Friday, February 19, 2021


 Inklings1 March 2021

Observations, Ideas, and Opinions on Math Education by a Contrarian in 2021, a Divergent View 😎

Here are two of my favorite passions: 

Note: Inklings1 is continued at Inkling2.

Passion 1. Teaching

Reform Math Dominates Math Teaching. It is not world-class. It puts our kids at risk. 

Practice does not cause talent; it improves performance. Kids need to practice-practice-practice to learn arithmetic well. They should overlearn (i.e., memorize) math facts for auto-recall, such as the multiplication table, "so that summoning up those facts [instant retrieval] during problems-solving becomes undemanding," writes Sanjay Sarma in Grasp, 2020. Practices such as getting the "right answer" or requiring students to "show their work" are not racist. Also, teaching math in a linear fashion with skills taught in sequence or valuing procedural fluency over conceptual knowledge is not racist. Math is hierarchical. You can't skip around. There is a sequence of topics with prerequisites (Robert M. Gagne), so it is highly structured. Also, not being able to calculate quickly using standard algorithms hinders the problem-solving process, observes W. Stephen Wilson, a mathematician. 3-3-21, 3-4-21

Getting ready for algebra starts in 1st grade when 3 + 8 = 11 becomes  + 8 = 11 or 11 - ❏ = 8, where box is a number that makes the equation true. Box is a variable like x in algebra. Included in my Teach Kids Algebra program (TKA) are true and false equations, think like a balance (equality idea), variables, solving equations in one variable, input-output model and function rules, plotting (x,y) points in Quadrant Iconventions (algebraic rule for substitution), equation-table-graph to represent simple functions, formulas (perimeter of polygons), and more. 

In 2019, I introduced integers in 2nd-grade TKA using a number line to calculate. But, in the AAAS Science--A Process Approach (SAPA 1967), integers were introduced in 1st grade, where one of the tasks was to arrange the numeral cards from -5 to 5 in order. How many steps were there from negative three to four? First-grade students were asked to start with an integer, let's say -4, then take one more than it (-4 + 1, which would be -3) or one less than it  (-4 - 1, which would be -5). Children gained a number line understanding of integers. (When I was Head of Science at a K-8 private school in the late 60s, I taught the Xerox kits of SAPA K-6 and integrated SAPA ideas with SSSP Time, Space, & Matter curriculum in grades 7 and 8.  3-7-21  

Algebra ideas are accessible to very young children through traditional arithmetic via worked examples. TKA also reinforces the importance of mastering traditional arithmetic, especially math facts and standard algorithms (i.e., calculating skills) used in problem-solving. In short, I fuse basic algebra ideas into arithmetic. Moreover, 1st and 2nd-grade students learned all these things in 7 hours of instruction. Some TKA ideas are from The Madison Project (1956). 3-6-21

Note: Asian children are taught mechanics first with an explanation later, and it works. We used to do that!

All kids need to do math, not be shielded from it, explains John McWhorter, a professor at Columbia University. In my opinion, if you can't do it, then you don't know it. Learning real math well requires a lot of "basic knowledge and technique," observes Ian Stewart, a mathematics professor. Math is universal. There is no such thing as "white" math. Yet, some of these normal practices (e.g., getting the right answer, showing your work, etc.) are said to be racist and are slowly creeping into progressive classrooms, which is scary. Well, they are not racist! 
A 4th-grade TKA algebra student is  
figuring out rules for input-output tables. 
The rule "multiply by 2, add 5" can be written as y = 2x + 5.
It's critical thinking!

