Friday, November 27, 2020

Radical Ideas 3

My Musings
December 2020
     Merry Christmas!

Happy Chanukah
to my former students at TDS. Be the light!

Welcome to Radical Ideas #3 
This website voices my opinion.   
(Radical Ideas 2 is still available but will fade slowly in 2021.)

Note: In January, Reflections 2021 will start.

HELP! Alice falls down the rabbit hole of reform math. 
Is that Mock Turtle? No, it's JB.
Reform Math crusaders have crushed
traditional arithmetic. They are wrong!
Remote learning puts students further behind.
2020 has not been a good year for kids!
Remote Is a Failed Solution (Model Credit: RemiT)

To think math, kids must know math facts! 

"Study hard enough to become Smart enough." (S. Korean Motto)

"Math is not a talent. Being good at math is a product of hard work." (Peg Tyre)

***** After eight years of Common-Core-based reform math, a meager 24% of 12th graders were proficient in math (NAEP 2019), an awful statistic. For decades, education in the U.S. has become a "fortress of progressivism," says Rick Hess. And, the leader of the revolution, in my view, has been Jo Boaler. By 4th or 5th grade, the reform math of the progressive movement has put our kids at least two years behind their peers from high-performing math nations.  American schools are tech-rich, not content-rich. Tech trumps Content. What a colossal error. 

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 School) 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.  

Researcher William Schmidt explains, "What you teach is what you get." Teachers should teach math knowledge to prepare kids for the next grade levels, not just to get through their grade level. William Schmidt calls this a "systemic failure" and an "injustice in our schools." Schmidt and McKnight emphasize equitable—not identical—learning opportunities in Inequality for All. I have long questioned the idea that all students will have the same (identical) opportunities to learn content.

"Perhaps as students sit in 2021 classrooms, they will daydream about the fun they had during remote instruction at home. I doubt that also," writes Larry Cuban in his blog post, Classrooms of the Future. Let's face it. 2020 has been a terrible year for kids and parents. 

Help! HELP! Reform math does not prepare students for college-level math because it downgrades essential arithmetic and algebra skills. Jo Boaler, a radical advocate of reform math, champions a crusade to crush traditional arithmetic and most everything Old School. The reformers want to stamp out Old School, including standard arithmetic, memorization, and practice-practice-practice (drills), even though "practice helps memory," writes cognitive scientist Daniel T. WillinghamMany of the reforms advocated by Jo Boaler and her squad make no sense. 

1877 Arithmetic in America

We have fallen behind. "If 12 peaches are worth 84 apples, and 8 apples are worth 24 plums, how many plums shall I give for 5 peaches?" If you cannot figure out the correct answer within 1 to 2 minutes and know you are correct, you do not know basic arithmetic or apply it. No calculators. The "peaches to apples to plums" problem was a mental arithmetic problem for 3rd and 4th graders in 1877. [From Ray's New Intellectual Arithmetic: 3rd & 4th Grades, 1877] 

Students should practice the most critical, foundational math skills to mastery (automaticity), not just for proficiency on the state test. In short, the most important skills (20%) should take up the most instructional time (80%). It should include place value, rules of arithmetic, single-digit math facts, and standard algorithms for calculating, along with important ideas (e.g., variable, equations, reciprocals, etc.) and uses (e.g., perimeter, area, percentages, averages, etc.). Good calculating skills are needed for problem-solving. Unfortunately, Common Core reform math has marginalized calculating skills such as learning standard algorithms. student cannot learn standard algorithms without first memorizing single-digit math facts. 

In contrast to reform math, standard calculating skills should be learned first!. Parts of algebra, measurement, and geometry should also be taught in 1st grade on up. Calculating skills are built on memorized single-digit number facts and standard algorithms. Children should also learn fractions, percentages, ratio/proportion, equation solving, and other key ideas and uses.   

In contrast to Boaler and her reformers, "Task performance improves with practice seems so universally observed," points out Daniel T. Willingham, a cognitive scientist. (It is an empirical generalization.) Contrary to reform math advocates (e.g., Jo Boaler and her cohorts), kids should first master math facts and traditional arithmetic algorithms starting in the 1st grade. Memorization, repetition, practice drills, and flashcards for instant feedback are good for kids. Confidence grows with practice, repetition, and review. Put away the manipulatives and use flashcards. 

