Inklings2

😎Inklings2: Observations, Ideas, and Opinions on Math Education by a Contrarian in 2021, a Divergent View. Inklings3 will open soon. 

My Impressions
March 2021

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➜ Calculating skill is one of the five basic building blocks of elementary school arithmetic, explains W. Stephen Wilson (Elementary School Mathematics Priorities). He writes, "You cannot teach mathematics without the place value system, standard algorithms, and our other building blocks." In short, the basics are for everyone. 

Tom Loveless notes, "The real problem is that the research evidence that latest reforms boost reading or math achievement is spotty at best." 

➜ The problem has been deeply rooted in our curriculum and instructional methods for decades. It will be hard for teachers to change from discovery/inquiry or project-based group work to knowledge-based schooling and explicit explaining of carefully selected worked examples. Ian Stewart, a mathematician, points out, "Mathematics happens to require rather a lot of basic knowledge and technique." Marginalizing the standard procedures (i.e., technique) has been the wrong approach. And implementing constructivist "minimal guidance methods" of instruction has paved the way for minimal learning, explain Kirschner, Sweller, and Clark ("Why Minimal Guidance During Instruction Does Not Work...", 2006). 3-23-21 

➜ Minimal Guidance = Minimal Learning!

➜ Delay, Delay, Delay ...

➜ Common Core (CC) and state-standards-built-on-CC delay the standard algorithms of whole numbers. It's called reform math. Also, CC is not world-class math. We were misled into thinking that CC math was internationally benchmarked. It wasn't. I pointed it out in 2010 when I compared CC's 1st-grade math draft to Singapore's 1st-grade math syllabus. In 2011, my Teach Kids Algebra (TKA) 1st and 2nd-grade curriculum (7 Lessons) was a strong reaction against Common Core reform math. No matter the grade level, my best students in algebra had a good command of single-digit math facts, place value, standard algorithms. By 4th grade, TKA students studied a more structured approach to solving equations (unpacking/inverse), fractions/decimals, and percentages.  

➜ Even though my algebra program (Teach Kids Algebra) is considered acceleration, I teach in mixed classrooms and expect most students to learn at least 70% of the pre-algebra skills. But the expectation is not enough. I also give students help and encouragement during each session. TKA students do not work in groups or use calculators. They are explicitly taught via worked examples. TKA is STEM math for young elementary school students and builds on traditional arithmetic. There are rules to learn, including the "substitution rule" for variables. In the 3rd and 4th grades, TKA students also learn inverses and unpacking to solve equations, although I often start the undo or inverse idea in 2nd grade (e.g., 12 + 4 - 4 = x).

I do not Zoom. Zoom is not school. 3-26-21

Students are not learning enough basic arithmetic early enough. For example, they don't master multiplication in the 2nd and 3rd grades like kids in top-performing nations. The problem will persist as long as teachers focus on strategies and alternative algorithms (aka reform math) and use inefficient minimal-guidance methods (e.g., group work, discovery, project-based, etc.) and test prep instead of the explicit teaching of traditional arithmetic from the get-go (1st grade on up). Often, standard algorithms are delayed by Common Core or State standards mainly built on Common Core.

By 4th grade, students should focus on fractions and fraction operations--the international benchmarks, but, instead, they are still on addition and subtraction. Unfortunately, too many students come to 4th grade or 5th grade without knowing the multiplication facts, a 2nd-3rd grade benchmark. Over the decades, I have seen this in many classrooms. 3-19-21

From Jill Barshay at the Hechinger Report (3-22-21) about reading comprehension:

• Generally, children posted higher comprehension scores after reading a print version of a picture book compared with a digital version.
• Digital enhancements, such as games, pop-ups, and sounds, can distract children from the narrative storyline. Built-in dictionaries were bad for comprehension but good for vocabulary development. 