The idea that black kids don't need to master basic math and algebra is absurd because "precision" is racist. Black and Hispanic students can excel. I see them in my TKA lessons. Still, "culturally relevant" instruction is a basic tenet in the Dismantling Racism in Mathematics Instruction movement. It is already in many school districts. It's real, so be on the lookout for it. A focus on getting the right answer and showing your work is not racist or white math. They are fundamental in teaching mathematics. 3-3-21, 3-8-21

Gap Closing & Fallacy of Fairness: What's happening? Progressive education leaders have elevated the holy grail of equity-diversity-inclusion as a replacement for academic achievement by "lowering those at the top," an equalization. Gap closing should not be an educational goal, observes Sandra Stotsky (The Roots of Low Achievement, 2019). Equalizing down to level gaps is a "fallacy of fairness," explains Thomas Sowell (Discrimination and Disparities, 2019). Still, progressives insist that equal opportunities should equate to comparable outcomes, but they don't. How can they? Similar inputs do not produce matching results. 

Sowell explains, "Asian American students spend more hours studying than either white or black American students. How surprised should we be that academic outcomes show a pattern of disparities ... [that are linked] to the pattern of disparities in the amount of time devoted to school work?" Such facts are seldom covered by the media. (Also, remote and its variants are failed strategies. Kids are learning significantly less than before. Closing schools was a bad idea. Bad ideas happen when state leaders lack wisdom)  2-26-21, 2-28-21, 3-4-21

My Premise: Children can learn so much more than is currently taught. For example, first-graders in 1877 learned more arithmetic than today's 1st graders. They could also read. Also, I have shown that very young children can learn algebra ideas via arithmetic. I have taught my algebra project--Teach Kids Algebra (TKA)--in the 1st-2nd-3rd-4th-5th grades, off and on, since 2011. It is STEM math for elementary students. Ideas illustrated below were introduced in my TKA classes with many other STEM skills, such as linking equations to (x-y) tables to (x,y) graphs, even in 1st grade. Almost all the students in my TKA program are minorities (90%), and I find very bright kids that way, both black and Hispanic. They can be high achievers, too. In TKA, I fuse key algebra ideas into standard arithmetic. Thus, learning arithmetic well is an important step to algebra. 

We underestimate what very young children can learn. Elementary school children, starting in 1st grade, can learn much more content than is usually taught, and they can learn it earlier and faster but only with the proper background or prerequisites in place (Gagne).  
3rd Grade TKA - STEM Math
Answering questions and giving individual feedback 
are important in TKA. 

😎 TKA: Teach Kids Algebra ProgramClick for more information about TKA.

1st Grade TKA: (❏ + ❏) - 3 = 5 (Box must be the same number: Algebraic rule for substitution.)

1st-2nd Grade: (❏ + ❏) = 7 (Box must be the same number: Algebraic rule for substitution.)

2nd-3rd Grade TKA: 12 + 4 + ❏ = 12,  ❏ = ? 

3rd Grade TKA: 12 x 4 x ❏ = 12,  ❏ = ?  

3rd Grade TKA: 5 ÷ 1/2 = 5 x 2 (True: 10 = 10)

4th Grade TKA: Is ❏ x (❏ + 1) = (❏ x ❏) +  an identity?

(Note: The identity idea is from Dr. Robert Davis' The Madison Project, 1956.) Students have learned some identities that have familiar names, such as the commutative law or the distributive law. 

✔︎ Notes for (❏ + ❏) - 3 = 5, Grade 1. The reasoning goes like this: If the right side is 5 (given), then the left side of the equal sign must also make 5 (the idea of equality: "think like a balance"). How do you make 5 on the left side? You must obey the algebraic rule for substitution, so  (box) has to be the same number (doubles), and in this case,  = 4 by guess and check. Thus, (4 + 4) - 3 = 5, which is 8 - 3 = 5 or 5 = 5 (True Statement) by mental arithmetic. Yes, 1st graders can follow a chain of reasoning. It's called "critical thinking," explains W. Stephen Wilson, mathematician. Does (6 + 6) - 3 = 5? No, 12 - 3 is 9 and 9  5. Therefore, (6 + 6) - 3 = 5 is a False Statement. Note:  is an inequality symbol that means "does not equal to." Click First Grade. Also, students must follow the order of operations!

