## Tuesday, August 13, 2019

### ThinkAlgebra Main Page

Welcome to the Main Page of ThinkAlgebra
February 6, 2020

Note. Jo Boaler wants to ban memorizing multiplication facts and standard algorithms (i.e., traditional arithmetic) to equalize outcomes and curb math anxiety. Many of her ideas are defective, yet they are found in almost all classrooms in the U.S. Consequently, students are weak in paper calculating skills, still calculating skills are essential for solving math problems. The views expressed here are my opinion.

Special Insert 4-2-2020
For Parents
I often hear teachers say they teach for understanding, critical thinking, creative/innovating problem solving, and so on, but these ideas are not easily measured and false goals. Teachers seldom say they teach content knowledge that enables or prepares the student for critical thinking in math. Knowledge must precede skill, says Daniel WillinghamFor example, you can't translate Latin if you don't know Latin. You can't calculate percentages without knowing many about percentages, fractions, and ratios, setting up and solving ratio equations. Most elementary students are taught a content-free curriculum using progressive instruction methods, such as group work and discovery/inquiry learning. The teacher is a guide-on-the-side, not the academic leader in the classroom--a drastic change in the teacher's role.

In short, students are taught progressive pedagogy and ideology, not pertinent content knowledge. Then, there is test prep. (Don't get me started on it.)

Students are weak in calculating skills (i.e., traditional arithmetic). Get some workbooks at Barnes & Noble. If the child is rising to the 6th grade, get the 6th-grade level practice books, and teach traditional arithmetic (TA) to your child. TA, not reform math, is needed to prepare for prealgebra in 7th grade and Algebra-1 no later than 8th grade.

Kumon has a good TA practice book for prealgebra (grades 6-8). Topics include fractions, decimals, percentages, exponents, order of operations, positive and negative numbers, algebraic expressions, and more. There is a reason for each step. In short, a proof.  Learn to write the intermediate steps and give the reason for each step.

Example Problem: 60 ÷ 12 + (8^2 - 5^2)
60 ÷ 12 + (64 - 25) Apply the powers rule (Order of Operations)
60 ÷ 12 + (39)        Subtract inside the parentheses (Order of Operations)
5 + 39                     Do division before sums (Order of Operations)
44                           Apply the addition rule (Order of Operations)

The Art of Problem-Solving (AoPS) publishes Prealgebra, which starts with arithmetic ideas, then to thinking in algebra. For example, subtraction in algebra is defined as a + (-b). Thus, to subtract a number, add its opposite. Example: 4 - 5 = 4 + (-5) = -1. I normally teach it in 3rd and 4th grade, even to precocious or advanced 2nd graders. I call it the add-opp rule (1). Subtraction is not commutative, so we change subtractions in algebra to additions, following the commutative rule. It is more complicated than this. Also, recall that the ^  computer symbol means raised to a power. Thus 6^3 means 6 x 6 x 6 = 216.

5 - 7 = 5 + (-7) = -2
There is a reason for each step. In short, a mini proof.
5 - 7            Given
5 + (-7)       Change subtraction to addition by applying the Add Opp Rule
-2               Apply the Integer rule for the sum of two integers.

The idea of proof is essential in arithmetic but seldom taught.
Learn to write the intermediate steps. (Error Corrected)

(1) I first encountered the add-opp rule in the earliest editions of Transition Math from UCSMP, which was in the early 2000s at an independent School. I used TM in my regular 7th-grade classes as a stepping stone to Algebra-1. Also, UCSMP provided the textbooks for three schools, an independent school, a catholic school, and a public school through a grant at the University of Delaware. The teachers met once or twice a month to give feedback on using TM.

End of Insert
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A concerned parent wrote, [We are in the] "dark ages of K-12 mathematics instruction." Indeed, academic progress has been shut down by radical progressives.

#### Guess which students were the best problem solvers in mathematics? It was the so-called rote learners of the East Asian nations. These kids know math stuff in long-term memory. "Thinking comes from memory."

Excellence, with rare exceptions, is not in the game. It is terrible what the math reforms have done to our kids. The latest round is Common Core, state standards, reform math, and progressive pedagogy. The standards are not world-class, and the instructional methods are substandard, too. Starting in the 1st grade, our kids are not learning basic arithmetic. For example, they are not required to memorize math facts or learn/practice the standard algorithms. If other nations can do it, then why not the U.S.?