➜ LT: For vocabulary development, which is the missing link in reading comprehension, students would benefit from a workbook written specifically for that purpose, such as Vocabulary Workshop from Sadler. Many Catholic schools and other independent schools use Sadler. Also, content-rich history lessons from Core Knowledge (CK) with vocabulary in the sidebars (e.g., Ancient Greece and Rome) are excellent, too. Vocabulary lessons are in the CK Teacher Guides. And it is all free via pdf. (Note: Core Knowledge or CK should not be confused with Common Core or CC. Also, CK focuses on content-rich history and geography, not social studies.) 3-22-21

3-18-21

"Teaching is very easy if you don't care about doing it right and very hard if you do." Thomas Sowell 

"I don't believe I can really do without teaching." [When new ideas are not coming to mind, I must have something to do.] If I am teaching, "at least I am doing something and making some contribution." It never hurts my teaching, says Feynman, to review the fundamentals in my mind! "Is there a better way to present them?" Richard Feynman

"Fairness as equal treatment does not produce fairness as equal outcomes." Thomas Sowell

"But in recent times, virtually any disparity in outcomes is almost automatically blamed on discrimination, despite the incredible range of other reasons for disparities between individuals and groups." Thomas Sowell (Discrimination and Disparities, 2019)

The idea of equal outcomes is nonsense, yet it is the goal of radical education leaders whose strategy has been to "equalize downward, by lowering those at the top. It is a crazy idea taught in schools of education," writes Thomas Sowell. It is biased.

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Clogging

"Working memory is limited. So "whenever something clogs up working memory, your all-around problem-solving abilities take a hit ... Learning via worked examples, instead of solving problems for yourself, is one potential way past such working-memory roadblocks," writes Sanjay Sarma, MIT (Grasp, 2020). One major roadblock is not having auto recall of multiplication tables, which should be "overlearned" to stick in long-term memory and be instantly available in the problem-solving process in the working memory.  3-21-21

Furthermore, in my opinion, remote or hybrid is "pretend" education. "71% of parents think their kids learned less than they would have had schools remained open ... [while] 72% of parents say they are satisfied with the instruction and activities provided by schools during the closure," according to surveys conducted by EducationNext (Winter 2021). (Really? Are parents satisfied with their kids learning less?) 3-21-21

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Give teachers a break. It is not their fault the State adopted Common Core reform math knowing it was not internationally benchmarked (i.e., not world-class) or shut the schools for inferior remote and hybrid instructional methods. 3-16-21 

"We don't have to understand the mechanics of the universe to go about our daily lives," suggests physicist Carlo Rovelli. And we don't have to understand calculus either, even though it unlocks the secrets of the universe. After all, Issac Newton, who invented the calculus, didn't understand its underpinning other than solving physics problems. So, let's think of mathematics as a tool to solve problems, which requires knowing calculating skills, ideas, and uses. Richard Feynman pointed out that there are not enough word problems or questions in elementary school math textbooks. 3-21-21

Standard arithmetic is not taught for mastery. This must change!

"If we want high-skill, high-wage jobs, then that is impossible to do with the education and job training system we have." (Tucker's Lens, 2-24-21)

American educators have not faced the reality that our educational and vocational structure (i.e., K-12 to Community College) is broken and, in my opinion, often lacking in quality and rigor. For example, Common Core math standards, now State math standards, are below international benchmarks, starting in 1st grade. So why did almost every state adopt Common Core in the first place? Many radical progressive leaders continue to claim that America is not a nation at risk. But, the facts don't support the claim.

After eight years of Common Core, only 24% of our 12th graders were proficient in math (2019 NAEP). Surprise! No great leap in achievement. Many AP courses are not equivalent to real college-level courses, primarily in the STEM areas. Surprise! For example, some colleges and universities, such as the University of Texas, do not accept AP calculus for STEM credit. (STEM students are required to take UT's calculus courses and for a good reason.) Frankly, I don't trust the quality or rigor of numerous so-called online college courses, much less online degrees. It is too easy to cheat. For example, you can get or hire someone to help you with an online test. (I know. I was asked to help with the final exam in a required statistics course.)  

Note: The superintendent of a local school district had this to say about returning to in-person classes on March 22nd for the last two months of school. I paraphrase: We don't expect students to make gains or make up lost learning. We expect a return to normalcy. In fairness, the super has been very supportive of teachers. 3-15-21

Nearly 80% of K-12 parents favor in-person schooling in their communities, writes Megan Brenan (Gallup) 3-11-21. I wonder if it is in-person, hybrid, or a combination of the two?