Below: A 2nd-grade TKA task in 2019 was: Given the equation, y = x + -4, make an x-y table of values for x = 4, 5, 6, 7, 8; then plot the (x, y) points on the graph. Yes, that is negative 4, so use an integer number line if needed. Also, the Equation-Table-Graph ideas of functions are a recurring theme in all TKA classes. Kids can learn so much more than the current reform math curriculum. Moreover, preparation for algebra, such as equation structure (expression = expression), the idea of variable, input-output, rules, and so on, begins in the lowest elementary school grades. Getting ready for algebra starts in 1st grade when 3 + 8 = 11 turns into  + 8 = 11 or 11 - ❏ = 8, where box is a variable that stands for a number to make the equation true. It is solving an equation for the unknown. Click here for more information about the seven lessons I gave to 2nd-grade students. 

TKA 2nd-Grade STEM Algebra Skills 
Practice/Review Sheet

Reform Math Dominates Math Teaching. It is not world-class. It puts our kids at risk. I read parts of the new California Math Framework: Equity & Engagement. I could not believe that cultural-relevant lessons were stressed. Real math is not racist and should be taught to all students, including black students. Many of the assumptions about equity and engagement in the California math framework are not back by evidence. 3-3-21

The problems with reform math start in 1st grade. Kids need to learn content, both factual and procedural knowledge, but they don't know nearly enough. They are weak in calculating skills, especially the standard algorithms which are not stressedW. Stephen Wilson points out that the standard algorithms are essential and one of the Five Basic Building Blocks in elementary school mathematics. 

Engagement is not the same as learning something. Teachers spend too much time on so-called engagement activities and not enough time on basic instruction. Equity dumbs down curriculum by lowering those at the top, explains Thomas Sowell, a black scholar. Equalizing down suggests that some students get special treatment, others not, which, I believe, is a bias against students who study and work hard to achieve, such as many Asian students. It is not only arithmetic and algebra; it is also chemistry and physics and the humanities. 3-2-21

Sandra Stotsky (The Roots of Low Achievement, 2019) reports that there has been a reduction in academic demands. "Educators still don't know how to turn massive numbers of low achievers into higher achievers," observes StotskyGap closing by leveling academic standards to the floor is an "unworkable educational goal," writes Stotsky. Teachers want to keep their jobs and have good evaluations so, to artificially close gaps, they often engage in rampant grade inflation or give group grades. Furthermore, "They may teach less content to higher achievers to equalize achievement between higher and low achievers or ignore what college teaching faculty say college readiness means," writes Stotsky.

✔︎ In my opinion, Jean Piaget's constructivist approach is responsible for many problems in elementary school arithmetic and algebra. Like Dewey, he advocated working in groups and engaging in discovery or project-based learning, which significantly reduces the teacher's role to a discovery facilitator. Students don't have to learn content. It's not important, but it is.  3-3-21

▶︎ Sanjay Sarma (Grasp, 2020) writes that these constructivist strategies (e.g., discovery, project-based, etc.) "do not support the cognitive processes necessary for learning and may even smother them." John Sweller formulated cognitive load theory (working memory). Working memory has limited space to operate on information. In short, students learn very little content in discovery lessons or project-based and other constructivist activities. His research indicates that "general problem-solving skills are essentially unteachable" and a gross waste of instructional time. In short, critical thinking skills (called problem-solving in math and science)--independent of content--do not exist. Sweller also points out that using worked examples is significantly better in solving arithmetic and algebra problems than "problem-solving by yourself." 3-2-21

Critical thinking is the product of domain-specific knowledge. 

