Observations, Ideas, and Opinions on Math Education by a Contrarian in 2021, a divergent view 😎
Musing
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| Surely You're Joking, Mr. Feynman |
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.
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| Tech Problems with Zoom, Internet speed, and hardware plague remote and hybrid learning. Many students and parents are frustrated. (Credit: GabbyB) |
✍️ 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)
✍️ 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.
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| 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.
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.
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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.
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| 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) |
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| 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. |
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)
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!
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| Richard Feynman at Cal-Tech Channeling Feynman: "You don't know anything until you have practiced! |
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| MariaB 7th 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.
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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 + -3. Thus, 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: 5 ÷ 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?
Novices need guidance, encouragement, explained examples,
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.
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| 2nd-Grade TKA Algebra Class STEM Math 2019 |
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 underprepared. Peg 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.
7. Problem-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.
Sameness
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 Murray, Real Education, 2008) and is mostly genetic (Robert Plomin, Blueprint, 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
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| LT/ThinkAlgebra |
We don't push kids hard enough.
More coming soon, including clarifications.
Comments can be made to LT at
ThinkAlgebra@cox.net
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