Jo Boaler, a "math education" professor, writes, "[After the NCTM 1989 standards] "Teachers were instructed to be facilitators rather than lecturers and to have children work in groups." Rebuilding the role of teachers as facilitators is a major change in the classroom, but, I think, it is the wrong direction. [This is a foolish, radical idea. The teacher needs to explain ideas and techniques--not be a guide on the side. And, the students need to practice for mastery. The problem today is that many K-8 teachers don't know enough mathematics to explain content well enough (Dr. H. Wu, UC at Berkeley). Good textbooks, like Dolciani Algebra-1, are gone. Or very hard to find. I blame the NCTM, schools of education, federal and state departments of education, radical professors like Boaler, and others.

Listen to a real mathematician about technology use.
Dr. W. Stephen Wilson (Johns Hopkins Univesity)
"I have not yet encountered a mathematics concept that required technology to either teach it or assess it. The concepts and skills we teach are so basic that technology is not needed to either elucidate or enhance them. YOU CANNOT REDEFINE MATHEMATICS." NCTM Reform math, which started in late 1989, is an attempt to change basic arithmetic and the teacher's role.

Also, kids don't need expensive TI-graphing calculators or use laptops/tablets to learn higher-level math. In my opinion, an AP Calculus Course is not the same as a university Calculus 101 course in which calculators are often not allowed on exams. It shocks incoming students who have been weaned on calculators since elementary school (reform math).

Dr. Bas Braans, physicist, regarding instructional practice, writes, "The teacher teaches, and the students practice. This has to be the core of effective instruction."

New Middle School, UPSD (PA), is based on ideas from Disrupting Class and Jo Boaler's reform math
The classrooms of future learning will be failures--just like open classrooms. There was little evidence to support Clay Christensen's book Disrupting Class (2008) or the wild prediction that half of the high schools would switch to customized software by 2018. "A key step toward making school intrinsically motivating is to customize an education to match the way each child best learns." Sounds great! Really? It is the same old, failed progressive rhetoric. I don't see how watching videos at school or at home or staring at a screen for much of the day will spark motivation. The novelty will quickly wear off. Contrary to the claims made by large education companies, the software that works either does not exist or misses the mark, and it is 2020. Ideas such as this one are overhyped

What adults think is not the same as what kids do. Kids are novices, not experts.
All students need to learn traditional arithmetic (TA) to advance in math. Under Disruptive Class and reform math, most students would fall behind. Individually Prescribed Instruction (IPI) was tried in the early 70s and failed at the elementary school I had visited at an NTCM Conference in DC. Test scores went down, not up. Elementary students didn't cover much of the curriculum. The real problem in our schools is that by age 10, students are not learning basic arithmetic through long-division, partly because they are taught reform math via progressive pedagogy (e.g., group work, discovery learning, etc.). Starting in the 1st grade, students do not memorize math facts or practice the mechanics of standard algorithms like the Asian nations and some schools in the U.S., especially certain private schools. Also, elementary students are weak in fractions-decimals-percentages, which are 4th- and 5th-grade content. Many middle school students do not learn fractions well either. Thus, they have trouble with algebra that builds on symbolic manipulation and representations. Sadly, modern classroom pedagogy disregards memorization, practice, and drill. Kids work in groups and often do discovery learning, which is an inefficient method, taking up valuable instructional/practice time.

1. PISA test scores put our high school students near the bottom.
The problem starts in lower elementary school in the 1st grade.

2. Algebra for elementary students, grades 1 to 4. Algebra ideas are fused to basic arithmetic. Teach Kids Algebra (TKA) is a singular, innovative algebra project for typical students in grades 1 to 4. Students use arithmetic to learn basic algebra ideas such as variables, equations, functions, negative numbers, graphs, etc. The TKA curriculum is designed to challenge students at a much deeper level than traditional elementary school mathematics or reform math. TKA is STEM math for elementary school students. The Hechinger Report wrote a blurb about TKA in its Future Learning blog (mid-January 2020). LT, Founder & Guest Teacher of TKA.

3. My Reflections 2020
"Factual knowledge must precede skill." (D. Willingham)
Teachers want students to learn many different skills, such as critical thinking, but Bloom's Taxonomy's upper-level thinking skills require extensive factual knowledge (Willingham). Learning (memorizing) math facts may be dull and not much fun at times, but it is essential, especially in arithmetic starting in the 1st grade.

Daniel T. Willingham, a cognitive scientist, explains, "Factual knowledge must precede [higher thinking] skill." Thus, "The ability to analyze and to think critically require extensive [domain-specific] factual knowledge." Bloom's Taxonomy and Polya's book (How to Solve It) have been wrongly interpreted to disdain factual knowledge. Knowledge is not inferior to evaluation; it is essential for evaluation. Also, the so-called "mathematical practices," "progressive pedagogy," or "state tests" should not drive the math curriculum; math content should drive the math curriculum.