▶︎ EducatonNext Survey

According to a survey by EducationNext, "Respondents categorize 61% of teachers as either excellent or good and 14% as unsatisfactory." The opinions are incongruous with national and international assessments, which show that most students are not proficient in reading, writing, and math (e.g., NAEP 2019). Then again, surveys are not that reliable. In the real world of facts, not opinion, teachers are not teaching mathematics well, starting with basic arithmetic. For example, after eight years of Common Core reform math, only 24% of 12th graders were found "proficient" in math. 3-2-21

David G. Bonagura Jr. writes in the Wall Street Journal, "Contrary to today's education theories; memorization is critical in the classroom and life." Memorizing math facts and standard procedures is good for kids. It enables them to solve math problems.


Instructional Note  ✔︎  Division can be thought of as multiplying by the reciprocal. To divide by 1/2, multiply by its reciprocal (2). Morris Kline writes, "The operations with fractions are formulated to fit experience." They must also be efficient. Thus, if we slice 5 oranges in halves, we have 10 (halves), which is why 5 ÷ 1/2 = 5 x 2 is true (10 = 10)

Note: 1/2 and 2 are reciprocals because their product is 1. Also, 3/4 and 4/3 are reciprocals because 3/4 x 4/3 = 1. Therefore, 12 ÷ (3/4) is the same as 12 times the reciprocal of 3/4, or  12 x (4/3)= 16.) In short, to divide by 3/4, simply multiply by its reciprocal (4/3). 

Numerical check: 16 sets of (3/4) are 12 or 16 x (3/4) = 12. It always works! (Yes, even for 8 ÷ 4, which is 8 x 1/4 = 8/4 or 2. How many sets of 4 are in 8? 2 sets of 4.) There are many fractions in algebra, so students must know the basics of fractions and how they work. 

[Also, read SmartKids  All the standard algorithms for whole number operations are based on single-digit number facts, which must be memorized. We make up algorithms to fit our experience in the real world, but they must be efficient for practical use. If we take 10 oranges and split each orange in half, then we would have 20 halves. That is, 10 ÷ 1/2 = 20 must give the same answer as 10 x 2 = 20; therefore, 10 ÷ 1/2 = 10 x 2. From this abstract idea and others, such as the idea of reciprocals (1/2 and 2 are reciprocals because their product is ONE), we should be able to make up an efficient algorithm for the division of fractions that always works in the real world. We have: To divide by a fraction, multiply by its reciprocal, which is the standard algorithm. It fits our experience. The proof, however, is left to mathematicians, not to novice kids. From SmartKids] 3-8-21


Beware!  Beware of harmful trends that give special treatment for some and not others, such as culturally relevant teaching methods, equitable math, and the equity agenda of racial justice. One would think that arithmetic and algebra would escape such nonsense, but it is not the case. The far-left extremists say that "white" math is used to "uphold capitalist, imperialist, and racist views," reports Sam Dorman at Fox News. Really? The extremists also claim that "getting the right answer" or "showing your work" to get the right answer is "evidence of white supremacy." How Stupid! These far-left radicals seek to "undo racism in mathematics." But, in my opinion, their bold assertions destroy their credibility. Still, many educators are willing to believe or accept this junk. 

▶︎ Algebra is deeply rooted in standard arithmetic, not reform math. The problem has been that standard arithmetic is not taught well or marginalized. Instead of traditional algorithms, for example, children are taught many nonstandard, alternative algorithms that lead nowhere and are often more cumbersome, complicated, and inefficient. Also, beginning in the 1st grade, the content in reform math is below international benchmarks. It's not world-class! By the 4th or 5th grade, our students are about two years behind their peers from top-performing Asian nations. 

▶︎ Progressive educationists claim that racial disparities are discrimination, that achievement is often called "privilege." Achievement is not "privilege." It's hard work, and disparities are not always discrimination, points out Thomas Sowell--a black scholar. Still, teachers assert, "We know best how to educate children." Equity is reached when outcomes are the same for all students. Really? I think not! Why? The elite educationists can't explain why 76% of 12th graders are not proficient in mathematics at the 2019 NAEP standard after eight years of Common Core reform math. Also, critical thinking is not a generalized skill, as many educationists believe. Critical thinking is domain-specific, but teachers are not taught this in ed school. Why? It doesn't fit their utopian vision of the future: sameness.  