Scores: Reading and math scores should not be going down or stagnating. We are spending all this money on education, and the results are mediocre if that. Most kids are not proficient in math or reading at the state or international levels. What we have been doing in the classroom has not worked well. Richard Feynman was critical of elementary school math and science textbooks. He once observed, "So we really ought to look into theories that don't work, and science that isn't science." Also, "None of the textbooks said anything about using arithmetic in science." Nothing has changed!

Richard Feynman

Dr. Richard Feynman ("Surely You're Joking, Mr. Feynman!" 1985) is a Nobel Winner in Physics. He writes, "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences or inequalities. So if education increases inequality, is this ethical?"

➜ "Factual knowledge must precede skill," writes cognitive researcher Daniel T. Willingham. By factual knowledge, he means math knowledge such as concepts, facts, and standard algorithms (often called calculating skills). By skill, he refers to higher-level thinking skills often called "critical thinking," which is at the top of Bloom's cognitive taxonomy. Therefore, it must be more important than knowledge--a gross misinterpretation of Bloom. In short, a student cannot do problem-solving in math without first knowing the math (i.e., knowledge). Why are teachers glossing over, even skipping the knowledge foundation and jumping into critical thinking? It's taught in ed schools, but it's backward and doesn't work!

Start with Knowledge!
Knowledge is needed for applying and reasoning.

[Note. The idea that all children have the same ability to learn is nonsense. "Children differ [widely] in their ability to learn academic materials," yet schools and educators often set aside this fact and treat all children the same, according to Charles Murray (Real Education). Consequently, we end up with mixed-ability classrooms (equity-diversity-inclusion policies) in which many children go unchallenged or shortchanged in mathematics. Children don't have the same ability but get the same opportunities to learn, not identical opportunities.

If students of similar math achievement are put in the same math class, they can excel and learn the math content. Also, I do not think we can catch kids up who have low academic ability because these students need a specialized curriculum that matches their abilities); however, effective intervention programs for incoming 1st-grade students with weak number skills, such as the safety net programs in Singapore, can help. It's called tracking!

Tracking in Singapore

Singapore has a pull-out math class with an expert teacher for incoming 1st graders weak in math skills. The program lasts up to two years to catch students up and continues in the 5th and 6th grades when the math content becomes much more complex. Singapore tracks students to remediate observed weaknesses. The 6th-grade exit test tracks students by math achievement to specific middle schools.]

➜ In my opinion, "equal opportunity to learn" does not mean learning equal content. Also, according to William H. Schmidt, "By international standards, our eighth-grade students are exposed to sixth-grade mathematics content." All students should have the opportunity to upgrade their content knowledge. But why are we teaching 6th-grade math to most of our 8th graders who should be capable of much more? The reason is simple. American students are not learning nearly enough basic arithmetic early enough.

Students lack good calculating skills. 

Teachers are often asked to teach math programs that do not stress the automaticity of math facts and standard procedures, i.e., arithmetic fundamentals. The lack of fluency in basic arithmetic often starts in 1^st grade with addition and subtraction. Many educators endorse a misguided notion that kids will learn facts and procedures when they are ready. In reality, students without essential background knowledge will struggle in math up the grades. In short, students lack standard calculating skills, one of the five most important priorities of elementary school arithmetic, explains W. Stephen Wilson, a mathematician. Competence in Computational Skills is the backbone of a good math program, explains Robert B. Davis, The Madison Project: skills, ideas, and uses). If you can't calculate, then you can't solve math problems. 

Traditional Arithmetic trains a young child's mind for higher math if the fundamentals are automated, especially the early memorization of math facts and the learning of standard algorithms, starting in 1st grade with Addition,  which I think all students can learn. 

1st-grade Arithmetic: 67 + 85 can be rewritten as columns to stress place value, putting ones under ones, tens under tens, etc. The 7 and 5 make 12, which is 1 ten plus 2 ones or (t + 2). Record the 2 in the ones place as shown, carry the ten to the tens column (carry mark), and then add the tens. The standard algorithm is built on place value. 