"The fundamental laws that govern the universe speak to us only in the language of mathematics. The key to understanding this language is calculus."  (Eugene Khautoryanstky)

Sadly, the educational establishment proclaims that diversity or multiculturalism should upstage quality and excellence, that mediocrity on national and international tests is better than soaring to the top. Even our best students are unexceptional compared to just average math and science students in Singapore or South Korea. Excellence, except in rare cases, is no longer in the game in the U.S. We will regret it later on. Also, substituting a quantitative reasoning course is not the same as doing a real math course, such as Algebra-2, Trigonometry, or Precalculus. Again, it is dumbing down the math. An AP calculus course is not a real college calculus course, and an AP statistics course is a TI-84 calculator course. AP courses just don't cut it and should not be used as a substitute for real college credit in math.

We need to reverse course and prepare more students for calculus. We must have a bedrock math curriculum that puts more students into Algebra-1 by the 7th or 8th grade and solid calculus courses for high school students who are prepared. The obstacle is that we don't have it. We are not even close. We were a little closer before Common Core. There are exceptions, of course. Today, some schools ignore Common Core, state standards, state tests, the NCTM, fads that are commonplace in the public schools, and Piaget. Stop blaming schools, parents, society, and money for our shortcomings in real math education. Indeed, reform math should have been booted out of the classroom decades ago, but it is still the status quo in education, and it has hurt our kids, blocking them from better career choices.

 Will our kids master enough math to get to the STEM or math-related fields?

In the real world, students should memorize multiplication facts for auto recall from long-term memory and master the mechanics of the standard algorithms for both multiplication and long division no later than the 3rd grade. Instead of the explicit teaching of traditional arithmetic that kids must know to advance, students are taught a hodgepodge of often more complicated ways of reform math via group work, which is a waste of instructional time. Natalie Wexler (The Knowledge GAP) writes, "The more you know about a particular topic, the better able you are to think about it critically." Indeed, content math knowledge in long-term memory enables critical thought or problem-solving in math.

Charles J. Sykes writes, "For most Americans, the teaching of arithmetic is a basic test of common sense. There is a nearly universal sense that 4 x 8 = 32--and that this is something that children ought to learn, even if some of them think it is hard or irrelevant. One learns math by learning math, and that takes hard work. You cannot 'think' your way to the solution of an algebra problem without knowing algebra." Likewise, you cannot translate Latin without knowing Latin. Also, you cannot work a percentage problem without knowing percentages and having experience solving percentage problems.

The same is true for basic arithmetic problems, such as multiplication, division, fraction, or percent problems. Hence, you cannot do or apply basic arithmetic without knowing it in the first place. In short, thinking is domain-specific and involves a bedrock of knowledge in long-term memory. The reform educationists do not get this. They screw up your child's education by insisting that knowledge is not that important. Progressive pedagogy isn't a matter of common sense; it is a matter of ideology, which has been a huge mistake. Progressive pedagogy disdains memorization and learning factual knowledge in long-term memory.

Time-on-Task (We are Losing the Race)
U.S. high students spend substantially less "time on task" on academic subjects (1460 hours) compared to Japanese high school students (3170 hours) and other nations, according to Charles J. Sykes (Dumbing Down Our Kids). Japanese students learn much more math and science than U.S. high school students. Sykes writes, "Only 15% of college faculty members [in the U.S.] say that their students are adequately prepared in mathematics and quantitative reasoning." The problem has been that American students are not learning nearly enough basic mathematics (e.g., algebra, trig, and precalculus), beginning with 1st-grade arithmetic at the Singapore level. Also, Common Core and most state standards are below world-class. Our kids start behind because Common Core math and most state math standards are below world-class benchmarks

In contrast, Teach Kids Algebra (TKA) lessons, grades 1 to 4, show students not to fear algebra, that, with effort, they, too, can learn some fundamentals in long-term memory even in the 1st grade, which is what has happened. Algebra follows the same rules as arithmetic. Like arithmetic, algebra is rule-driven.

Peg Tyre (The Good School) writes, "Many teachers hold on to Piaget's idea that children need to "grow into" abstract thinking. But now research suggests that children often don't know math at an early age, not because they are not developmentally for it but because they haven't been exposed to it. What children are "ready for" is mostly contingent on prior opportunities to learn." Teachers need to teach more content early on--from algebra and measurement to geometry and especially basic arithmetic, including long-division and fractions.

