Radical Ideas 2
After eight years of Common-Core-based reform math, a meager 24% of 12th graders were proficient in math (NAEP 2019), an awful statistic. Common Core reform math doesn't work, so why are we still teaching standards and progressive ideas based on Common Core? When big decisions are made in education, the mistakes are not small, and the unintended consequences run deep, such as 76% not proficient in math. Common Core has not made our students better in math.
|Remote makes some kids sad, |
angry, and frustrated.
Remote often hurts kids more than it helps. It is a bust. How many kids are self-motivated to pay attention and stare at a screen much of the day and do homework?
|Algebra in 1st grade: Teach Kids Algebra|
✍️ Educators need to "make learning easier: more user-friendly and far more accessible," writes Sanjay Sarma at MIT (Grasp, 2020) and abandon the idea that "serious learning should be difficult." Reform math is an excellent example of making arithmetic harder to learn with at least five multiplying methods and little attention paid to standard algorithms or memorizing math facts. Toss into the mix the so-called standards for mathematical practice, social-emotional and self-esteem stuff, minimal guidance instructional strategies (i.e., group work, discovery learning, etc.), and a lot of grade inflation. What a mess! Is it any wonder that most kids never become good at arithmetic, which is the backbone of algebra?
|Chris Ferrie has the ability to explain complex science to little kids.|
The more I know, the more I can learn, the faster I can learn it,
the better I can think and solve problems. WOW, isn't cognitive science great?
- To learn something is to remember it.
- Engagement is not the same as learning.
- Critical thinking is difficult to measure.
- Practice is necessary to improve.
- The spiral curriculum of J. Bruner failed.
- Practicing math facts will help with long division.
- Children need feedback so that they can make corrections.
- We have good tests that measure content knowledge.
- Our ability to measure creativity, collaboration, or critical thinking is limited.
- "Thinking well requires knowing facts. Factual knowledge must precede [higher level thinking] skill."
- Opinion is not science.
- Your mind is lazy and doesn't want to think.
- You have to force yourself to recall a fact or a procedure in arithmetic or algebra.
- To learn effectively, students should quiz themselves at home and school on new content Flashcards give instant feedback. (Stanislas Dehaene, How We Learn, 2020).
- "Your ideas will never be more effective than your ability to make others grasp them." (Thomas Oppong, 50/50) In short, you have to explain complex stuff so those very young children can begin to grasp it. Many teachers cannot teach content well. (This is similar to a saying by the late Richard Feynman, who, in physics, was the "Great Explainer.")
- With a good education, you increase differences. (Feynman)
- "Nothing is fun until you are good at it. Rote repetition is underrated in America" (Amy Chua)
- A Chinese mother believes that "Schoolwork always comes first; an A-minus is a bad grade; Your children must be two years ahead of their classmates in math." And, "No matter what, you don't talk back to your parents, teachers, elders." (Amy Chua, Battle Hymn of the Tiger Mother, 2011).
Here is a good idea: dividing by a number (other than zero) is multiplying by its reciprocal. It is basic arithmetic that should be taught in 4th grade with fractions. Thus, 8 ÷ 4, by definition, is 8 x (1/4). Therefore, a division is actually a multiplication by reciprocal. Zero doesn't have a reciprocal because the product of reciprocals must always equal 1 by definition (5 x (1/0), oops, one can't divide by zero. (Model: GabbyB, a Middle School Student)
- Equation Structure: Expression = Expression (x - 3 = 19)
- Equal-Arm Balance: Left Side = Right Side
- Solving (Guess & Check): x - 3 = 19 is true only when x = 22
- Balanced: 19 = 19
Educators often make learning math hard, but it isn't that hard when explained well and linked to basic arithmetic students already know. Students have a vast capacity to learn, but educators are not taking advantage of it. I decided to teach introductory algebra to 6 and 7-year-olds (1st and 2nd-grade students) at an urban, Title-1 school of mostly minority students. Skills, ideas, and uses were introduced with worked examples. I fused algebra ideas with standard arithmetic. For example, 5 + 7 = 12 is a true equation or math fact shown on the number line, while x + 7 = 12 is an equation in one unknown. What value of x makes a true statement? Solving an equation for an unknown (x) means finding the value of x to make a true statement. So mathematics such as arithmetic or algebra "requires a lot of basic knowledge and technique," points out Ian Stewart (Letters to a Young Mathematician, 2006). Students must learn proper "technique" as they gain "knowledge" beginning in 1st grade. How does a 6-year-old solve simple equations? They use Guess and Check, based on memorized math facts and rules from long-term memory. Note: I show inverse equation-solving techniques (Undo) in the 3rd grade, but I would like to try it in 2nd grade.