1st Grade by Christmas

 

Competence in Computational Skills (i.e., standard algorithms) is essential for problem-solving. It requires memorization and practice-practice-practice. 

Working memory is limited. so "whenever something clogs up working memory, your all-around problem-solving abilities take a hit ... Learning via worked examples, instead of solving problems for yourself, is one potential way past such working-memory roadblocks." One major roadblock is not having auto recall of the multiplication table, which should be overlearned to stick in long-term memory and instantly be available for the problem-solving process in working memory. The same is true with other single-digit math facts, standard algorithms, and fraction operations.

Note: Under a remote or hybrid system, "My child is learning less." 

It's "pretend" education, which is incongruous with the EducationNext survey of parents who say they are happy with their child's education. Then again, surveys are opinions, which can vary from month to month. 

 

😎 Basics are not enough. Teach content that prepares more kids for algebra in middle school by restoring real education--not group work with minimal guidance methods of ineffective instruction. Moreover, students don't need to reinvent arithmetic. They are novices, not experts or little mathematicians. Algebra should start in 1st grade. Also, you don't make drawings to do arithmetic. Who does that? "It's not that Asian kids overachieve. It's that American kids grossly underachieve," explains Richard Nesbett, Intelligence and How to Get It. 

The Asian culture of working hard and persistence is something we could and should emulate. Intelligence, in Confucian terms, is a matter of hard work, not IQ, notes Nesbett (Schooling Makes You Smarter). For decades, the achievement bar for math has been set too low for American students. Most State math standards are below international benchmarks. For example, many kids struggle with whole number operations, such as addition (Common Core), when they should be learning all four fraction operations in 4th grade, an international benchmark. 

Instead of Piaget, we should have followed Jerome Bruner
like top-performing nations.

Professor W. Stephen Wilson points out, "Avoiding hard mathematics with younger students does not prepare them for hard mathematics when they are older." Indeed, he argues correctly that the lack of instant recall of multiplication facts permanently slows students down."

Note: If we were good at teaching arithmetic, then our kids would be among the best in the world, based on the billions and billions ($) we spend on schooling. They aren't. We have been on the wrong path for decades! Knowledge is the foundation for higher-order thinking in mathematics, so the first priority should be mastering fundamental mathematical knowledge (conceptual, factual, and procedural), which is what top-performing nations do. Often, weaknesses in elementary math don't show up right away but suddenly pop up in middle school or high school when students hit a wall and struggle with higher-level maths.

➜ Teaching kids to score high on a state test is not the same as teaching content that prepares students for algebra by middle school and beyond. We are merely teaching the minimum, just enough to get by if that. Common Core, which embraces reform math, won't change this. Preparing more kids for algebra by middle school was the highlight of the National Mathematics Advisory Panel report (2008). Still, Common Core and state standards did not embrace the premise by postponing most algebra standards to high school. It is a huge mistake to ignore standard arithmetic in the early grades. 

Issac Newton

Math is a tool to solve problems. But, we don't teach math as a tool. Think, Issac Newton! Newton invented calculations [calculus] that solved physics problems without a deep conceptual understanding of the procedures he used to solve the problems. The conceptual basis for calculus didn't come for another 250 years (limits). WOW! An incomplete understanding of, let's say, the long division algorithm should not a big deal for novices. Likewise, an incomplete understanding of calculus did not diminish Newton's worth or prestige as a super mathematician and physicist.

4th Grade Algebra (TKA)

The 4th graders in my Teach Kids Algebra (TKA) class find how the height of the "tennis ball bounce" varies according to the height the ball is dropped. [Discussion: The height of bounce is directly proportional to the height of the tennis ball drop.] After calculating the averages of the trials of the ball bounce from several heights, students plotted the results, drew the best fit line, figured out the slope of the line, wrote a linear equation that models the relationship (y = kx), and made predictions. Calculators are not used in TKA. The equation is a direct proportion and starts at (0,0)--without a bounce, there isn't a measurable height. When a prediction was made, the team set out to confirm it. Sometimes, the prediction (made by extrapolating beyond the known data) didn't match the experimental result. If the prediction does not agree with experiment, then the equation is wrong (Richard Feynman). [Discussion: Name the factors that could have interfered with accurate measurements or slope calculation.] 