 Students cannot do high-order thinking in math  without knowing the math.
"Critical thinking, which is "problem-solving" in math, is not possible without a bedrock of knowledge and technique in long-term memory. Hence, math education in many progressive classrooms is backward and ineffective because the teaching is wrong. Daniel Willingham writes that critical thinking, which is one of the biggest education trends, isn't appropriately taught in schools (Barshay). According to Jill Barshay (Proof Points, Hechinger Report), Willingham says that "teachers should teach old-fashioned content knowledge instead of abstract critical thinking skills that don't transfer between subjects and disciplines."

Natalie Wexler (The Knowledge Gap) points out that Bloom has been grossly misinterpreted: "Bloom's pyramid meant that knowledge and comprehension are prerequisites for higher-order thinking." Skipping the knowledge phase and rushing straight to higher-level thinking skills (at the top of the pyramid) has been a counterproductive school policy. Hence, American students lack math skills and stumble over simple arithmetic. If you can't calculate it (e.g., perimeters), then you don't know it.

 Students need a bedrock of knowledge to do critical thinking!

Matt Parker (Humble Pi) writes, "Our whole world is built on math, from the code running a website to the equations enabling the design of skyscrapers and bridges." We don't notice it until a mistake is made. "Math is easy to ignore until a misplaced decimal point upends the stock market, a unit conversion error causes a plane to crash, or someone divides by zero and stalls a battleship in the middle of the ocean."

We need more high school students who are ready for Calculus. At the least, the students who want to graduate with a bachelor's degree at a four-year university should take a precalculus course in high school. Preparation for the future starts in 1st-grade arithmetic--not middle school or high school. Common Core or state standards miss the mark badly.

College success begins in 1st-grade arithmetic.
1. Not all students have college-level abilities. (C. Murray)
2. Likewise, not all students should take calculus.
3. Likewise, a beginner swimmer does not have the skills to take a senior lifesaving course.

But I think, most students can learn arithmetic and algebra at acceptable levels with study, practice, and effort when they are taught explicitly, not reform math using minimal guidance methods, group work, or test prep.

 In many U.S. classrooms, kids learn very little substantive math content, starting with arithmetic in 1st grade. Kids must know stuff to move forward in math, but many don't. Academics should come first. (Model: RemiB)

Note. To educators, policymakers, and parents: Read The Knowledge Gap, The Hidden Cause of America's Broken Education System--and How to Fix It, by Natalie Wexler. Wexler observes that the radical ideas of the 60s still dominate
today's classrooms as constructivism: "Teachers should engage in as little direct instruction as possible; children should learn through a natural process of discovery and what they were learning wasn't that important. Those theories persist today as constructivism. Constructivism, like [child-centered] progressivism, takes a dim view of memorization" and practice. The reformers say that knowledge isn't that important. Really? The reformers are dead wrong. Thinking and innovation depend on knowledge in long-term memory. We have known this for centuries. Thinking without content knowledge is empty. (I. Kant, 1724-1804)

"Content knowledge is crucial to effective critical thinking," writes Jill Barshay for The Hechinger Report"Critical thinking is all the rage in education. Schools brag that they teach it on their websites and in open houses to impress parents. Some argue that critical thinking should be the primary purpose of education and one of the most important skills to have in the 21st century, with advanced machines and algorithms replacing manual and repetitive labor." But.

"But a fascinating review of the scientific research on how to teach critical thinking concludes that teaching generic critical thinking skills, such as logical reasoning, might be a big waste of time. Critical thinking exercises and games haven't produced long-lasting improvements for students. And the research literature shows that it's challenging for students to apply critical thinking skills learned in one subject to another, even between different fields of science."

Critical thinking, such as problem-solving in trig, is domain-specific. You can't solve a trig problem unless you know some trig. You can't translate Latin unless you know some Latin. And, you can't work percent problems unless you know the arithmetic of percentages. You need content knowledge, lots of it. The more knowledge you have in long-term memory, the better thinking you can do.

Gifted classes do not help talented students advance faster because "the teaching" in the classroom often contradicts scientific evidence that critical thinking is a product of knowledge. Thinking that is independent of knowledge is a myth. Don't expect talented and gifted children to advance much by giving them grade-level math to pass a state test. Most U.S. children are not learning nearly enough math content--not even the best students in so-called gifted programs.

According to the survey, the gifted curricula' two highest priorities were critical thinking skills and creativity/creative thinking. Accelerated mathematics content and accelerated English language arts content were near the bottom of the list. It is backward.

According to cognitive science, content knowledge is crucial to practical, critical thinking," writes Jill Barshay. It means that higher-level thinking is impossible without content knowledge in long-term memory, which should be a priority in talented and gifted programs, but it is not. In my opinion, the primary justification for many talented and gifted programs contradicts cognitive science. The lack of acceleration is why some children in gifted programs are blocked from moving forward rapidly. Some kids rise to the top despite the schools, not because of them. They get private tutoring or enroll in Kumon.