▶︎ Moreover, for 50 years, the stock excuse for low performance is the lack of adequate funding. Really? Billions and billions and billions have been poured into schools. Teachers have not figured out how to change low-performing students into high-performing students without lowering the bar to the floor. Equalizing downward by lowering those at the top is a crazy idea taught in schools of education, observes Thomas Sowell, who brands it as a fallacy of fairness. Equalizing down is a special treatment for some but not others. 

▶︎ Ying Ma (Fox News) writes, "The equity agenda calls for racial justice, but inevitably inflicts injustice on groups it does not favor." Thomas Sowell labels it for what it is, a fallacy of fairness. It mistreats students who work hard to achieve and marks them as privileged, which is wrong. Radicals in education often use terms like diversity, equity, and inclusion, but are these not ideas that give special treatment for some and not others? Why should most federal funding be diverted to low-performing students who don't make much progress and little for high-performing students who are not academically challenged with a lot of group work in the name of equity, diversity, or inclusion?

Thomas Sowell, a black scholar, explains that many disparities are not discrimination. Outcomes can never be the same! 

The Supreme Court decreed that racially segregated schools were "inherently unequal" and proclaimed mandatory busing as the means to integrate schools, but "seating black school children next to white school children resulted in no general educational improvement," writes Thomas Sowell. The Supreme Court was wrong, explains Sowell, a black scholar. At the time, he says, many racially segregated schools were not inherently inferior. 


1877: First Grade in 1877 was far different from 1st Grade today. In 1877, Kids learned to read and master the classic curriculum of arithmetic. 

"In America's one-room schoolhouses, Ray's Arithmetic was used alongside the McGuffey Readers" (e.g., McGuffey's First Eclectic Reader, Revised Edition, K-2.). 

▶︎ Ray's Primary Arithmetic (© 1877) combined Grades 1 and 2 into a tiny 4.5" x 7.25"  book of 94 pages and 89 Lessons. Primary Arithmetic "covers all four basic functions: addition, subtraction, multiplication, & division in single digits with word and money problems." Below is a typical addition word problem from Lesson XXIII. First-grade students read the word problems and use arithmetic to answer the questions. Also, there are many drills in the 1st-2nd grade compact book. Ray's Primary Arithmetic was arithmetic, not reform math. 2-19-21

Note: School children do not need to prove why an algorithm works. Nor do they need to learn many different algorithms for an operation or write explanations. They need to know math facts and axioms, perform standard procedures efficiently, and apply them to solve problems (cases). In my opinion, educators spend far too much class time learning dead-end algorithms and "understanding" often at the expense of mastering standard algorithms, which require memorization and extensive practice. 

1st Grade: Ray's Primary Arithmetic, Lesson XXIII

At the end of the first marking period, 1st-grade students could read the problems and use simple arithmetic to answer the questions. Later, the problems were more complicated and involved multiplication: Multiply 3 and 7, then subtract 6. 

▶︎ Certain progressive educators claim that getting the "right answer" or "showing your work" to get the right answer is "evidence of white supremacy." How stupid! The algebra program I present to 1st-grade through 5th-grade at a Title-1 urban school, with 90% minorities, requires students to get the right answer and show how they got the correct answer. Kids do not work in groups. Mathematics is based on facts, not opinions. I think some teachers hate math. Learning arithmetic well is hard work. Sadly, memorization and practice have fallen out of favor in many progressive classrooms. 2-18-21

Progressive educators are wrong! Critical thinking skills are not independent of domain knowledge. Students who study math should develop a toolkit of facts, procedures, rules, and cases in long-term memory. These fundamentals are the knowledge required for critical thinking in math (aka problem-solving). There are no shortcuts. 