Second and third graders also need to memorize the multiplication table. Thus, according to Edward Thorndike, 3 x 7 is associated with 21 and is easily shown on a number line. For example, a linear equation in y = mx + b form can be related to an x-y table of values, making a picture on a graph by plotting (x,y) points—one thing associates with another, with another, and so on. All three models of a function are interrelated. (Note: Slope/similar triangles, the meaning of y = mx+b; quadratic and exponential equations: 3rd to 6th grade).
I believe that most students can learn arithmetic and algebra at acceptable levels with study, practice, and effort when they are taught traditional math explicitly, not reform math using minimal guidance methods, group work, alternative algorithms, or test prep. We keep lowering the bar instead of encouraging students to do better by working harder, longer. For example, the math curriculum, based on Common-Core-like state standards, is below international benchmarks. Our 4th or 5th graders, on average, are two years behind their peers from Asian nations. In short, the math education that most kids get is substandard, starting in 1st grade. The goal should be the early mastery of basics, not state test-based proficiency. It has always been true that children need to memorize and practice (drill) to master the fundamentals of math in long-term memory, which requires a significant curriculum upgrade for all students.
|Common Core reform math confuses and frustrates students, holds them |
back and befuddles angry parents.
|Common Core math clutters the curriculum and a child's mind with complicated nonstandard algorithms that frustrate students and baffle parents.|
✍️ The Madison Project
Robert Davis, Syracuse University, wrote, "The Madison Project seeks to broaden this curriculum by introducing, in addition to arithmetic, some of the fundamental concepts of algebra (such as variable, function, the arithmetic of signed numbers, open sentences, axiom, theorem, and derivations), some fundamental concepts of coordinate geometries (such as a graph of a function), some ideas of logic (such as implication), and some work on the relations of mathematics to physical science. Arithmetic becomes evident as one sees it in relation to algebra and coordinate geometry."
The purpose of my Teach Kids Algebra (2011-2019) program, like The Madison Project half a century ago, seeks to expand the elementary school curriculum beyond arithmetic. It was also a reaction against reform math, which often downplays the importance of memorizing math facts and learning the standard algorithms for operations.
✍️ We should focus on the mastery of fundamentals, not learning for a test. In the U.S., instruction is the opposite. It is geared toward learning for the state test via a reform math curriculum that had failed in the past. No wonder our children can't do arithmetic or algebra well.
The goal should be the mastery of basics, not state test-based proficiency. Furthermore, math skills are domain-specific. Starting in 1st grade, some of the skills include memorizing the single-digit number facts, applying the place value system, using the rules, and learning the standard algorithms. These beginner skills are needed for doing arithmetic in 1st grade. So, why are our 1st-grade children not learning them?
- apply the addition standard algorithm to calculate sums quickly,
- find the perimeters of polygons (formulas),
- solve equations in one variable (12 + x = 25),
- build x-y tables from a linear equation (y = x + x + 2), and
- plot (x,y) points on graph paper, which were some of the action learning objectives that directed my teaching and led to student learning.