Another problem is that some students in gifted programs are below grade level in reading or math or both. So, how were students selected for gifted programs? Over the past 50 years, except for a few students in my 7th-grade honors math class at a private school in the early 90s, I have not taught a TAG or GATE class in which most of the incoming students were a couple of grade levels ahead in math and reading/writing vocabulary. My honors math class was a much better fit than TAG or GATE.

See TAG & GATE

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We know that practice helps memory. The bottom line is that many U.S. students are not learning enough arithmetic and algebra because the math content has been downgraded to "fit" low achieving students. Memorization, practice, and review have fallen out of favor in progressive classrooms where reform math is taught. Educationists often make generalized assumptions, that is, "statements that are not grounded in observations of the world," and which are often interpreted in many different ways, writes Daniel T. Willingham, a cognitive scientist: learning is social, everybody learns differently, knowledge is constructed, and learning in nature. These assumptions are too general to be of practical use in the classroom; they are often misapplied.

 Start at the bottom (Knowing), not the top!

Also, Wexler points out that Bloom has been grossly misinterpreted: "Bloom's pyramid meant that knowledge and comprehension are prerequisites for higher-order thinking." Indeed, the focus in the classroom should be knowledge first, but it isn't. Today's prevailing pedagogy jumps right into critical thinking (i.e., higher-order thinking at the top of the pyramid), often skipping the knowledge foundation needed to support thought. She notes, "Cognitive science has since provided scientific support for Bloom's intended approach; we can think critically only when we have factual information at our fingertips--or to be more precise, stored in our long-term memory." Teachers are not taught the cognitive science of learning in ed school or professional development.

Sandra Stotsky's solution to the "low achievement" problem is to teach content knowledge to all students, not just low achievers (The Roots of Low Achievement, 2019). Stotsky quotes Laurence Steinberg: "We think the school reform movement has been focusing on the wrong things." It should not focus solely on low achievement. It should focus on achievement for all students. But, first, math content needs to be upgraded to world-class. Repeated efforts to close the achievement gaps have been nothing more than "equalizing downward by lowering those at the top," asserts Thomas Sowell, a black scholar, and a renowned economist. It is not fairness. Sadly, "gap closing" through equity has been the main thrust of public schooling for decades. Also, boosting test scores is not the same as mastering essential academic knowledge, yet test prep has dominated classrooms since 2002. Like critical thinking and digital gadgets, engagement has been all the rage in the classroom. But, too often, engagement pushes out time for learning. Engagement is not the same as learning.

According to Stotsky, some of the "classroom practices" that reduce academic demands are: block scheduling (reduces time on task), some types of mastery learning (smarter kids have to wait for lower kids to catch up), group projects or group work is inefficient and reduces class time for learning), homework (many students don't do it, so homework is cut back), and practice (kids don't practice enough, so they absorb less). There are others.

In short, the public school system is set up to reduce, distort, or omit content knowledge. Equal opportunity does not cause equitable outcomes (Thomas Sowell). Professor Stotsky points out, "There are thinking skills to be developed in every subject in K-12, but not at the expense of the basic knowledge that becomes the basis of thoughts.I do not blame teachers because they are doing what they were taught in schools of education and professional development.

Arithmetic Has Eroded over Time.
How many 21st Century 4th-grade students can figure out 4th-grade level math from the 19th-Century? Below are two routine problems that involve percentages and ratios.

19th-Century 4th-Grade Basic Arithmetic in America
1. Find the interest of \$60 for 4 months, at 5 percent.
2. If 12 peaches are worth 84 apples, and 8 apples are worth 24 plums, how many plums shall I have for 5 peaches?
(Ray's New Intellectual Arithmetic, 1877, which combined 3rd and 4th-grade arithmetic into one compact 140-page book: 4.5 x 7.25 inches.)

Multiplication Scope & Sequence the Way It Should Have Been
Note. Extensive practice and review dominate the first four years of schooling.
1. In 1st grade, students learn multiplication as repeated addition: 3 x 4 = 4 + 4 + 4 = 12. (The mechanics of the addition and subtraction standard algorithms are introduced and practiced.)
2. In 2nd grade, students memorize half of the multiplication table for instant recall and use multiplication ideas to solve routine word problems.
3. In 3rd grade, students memorize the rest of the table for instant recall and learn the standard algorithms' mechanics for both multiplication and long division. Students solve routine problems involving these operations.
4. In 4th grade, the focus should be on fractions, decimals, percentages, and ratios/proportions, with extensive reviews of the four primary whole number operations.