4th-Grade TKA: Ball Bounce

In my Teach Kids Algebra (TKA) program, 4th graders did a ball bounce experiment from 5 different heights, 3 trials per height, recorded data, calculated averages, graphed it, figured out the best fit line, the slope, and wrote a linear equation in (y= mx + b) form. There was a lot of discussion about measurement error and why the equations differed from group to group. Then the kids tried interpolating and extrapolating values in their equations. They tested the new values to see if the equations worked. When the new observations didn't match the predictions made from their graphs, they knew something was wrong. Either the measurements and calculations were wrong, or the equation of the best fit line was wrong, or both. More class discussion, etc. (2012-2013)
 

Teach Kids Algebra (TKA) introduces advanced content to very young children, from integers to equations [functions] in two variables (y = mx + b). Educators should be teaching more than just basics. Starting in 1st grade, elementary school teachers should be preparing students for algebra by middle school. There is no set curriculum for this, so I made it up and wrote my own classroom lessons [curriculum] and taught it. I fused algebra to arithmetic to make algebra accessible to very young children. The students were in Title 1 urban schools.

Children don’t need to reinvent arithmetic; they are not little mathematicians. They are novices. Understanding does not produce mastery; practice does.

➜ Problem-solving is domain-specific.

Polya poses a problem.

The length of the perimeter of a right triangle is 60 inches, and the length of the altitude perpendicular to the hypotenuse is 12 inches. Find the sides?

A typical student cannot solve this problem without substantial knowledge of high school mathematics (algebra and geometry). But, isn't prerequisite knowledge necessary for any math problem, at any level, even for routine problems? Yes, knowledge precedes thinking, says Willingham, a cognitive scientist. Knowledge first! Why are teachers glossing over the knowledge foundation and jumping into critical thinking? It is backward!

Students should start with basic arithmetic and routine problems in mathematics to build a storehouse of knowledge and experience in long-term memory before moving to more complex problems that require more insight. The idea that students can do problem-solving (i.e., critical thinking in math) without fundamentals in place is illogical and backward, yet this is what many teachers think or are taught to think in education schools. In contrast to reform math, elementary school students should focus on mastering basic arithmetic and routine word problems, grade by grade. It requires a lot of practice and review. In short, solving problems requires domain-specific knowledge. The same is true for science, but the thinking in math (deductive) is different from the thinking in science (observation-inference), and so on. While math is absolute, science is subject to change as new observations become available.

I pose a chemistry problem.
Calculate the grams of hydrogen required to produce 82.000 grams of ammonia from nitrogen and hydrogen gasses.

Would you attempt to solve this routine chemistry problem without knowing the fundamentals of high school chemistry and algebra? Of course not! Solving chem problems requires domain-specific knowledge. Moreover, learning to solve routine problems, whether in chemistry or elementary school arithmetic, presupposes both knowledge and practice.   

I pose a Latin problem.
Ego vos hortor ut amicitiam ombibus rebus humanis anteponatis. Sentio equidem, excepta sapientia, nihil melius homini a deis immortablibus datum esse. 

Would you attempt to translate Latin without knowing the fundamentals of Latin? Of course not. Translating Latin requires domain-specific knowledge. 

What do I think? Teachers should teach, not facilitate. To teach math well, K-8 teachers must know mathematics through precalculus. Moreover, they should know college-level chemistry and algebra-based physics to grasp how arithmetic and algebra are used. The methods of instruction must be effective, not faddish or minimal guidance. We need smart and wise teachers in the classroom to produce smart and wise students. Schools of education have been producing mediocre teachers if that. Academic coursework for wannabe teachers, especially in higher mathematics and science, is weak. Blame education schools with low standards.  

In 2009, the Albert Shanker Institute wrote, "Advances in cognitive science make it clear that very young children are capable of much more academically than was previously imagined." This is not a new idea! We figured this out in the 50s and 60s with real instruction in the classroom--not from fads, ed school theory, progressive ideology, reforms without evidence, or beliefs without reason. 