Jo Boaler once said that she never memorized the multiplication table, but that didn't prevent her from becoming a Ph.D. in "math education," which is far below a degree in mathematics. She marginalizes math facts and technique. I think she did memorize stuff as a child. All 2nd and 3rd graders need to master the times table. Even though many kids exhibit some math anxiety, it doesn't mean they can't work through it. Most kids do. Timed tests are not the reason kids don't like math. Math is abstract, sequential (one idea builds on another), and harder to learn than other subjects such as history. Problem-solving becomes unachievable without the basics in place. Learning math requires a lot of practice to gain "knowledge and technique," writes Ian Stewart (Letters to a Young Mathematician, 2006)Math builds from one idea to another idea. It takes more time, practice, and review to learn mathematics than other subjects. Nothing is fun until you can do it well, notes Amy Chua

Boaler is the "public face of K-12 mathematics reform," writes Sam Scott (Stanford Magazine, 4/27/2018). I disagree with Boaler on many points, one being that speed isn't essential in calculating. For math facts that support standard algorithms, instant retrieval from long-term memory is necessary for problem-solving. In short, good calculating skill is a pillar of problem-solving. The trio for a good math program is skills-ideas-uses, writes MIT-trained mathematician Robert B. Davis who created The Maidson Project, 1957 for grades 3 to 6. The Madison Project (1957) and Science--A Process Approach (SAPA K-6 1967) inspired me to teach little kids algebra. In 1st-grade SAPA, four of the six process skills were math or math-related. Using math in K-6 science is essential, so where is it? 

SAPA: "Gagne has shown that if a learner attains behaviors subordinate to a higher behavior, there is a high probability that he can again that higher behavior. On the other hand, if he has not attained the subordinate behavior, there is near zero probability that he will attain higher behavior." The sequence hierarchy is essential in learning math, as a new idea builds on or connects to old ideas already learned. Learning new content has little to do with Piaget's stages of development or manipulatives. Learning new content is about knowing the prerequisites (i.e., background knowledge). 

"To think math, kids must know math facts!" 

Functions are just rules connecting two variables like Add 3y = x + 3. 

"The place value system, fractions, and standard algorithms all contribute greatly to algebra readiness." (Quote: W. Stephen Wilson, mathematician) 

American teachers overuse and misuse manipulatives. 

Mary Baratta-Lorton (Mathematics Their Way, 20th Anniversary Edition, 1995) writes, "During the beginning stages of concept development, abstract symbolization tends to interfere rather than enhance the understanding of a concept. For this reason, a great deal of this book deals with ideas that develop concepts without the use of any written numbers" Really? The assumption makes no sense, yet it is still an essential part of reform math. Manipulatives have been the rage since the early 1970s. The abstract algebra ideas I teach to very young students are not dependent on Piaget's stages or manipulatives. They require symbolics and supportive background knowledge, such as the auto recall of arithmetic facts--just the opposite of Baratta-Lorton. 

Note: First graders should write numbers to 100, use a number line for simple combinations, add 1 to get the next counting number (n + 1 = n'), the zero rule (n + 0 = n) the commutative rule that 2 + 3 = 3 + 2, which is very easy to demonstrate, the idea of balance for equations ( 5 = 5), negative integers (5 + -5 = 0), undo operations such as 12 + 16 - 16 = 12. (To undo "add 16", do the opposite, "subtract 16" to get 12 back again.) They should also learn the place-value system from day one, use equality and inequality symbols (e.g., 5  >  2), and the standard algorithms for adding and subtracting. First graders need to memorize addition facts and many subtraction facts. For algebra ideas, go to TKA (below). Singapore children also learn multiplication as repeated addition, such as 3 x 4 = 4 + 4 + 4 or 12. I introduce multiplication as repeated addition in my TKA (Teach Kids Algebra) program at the 1st and 2nd-grade level.

Marilyn Burns (About Teaching Mathematics K-8 Resource, 1992) writes that children make eight common mistakes in arithmetic; however, the errors originate from poor teaching of traditional arithmetic, not the arithmetic itself. In my view, the reason for these errors is that teachers taught reform math and used "manipulatives" and "cooperative learning" instructional methods (i.e., minimal guidance group work). It's okay to have an exploratory lesson now and then, not a steady diet of discovery or project or problem-based learning, which is inefficient in teaching arithmetic and math in general. On the other hand, explicit teaching via worked examples with ample practice and feedback works!  