We have a knowledge gap!
But, we pretend it's not there.
The fundamentals of arithmetic (e.g., number facts and the mechanics of standard algorithms) must be memorized and used in the early grades to solve routine problems. Reviewing math skills should be a large part of every lesson (Engelmann). Furthermore, the fundamentals of arithmetic launch the foundation (schema) for learning more math. My Summary of the Cognitive Science of Learning: The more I know, the more I can learn, the faster I can learn it, and the better I can think to solve problems. Also, learning is remembering from long-term memory. For example, if you cannot instantly recall 8 x 7 = 56, you have not learned it. Learning requires repetition and continual review!

Literacy and math literacy (numeracy) are not the same. Knowledge in math means math skills. Children need to calculate things. Also, math requires logic, background knowledge, the idea that one idea builds on another (prerequisites - Gagne), and an extensive vocabulary, such as evaluate, solve, prime number, area, polygon, square roots, exponents, expressions, equations, linear, improper fractions, meters-squared, and many more.

"In math, knowledge is skills," writes Wexler. For example, the skill of calculating sums to find the perimeter of a polygon is content knowledge. Also, in learning math, knowledge is cumulative; that is, one idea builds on another. Moreover, paper calculating skills should start in the 1st grade, as they are essential for problem-solving. If kids don't learn the fundamentals of arithmetic, they are blocked from more complex math, cautions Zig Engelmann. It is common sense.

Learning facts is a good thing! (Willingham)
In the 1st grade, students should learn basic arithmetic, parts of geometry, measurement, and algebra, along with recognizing common problem types. In short, students should learn conventional or traditional arithmetic that shapes content knowledge--not reform math and progressive theories via Common Core or state standards. Students must learn content to do critical thinking (i.e., problem-solving in math), which is domain-specific. Kids must know stuff in math to move forward, but they don't! Natalie Wexler (The Knowledge Gap, 2019) explains, skipping the step of building knowledge doesn't work." In schools of education, teachers are trained to marginalize "knowledge building" in reading and math. Many current school programs are not focused on knowledge building. Wexler explains that this is "the hidden cause of America's broken education system." Kids are not learning enough content. They don't practice enough.

The Abacus & Calculators
Richard Feynman writes, "With an abacus, you don't have to memorize a lot of arithmetic combinations: all you have to do is learn how to push the little beads up and down. You don't have to memorize 9 + 7 = 16; you just know that when you add 9, you push a ten's bead up and pull a one's bead down."

Likewise, with calculators, all you do is press keys.

 I am a novice--not a  little mathematician. (Model: Aryana)
Photo Caption: Attend to precision? You must be kidding!  I'm six and a beginner, not a pint-sized mathematician. Like any beginner, I need to memorize stuff, like 5 + 7 = 12, to build knowledge, not critique another student's reasoning. I need math skills in long-term memory to solve math problems. I can't calculate the perimeter of a rectangle without knowing sums, the concepts of rectangle and perimeter, and having some experience. Calculating something is an intrinsic part of problem-solving.
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Math education in elementary school is not about training children to be little mathematicians. Children are novices, not experts. Furthermore, we should not limit students to so-called grade-level content.

Content Knowledge, not presumed mathematical practices, should guide curriculum and instruction. Instruction should be straightforward, explicit, efficient, and systematic. Content should rise to world-class and be arranged hierarchically, according to the ideas of Gagne. Also, "reviewing" should be a significant part of every math lesson (Engelmann).

It is common sense that if kids don't learn the fundamentals in the early grades, they are blocked from higher-level math and future career opportunities (Engelmann).

 Typical 3rd-grade students in my weekly algebra enrichment program:  Teach Kids Algebra (TKA), 3/17/2012.
Schools need more retired teachers to volunteer in classrooms to mentor and teach mathematics to young children.

Knowledge has always been the best preparation for the future.
Ipsa scientia potestas est.
Knowledge itself is power. - Francis Bacon

How do we prepare children for the jobs of the future? In the same way, we have always trained students for the future(Read more below.)

Note. Extensions of this post can be found here:
Preparing Students for the Future.
Leonard Mlodinow (How Randomness Rules Our Lives) writes, "It might seem daunting to think that effort and chance, as much as an innate talent, are what count. Our degree of effort [persistence] is up to us." Content free is the wrong approach.

Unfortunately, some children do not value education as much as others or study as much as others, observes Thomas Sowell. Indeed, students of all colors need higher-level math courses to expand their career choices later on. Will our kids learn enough math to get into the STEM fields or the math-related fields? Probably Not! Only about 1/4 of graduating seniors are good enough in math (NAEP).