The "many levels" in the same math class and an emphasis on group work and minimal guidance methods of instruction, which are very common in elementary schools and many middle schools, have led to widespread underachievement and other unintended consequences. We expect too little from students. 

Elementary teachers of self-contained K-6 classrooms spend about an hour a day teaching math, yet, over the years, there has been little [flat] progress in US math achievement. One might ask, what are teachers doing in the math hour? Instruction has not been as effective as it needs to be. Teachers work hard, but there seems to be little payoff. The grouping of students for math class (heterogeneity) is part of the problem. The instructional time consumed by test prep is another. The "group work" way of thinking by many teachers is still another. A dumbed-down math curriculum and minimal guidance methods contribute greatly to poor student achievement. Teacher training is inadequate in math and science. 

Not That Good ...

The Finnish school system has been lauded as one of the best in the world. It's not. In TIMSS 2011, Finnish 4th and 8th graders scored below US students in mathematics. In 2009, Professor Olli Martio, University of Helsinki, pointed out in a mathematics journal that less than 30% of Finnish 9th graders (15-16-year-olds) can multiply 1/6 x 1/2 correctly. He attributes the decline in competency on curriculum changes that sidelined basic arithmetic in favor of calculators. In Finland, the national standards have eroded to a lax framework that allows individual teachers to decide what they will teach and how. Thus, many Finnish teachers no longer teach some of the arithmetic skills that can be done on a calculator. 

I tested typical Finnish questions with my two 5th grade TKA classes. No calculators were used by Finnish students. Also, TKA students never use calculators. The Finish students didn't do well. Maybe, American educators should think twice before emulating Finland. Not much has changed! 

Straight Talk

1. Confirmation Bias

In Being Wrong, Kathryn Schulz points out that "we believe things based on meager evidence." For example, experts tell us that technology in schools boosts math achievement. But is this guess true or false? We have had computers in classrooms for over 30 years. Where is the evidence? Is this a false claim? Schulz also defines confirmation bias as the "tendency to give more weight to evidence that confirms our beliefs than to evidence that challenges them." She explains that we fail to look for evidence that would contradict our beliefs. 

2. Integrity in Education

Richard Feynman explains the essence of good science: "If it disagrees with experiment, then it is wrong." Can this science idea be applied to education? I think so, but education is not physics. We have been relying too much on opinions rather than experiments. We in education have lots of guesses and programs built on guesses and assumptions; however, we do not conduct randomized sound experiments to test ideas. Indeed, many popular classroom practices have no basis in evidence. Feynman often talks about integrity in science. He writes, "If you're doing an experiment, you should report everything that you think might make it invalid." We never talk about this kind of integrity in education. Many education studies are flawed. It isn't easy to find a control group. Also, time-on-task is often ignored in educational studies, which makes them suspect. 

3. Minimal-Guided Instruction = Minimal Learning

Clark, Kirschner, & Sweller state that minimally-guided constructivist instruction (e.g., discovery, inquiry, project-based, problem-based, etc.), which is revered in schools of education and prevalent in today's classrooms, can "increase the achievement gap" and that the "failure to provide strong instructional support produced a measurable loss of learning." 

The researchers explain that the "aim of all instruction is to add knowledge and skills to long-term memory. If nothing has been added to long-term memory, nothing has been learned." If nothing has been remembered (retrieval), then nothing has been learned. In short, students must have the proper prerequisite knowledge in long-term memory to solve problems in math. Prior knowledge is essential. 

➜ The parent of a 3rd grader asked the teacher why the multiplication table isn't being taught. "We don't do that anymore; we teach critical thinking." The teacher's reply is the essence of reform math. And, it's wrong! One way for parents to fight woke teachers is to withdraw their children, which is what the parent did.  3-27-21

©2021 ThinkAlgebra/LT

 

Inklings1

 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: 

• Teaching 
• Photography: LT's Photo Page 

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 I, conventions (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 Program. Click 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 stressed. W. 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 Stotsky. Gap 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.

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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

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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. 

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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. 

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▶︎ 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. (Stotsky, The 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 Gagne, the 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 canopy. No 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. 

End

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.

Contact:  ThinkAlgebra@cox.net

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