The primary learning goal should be the mastery of necessary content in long-term memory, not proficiency on state tests. Also, Learning is remembering from long-term memory! "You don't know anything until you have practiced." (Richard Feynman)
Not knowing single-digit math facts impedes standard algorithms' learning, fractions-decimals, percentages, ratio/proportion, algebra, geometry, and measurement. For example, the addition algorithm must be automatic, correct, and efficient. Novices should first learn the standard algorithms, which means that very young children need to memorize single-digit math facts supporting the standard algorithms. The reform math's alternative algorithms should be avoided because they are cumbersome, inefficient, and dead ends. Also, they needlessly clutter the curriculum as extras and increase the students' cognitive load.
- "Practice does not cause talent; it improves performance." (Ian Stewart)
- "You learn only by mastery." (Zig Engelmann)
- "Mastery requires memorization and repetition." (Stanislas Dehaene)
- "You don't know anything until you have practiced." (Richard Feynman)
- "The building blocks of understanding are memorization and repetition. (Barbara Oakley).
The quest for sameness (i.e., Common Core, equity, etc.) has been a "fallacy of fairness in education." Sameness is a poor fit for most students. It dilutes content. In mathematics, there is an inverse relationship between equity and rigor of content. (Quote marks: Thomas Sowell) Reducing content, shrinking grade-level curriculum, lowering test cut scores, and inflating grades are pseudo schemes to fix education. It is nonsense!
✍️ Reform Math Is a Bust
Sometimes the easiest things in math are the most difficult to understand. Thus, much of the talk about understanding and alternative "reform math" algorithms called "understanding" algorithms is nonsense and a waste of instructional time. I would not worry much if your child doesn't understand the standard algorithms, especially long division. Most adults don't either. The most straightforward facts, such as 5 + 7 = 12, can be shown on the number line to Kinder children. Kids develop a "number line" understanding of algorithms that makes sense, while the alternative algorithms are complicated and complex and have little practical value. Kids are novices, not little mathematicians. Who multiplies 7 x 2.67 using the area model or the array model? The standard multiplication algorithm takes 15 seconds if that for 7 x 2.67. The caveat is that students need to automate the multiplication facts to support the standard algorithm. First-graders in Singapore learn multiplication as repeated addition. Thus, 3 x 4 = 4 + 4 + 4 or 12. Three jumps of four on the number line lands at 12.
Remote Learning Is a Bust
Liberal educators want to revolutionize education by creating a utopian as normal; however, I think we should aim for the old normal first and then figure out which changes benefit students by implementing learning science. Among the likely changes would be to increase class size as it accounts for only 1% variance on achievement testing (Plomin), sort kids by achievement in each major subject, upgrade the math curriculum to correlate with international benchmarks, focus on teaching traditional arithmetic well, not reform math, starting in the 1st grade, and jettison standardized testing.
Equity and academics vary inversely. Today's stress on equity has resulted in lower academic standards and achievement in primary subjects such as reading, math, science, and history. For decades, "equalizing downward by lowering those at the top" has been a widespread policy taught in education schools, writes Thomas Sowell. It is a "fallacy of fairness." We do not live in a perfect world.
As mathematician Keith Devlin describes it, functional understanding is "understanding a concept sufficiently well to get by for the present." It is "understanding that is defined in terms of what the learner can do" (i.e., apply it). Devlin also states, "I think many mathematical concepts can be understood only after the learner has acquired procedural skill in using the concept." Professor Devlin is saying that kids should have procedural fluency to foster functional understanding. But what does that mean? Performing math implies some level of understanding, but it can't be quantified. In school math (e.g., arithmetic, algebra, precalculus), the focus should be on the "how" first" Good math books have a bunch of worked examples in each lesson--the "how!" Most teachers spend too much time on understanding (the why). It is the wrong approach. The "why" should not come first.
If you cannot calculate something, then you don't know it well enough.
The more I know, the more I can learn, the faster I can learn it, the better I can think and solve problems.
Common Core reform math confuses and frustrates students, holds them back and befuddles angry parents.
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