 Kids would rather play games, text,  and do social media on their phones than study math, read Silas Marner,  or learn Latin. (Model: GabbyB)

Today, kids would rather text than learn, be a YouTube star rather than a computer scientist, or post photos on Instagram rather than read a book. For children 8 to 12, girls have less interest than boys in becoming an astronaut, athlete, engineer, or firefighter. Girls want to be doctors. (New York Life: Fatherly Survey). I am sure that preferences will likely change over the teen years. We need to get more girls into the hard sciences, mathematics, and engineering.

I often hear that we want kids to become innovators, but what we are not teaching is that the student must first become an expert in a noteworthy field of study. Texting, googling, and posting selfies do not make you an expert on anything.

Teach Content. Teach ContentTeach Content.

Students must learn content to do critical thinking, which is domain-specific. Thinking and knowing facts are intrinsically intertwined. Unfortunately, fact learning and memorizing are marginalized in many progressive schools, even though knowledge is the goal of learning and the basis of critical thinking. Indeed, "Factual knowledge must precede [thinking] skill," writes Daniel Willingham, a cognitive scientist. Starting in the 1st grade, math facts and the standard algorithms' mechanics should be mastered first, not cast aside as in reform math.

Note. Extensions of this post can be found here:
Preparing Students for the Future.
Needed are strong academic skills, a work ethic, and postsecondary education.

Knowledge has always been the best preparation for the future. The critical importance of gaining knowledge has been pushed aside. Only 25% of high-school seniors are proficient in math (NAEP, 2017), but only 3% are at an Advanced Level. We need more students who are advanced in mathematics.

Unlike American parents, Asian parents push their young children into math, and it shows: 54% of Singapore 8th graders scored at the Advanced Level (TIMSS) compared to only 10% of U.S. 8th graders. For decades, our kids have stumbled over simple arithmetic. Many incoming college students are unprepared for college courses and end up taking remedial math.

We should prepare more students for a solid precalculus course in high school. But, the reform math curriculum and the weak progressive instructional methods, including teaching the test, have driven underachievement, not preparedness. The progressive educationists do not take learning math seriously enough. They often criticize "memorization and practice" as poor teaching. Memorization and practice-practice-practice to get facts to stick in long-term memory are essential for problem-solving in mathematics.

Note. How do we prepare children for the jobs of the future?
In the same way, we have always trained students for the future. Kids need strong academic skills (i.e., math, science, reading/vocabulary, and writing/language) and a work ethic to get a good job now and in the future. Most students will need some form of postsecondary education or training. Knowing math plays a vital role. Indeed, a multitude of careers uses math.

 "It's not that Asian kids overachieve;  it's that American kids underachieve!"  It's not their fault. Model: Em)

Students must learn content to do critical thinking, which is domain-specific. For problem-solving in math, good paper-calculating-skills are required. Missing in many U.S. math programs is the early mastery of number facts and the practical calculating skills on paper (i.e., standard algorithms) that support concepts and enable problem-solving. Also, today's instruction is focused on a grade-level test. The curriculum is not world-class, and reforms such as minimal guidance instructional methods are inefficient.

Educators should concentrate on content knowledge, not test-based proficiency."It's not that Asian kids overachieve; it's that American kids underachieve!" Knowledge should be the primary goal of learning, but often isn't; it is the basis of critical thinking. Unfortunately, in progressive classrooms, academic achievement is far less important than getting along (i.e., doing a lot of group work).

"You learn only through mastery!"
When it comes to math, kids don't practice or review enough to master the fundamentals. The late Zig Engelmann pointed out that not teaching for the mastery of basics has been a considerable error in education. I believe that the central problem in schools today is "the teaching" in the classroom, which irks teachers and educators. But, the fact is that achievement in math has stagnated for at least a decade and remains far below the Asian level. American students continue to stumble over simple arithmetic, but little is done to fix the problem.

Engelmann observed that, for mastering arithmetic or reading, there must be a strong emphasis on review built into each lesson. Also, mathematics is cumulative: one idea relies on another. The learning of future lessons depends on the mastering of previous lessons.  Under reform math, the mastery of arithmetic fundamentals has not been the primary goal, starting with the memorization of math facts, the learning of the standard algorithms to calculate, and routine applications. You can't find perimeters of polygons unless you can do sums.

The learning of future lessons depends on the mastering of previous lessons. In short, mastering math or reading requires substantial practice-practice-practice and review-review-review.

 Students cannot do high-order thinking in math without knowing the math. (Model: Em)

Knowledge should be the primary goal of learning but often isn't. Knowledge is the basis of critical thinking, which is domain-specific.

