graders, there were 6 or 7 lessons. 

Algebra in 1st & 2nd Grade
It starts in 1st grade with missing addend equations, such as x + 4 = 12 + 7 with guess and check methods. 12 + 7 are like terms and should be added to make 19. If the right side is 19, the left side must sum to 19, too. Also x-y table building and graphing linear equations (e.g., y = x + x + 3) are 1st-grade content.
Click 2nd Grade TKA Algebra Class (2-20-19 to 4-17-19): 7 Lessons, one hour each. 2nd-Grade Task: Given y = x + x + -4, build an x-y table when x = 0, 1, 2, 3, 4 and plot the number pairs (x, y). (Yes, that is negative 4)
2nd-Grade TKA Algebra Class
STEM Math 2019

Kids can learn much more content than the current curriculum. Learning algebra has little to do with developmental appropriateness (Piaget's stages) but everything to do with knowing the prerequisites (i.e., background knowledge). In other words, the fundamentals of arithmetic are not developed well in elementary school, beginning in 1st grade.
Many seem concerned about education, but good intentions are not good enough. Our kids lag behind. They are not achieving like their international peers. Educationists blame parents, society, and money as scapegoats for poor student performance, but never themselves--the teaching in the classroom (i.e., curriculum and instructional methods). First, Common Core math and state math standards are not world-class. Second, the practices used to teach Common Core-based reform math are substandard, e.g., minimal teacher guidance (i.e., group work such as discovery learning). As the late Zig Engelmann would shout, "It's the teaching." For decades, classroom teaching has been directed by schools of education that prepare teachers. Herein lies the problem with progressive "math educators" like Jo Boaler, who heads the Education Department at Stanford University. Progressive education professors endorse and champion reform math practices.

Laurie Rogers (Betrayed, 2010) writes, "In reform math, children don't practice skills to mastery." Common-Core-based reform math is a product of progressivism. Other nations are out-educating us in mah and science. Our kids are underpreparedPeg Tyre (The Good Schools) sums it up this way, "Your attitude counts. Math is not a talent. Being good at math is a product of hard work. The harder you work, the better you will be." In short, we expect less from children than parents in high-performing nations. Parents should supplement school math at home and teach children traditional algorithms starting in the 1st grade and memorize math facts from the start. 

1. Thinking in math requires knowing math facts in long-term memory. Kids must know facts from memory to perform arithmetic. ​

2. Children learn through mastery. "Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning." (Zig Engelmann) Exactly!

3. Cognitive Science Summary:

The more I know, the more I can learn, the faster I can learn it, the better I can think and solve problems. LT

4. Learning is remembering from long-term memory. If I can't remember something, then I haven't learned it well enough. Much is taught, but little is learned.

5. Practice is good for kids; it makes them better at math. Practice builds conceptual knowledge, math skills, and competency. 

6. Content knowledge and supporting calculating skills are essential for problem-solving. ​Exactly! In mathematics, you can't live on concepts alone. To do problem-solving in math, good calculating skills are required.​​

​​7Problem-solving is domain-specific. ​There are no generalized thinking skills independent of domain content knowledge.

8. The progressive narrative of reform math, tech "solutionism," constructivism, minimal guidance methods of instruction, and one-track grade-level math (sameness) has screwed up math learning. Also, state standards that are based on Common Core math are not world-class. Children are not taught standard arithmetic for mastery; consequently, they stumble over simple arithmetic.

Content should drive the curriculum, not tests. 


For decades, the progressive education movement with its "toxic ​social vision" of "sameness" (Thomas Sowell) has dominated many classrooms and impeded student growth and achievement in both mathematics and reading. Progressive teaching encourages thinking without content knowledge, not memorization and mastery of content (i.e., the fundamentals), even though "thought without content knowledge" in long-term memory "is empty" (Immanuel Kant). 

Right Answer

For novices, math is often a "process of getting the right answer."

The answer is important, but so is the process. A student can get the correct answer by using the wrong process or reasoning. "​​Your conclusion is correct, but your reasoning is wrong," writes mathematician Eugenia Cheng (How To Bake Pi, 2015).

You won't like what I write here about ability.

No matter how long or hard I practice-practice-practice, I will never approach the performance level of 11-year-old Chloe Chua (Click), the Junior Winner of the 2018 Menuhin Competition. Chua's musical ability is off the charts. Even though my musical ability is unexceptional, it does not prevent me from listening to and enjoying classical music. "Practice does not cause talent," explains mathematician Ian Stewart. The talent must already be there (DNA)The talent, ability, or skill can be developed only through excellent instruction, lots of practice, and feedback. In short, practice unleashes talent and improves performance. It takes hard work!

You Are What You Inherit

Ability widely varies (Charles MurrayReal Education, 2008) and is mostly genetic (Robert PlominBlueprint, 2018 MIT). Plomin points out that "Genetics [DNA] contributes substantially to differences between people," You are what you inherit. The differences in school achievement are 60% genetic, reports Plomin. But, the percentages are not deterministic. 

We were not told that a century of genetic research had shown that the variation in school achievement is 60% DNA. Poverty is not the root cause of variation in school achievement. It is mostly DNA! Also, the assumption that high self-esteem produces high achievement is false. Like poverty, low self-esteem is not the root cause of low achievement. While we are not our parents, we get our genes (DNA) from them. 

Wrong Assumption 

If we provide the right environment in school and at home, students will do well academically. The progressive reform idea sounds great, but it doesn't work that way. As it turns out, school achievement is mostly genetics, not nurture. However, it does not mean that children with lower aptitudes in math can't learn the fundamentals of arithmetic and algebra at an acceptable level with proper instruction, hard work, effort, and lots of practice--even AP calculus, which is for average students who are prepared. The percentages are what is and do not predict "what could be," says Plomin. They are not deterministic. For more information, read DNA, Not Nurture

The percentages, such as 60%, are what is and do not predict what could be, says Plomin. They are not deterministic. Plomin does not say that intelligence is genetically fixed.  It's not. He writes, "Genetic influences are probabilistic propensities, not predetermined programming." 

Nurture: The 40%

Environmental influences are "unsystematic, random experiences over which we have little control," says Plomin. They are broadly defined as "nongenetic" If the differences in school achievement are 60% genetic, then the remaining 40% of the variance is nongenetic (i.e., environmental influences). So, nurture is important, too, but not nearly as much as we used to think. Even if the inputs are the same in school, the outputs will be different due to genetic variation. Genetic variation also explains why the achievement of children from the same family can vary substantially. "Equal opportunities do not create equal outcomes." 

Below: Screen Shot 2011 

ADD ONE - First-Grade Concept


We don't push kids hard enough. 

More coming soon, including clarifications. 

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