▶︎ After 8-years of Common Core reform math, only 24% of 12th graders are proficient in math (2019 NAEP), not a favorable statistic. Based on national and international tests, I have concluded that learning basic content has not been a top priority in progressive schooling. If it was, then scores would match the math scores of top-performing nations. Given the amount of money spent on K-12 education in the United States, we are not the best or close to being the best. For decades, we have imported talent from the Asian nations because not enough had developed here. Furthermore, many students who enroll in community colleges are typically placed in remedial math (high school algebra). Students who have trouble with algebra are the same students who had difficulty with basic arithmetic, especially fractions, ratio/proportion, and percentages. Unfortunately, many have not mastered the x-table and have been weaned on calculators since elementary school. 2-18-21

Dr. Scott Atlas points out, "We have children, young children, wearing masks, being separated, thinking they are an infection vector for everyone and that everyone is a danger to them ... It is almost insane." 2-16-21

It is an age of foolishness. (Charles Dickens)  What are we doing to our children? Also, there is "gap chasing" and a "fallacy of fairness." Still, I am optimistic! Things will get better.

What's happening? Progressive education leaders have elevated the holy grail of equity-diversity-inclusion as a replacement for academic achievement, which is an alleged privilege, via gap closing through equalization by "lowering those at the top. Gap closing should not be an educational goal, observes Sandra Stotsky. Equalizing down is a "fallacy of fairness," explains Thomas Sowell. Still, progressives insist that equal opportunities should equate to comparable outcomes, but they don't. Similar inputs do not produce matching results. 2-26-21

▶︎ The achievement gap hasn't closed. Are we chasing after the wrong goal? Sandra Stotsky thinks so because the reforms have not worked. (StotskyThe Roots of Low Achievement, 2019) For example, achievement in math has stagnated ever since states adopted Common Core in 2011. Thomas Sowell points out that "equalizing ... by lowering those at the top ... is a fallacy of fairness." Lowering the bar has not been a solution. More money isn't, either. 


▶︎ In the real world, academic ability widely varies, so all students can't possibly attain an identical level of achievement. Even with the same inputs, the outcomes will differ. Also, educators have not figured out how to change low achievers to high achievers. It must be magic! Today, the idea of equity is to keep all students at the same level of achievement, which is wrong. No student gets ahead. It is low expectations in the guise of equity! But, improving the curriculum for all students is a start in the right direction, as Stotsky points out

"Robert Gagnethe chief architect, was responsible for developing the systematic structure of Science--A Process Approach (K-6, 1967)." The entire curriculum was based on the theory that any learning act--such as a process of science--can be broken down into component skills. These skills can be arranged and taught in a hierarchical order from simple to complex. Furthermore, successive exercises in each process should build upon earlier exercises in a progressive sequence, and at the same time should introduce variations in subject matter." (Sanderson, Barbara A., and Kratochvil, Daniel W.)

Gagne's idea that the curriculum can be broken down to component skills taught in hierarchical order from simple to complex is not only applicable to science; it is especially applicable to arithmetic and algebra. In my opinion, a good science curriculum should have both process and content components.  3-7-21

Passion 2. Photography 

How I shoot photos: Click LT's Photo Page. I shoot test photos of school-aged children and young adults and photo illustrations for my math and science websites. 

Gabby, a teenager, uses concealer make-up. She is sitting in the shade under a canopyNo flash or reflector was used to light the face or eyes. The fill was from the cement floor. The white balance was set to Cloudy. Data: Canon 85 mm f/1.4 L Prime Lens, 1/1000, f/2.8. EC = 0. Most of my photos are verticals. I use the Canon EOS 5D Mark IV. Outdoors, I often use a Canon flash unit on the camera set to high speed. And I process images to give them more snap. 


It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness... Charles Dickens, A Tale of Two Cities.


©2021 ThinkAlgebra/LT

We shall miss you, Rush, 70, 2-17-21. God-speed!