Gaining knowledge makes critical thinking possible.
Daniel T. Willingham writes, "Factual knowledge must precede [thinking] skill." He explains, "The ability to analyze and think critically [i.e., skills] requires extensive factual knowledge." Fact learning is important!  Thought without content is empty (I. Kant).

Very young children can learn much more content than is currently taught in a one-track system. We should not limit students to so-called grade-level content.

In my opinion, the progressive reforms, which stemmed from the 1947 NEA Yearbook and, more recently, the 1989 NCTM math standards and the Common Core, have disrupted the teaching of mathematics, starting in the 1st grade.

Reform math educationists advocated the early use of calculators and group work (minimal guidance methods) and de-emphasized computation skills and the memorization of math facts. These are flawed guidelines with dire consequences.

Moreover, the push for technology in schools, such as tablets, laptops, and computers, has not boosted student achievement. Math test scores have been stagnant for at least a decade.

Calculator use often masks the weak skills. Today, students are permitted to use calculators for questions on the state test and the GED, SAT, and AP exams. Calculators should not replace knowing basic skills and standard procedures in arithmetic in long-term memory. Yet, this is what has happened in progressive classrooms.

The reformers are wrong!

"It's not that Asian kids overachieve; it's that American kids underachieve!" (J. Bempechat) We should focus on "the mastery of fundamentals, not teaching for a state test." In short, American kids don't learn nearly enough arithmetic. Common Core Math and state standards that are based mostly on CC Math are substantially below the Asian level. Our kids start behind in math and stay behind. Still, most parents rate their children's schools from good to excellent. Go figure?

 Learning math facts is useful for thinking in math! (Model: Jayne, 1st Grade)

Memorization is good for kids!
Fact Learning is good for kids!

Natalie Wexler writes, "Memorization isn’t antithetical to critical and analytical thinking; it’s what lays the foundation for it.” According to the National Math Panel (2008), it is critical in arithmetic for children to memorize math facts and compact procedures (i.e., the standard algorithms) to add, subtract, multiply, and divide. Facts in long-term memory boost thinking and problem-solving in working memory. Indeed, thinking in math requires knowing math facts and procedures in long-term memory.  "It's not that Asian kids overachieve; it's that American kids underachieve!" We need to reverse it.

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

Zig Engelmann's preschool. Factoring the expression 6A + 9b + 3C.

Zig Engelmann pointed out, "Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning. Concepts are difficult to teach when students have mastered only some of the facts some of the time."

In the 1960s, using old-fashioned, repetitive drills, Engelmann's pre-1st-grade students memorized number facts and multiples to do arithmetic, find areas of rectangles, solve equations, calculate fractions, and factor expressions. Being able to calculate something is an essential part of problem-solving. So is recognizing the problem type.

Eric A. Nelson ("Cognitive Science and the Common Core Mathematics Standards" 2017) writes, "When solving math problems, due to Working Memory limits, students must rely almost entirely on well‐memorized facts and algorithms." The late Zig Engelmannpointed out: "You learn only through mastery!"Kids are novices, not experts; i.e., they need to memorize stuff to build a storehouse of essential information to engage in problem-solving. Also, for beginners, solving a problem in math is usually not a matter of critical thinking. It's a matter of recognizing the problem type and practicing it with variations. (Oh, that's a subtraction problem, triangle problem, Pythagorean problem, multiplication problem, velocity problem, percentage problem, circumference problem, an area problem, and so on.)

Engelmann challenged the education establishment:" Education policymakers have a model of how things should be, but 'should be' is not reality. They believe that it is more important to preserve their flawed understanding of how kids learn than it is to provide effective instruction to kids."

If learning is remembering from long-term memory, we have not taught children to learn and master essential content. We should not limit students to so-called grade-level content.

Lastly, Robert Plomin (Blueprint) writes that the differences in academic achievement in schools are 60% genetics; however, the percentage should not be construed as deterministic. It's not. It's what we do with what we have that counts in learning. Read DNA, Not Nurture

"In education, you increase differences."

Richard P. Feynman was invited to a conference to discuss "the ethics of equality in education." He confronted the experts by asking this question. "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?" (Surely You're Joking, Mr. Feynman! by Richard P. Feynman, Nobel Prize-Winner in Physics)

"You don't know anything until you have practiced." (Feynman)
You don't get good at something unless you practice-practice-practice, then practice some more. Oh, did I say you need to practice?

Starting in the 1st grade, students should learn the math facts and the Standard Algorithms first, not a hodgepodge of reform math strategies or alternatives.

3rd Graders in my algebra program.
(Teach Kids Algebra)