Note: Typical 1st-grade students can learn the fundamentals of real algebra, given proper instruction. It is important to Teach Kids Algebra (TKA), which is STEM math built on traditional arithmetic starting in the 1st grade.

Teach Kids Algebra (TKA)

STEM Math for Grades 1-5
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 the standard algorithms. It starts in 1st grade.

Note: Giving immediate feedback is important when children are learning algebra fused to traditional arithmetic. It's STEM mathChildren need guidance and encouragement. For grades 3 to 5, lessons were once a week for the school year. For 1st and 2nd graders, 6 or 7 lessons.

TKA 4th-Grade Algebra, 2018-2019

  Bottom Line: Common-Core-Based Reform Math Has Failed
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 does not 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 reform math has not made our students better in math. Common Core reform math is a radical idea addressed on this website. Remote schooling has been a bust and another radical idea.

There is a math crisis in America because most kids don't know enough math. Our schools will not likely suddenly change to a no-nonsense, content-rich, world-class math model, such as the Singapore math program. (Singapore kids are tops in math.) Unless parents take charge of their children's education and homeschool them in traditional arithmetic, the children will probably become one of the 76% who does math poorly, affecting future opportunities. The alternative is to hire a top-notch $60/hour tutor. 

✓  Fareed Zakaria (In Defense of a Liberal Education, 2015 ) writes that Chinese students are "two years ahead in math because they've taken at least two more years of math. They spend 25 to 30% longer a year in school than their American counterparts. They also work hard." In contrast to the American education system, says Zakaria, "the Chinese system teaches basics very well" via memorization, practice, repetition, and drills. "The Asian system does teach you to work hard, to retain knowledge, and to perform under pressure--all of which are valuable skills. That may be the simple problem in the United States today--people were working less at school," when they should be working more. Also, there is rampant "grade inflation" in American schools, even in our best schools. Many of our best students would be just average if that in Singapore schools. Zakaria says that a liberal education should include humanities and science, especially chemistry and algebra-based physics, and mathematics, including arithmetic, algebra, precalculus, and proof-based geometry. "The greatest shift in liberal education over the past century has been the downgrading of subjects in science and technology." It's an alarming trend. Today, many see liberal arts as only the humanities, but a good liberal arts education includes math, science, and technology courses. 

American students spend about 41% of their school time on basic academics, according to the National Education Commission on Time and Learning. In 2005, Milton Goldberg & Christopher T. Cross updated the preface: "We know that American schoolchildren spend less of their school day receiving substantial academic instruction than students in most nations that outperform us in international comparisons." American kids are into sports and other nonacademic lessons or activities, while Asian kids focus on studying and after-school academics--not sports. American students are "Prisoners of Time." Not much has changed since the report was first published. 

First-Grade students should avoid using fingers to count or do arithmetic, which goes against reform math people who advocate it. At first, a 0 - 20 number line can be used to visualize 2 + 6 = 8, 6 + 2 = 8 or that 6 > 2 or 2 < 6, etc. Students should be taught to write expressions, equations, and inequalities with symbols. Math is a symbolic language. Children are novices and should memorize basic number facts, not calculate with fingers. In short, children should use symbols and memory for math facts, not manipulatives or fingers.
(Model: Jayne, 1st-grade student)

  When I first started teaching algebra lessons to 1st-grade students at an urban Title-1 school in the Spring of 2011, I had no idea what would happen, but I thought it could be done with the right sequence of lessons. I asked teachers to start drills on addition facts, which was a big help because I fused algebra to traditional arithmetic. "To think math, kids must know math facts!" I explained worked examples on the board, then gave students a practice sheet. I provided individual support as I walked around the room. Thus, my Teach Kids Algebra (TKA) idea was born. Most students were able to do the performance tasks, but not all. 

I found that very young children could learn much more math by giving them challenging tasks to complete. The problem was not that the content was too hard or developmentally inappropriate. Instead, it was that children didn't know the prerequisites, especially single-digit addition facts in long-term memory for instant recall, something I had taught to my 1st-grade self-contained class at an urban Title-1 school in the early 80s.

No group work. No reform math. No calculators. No Common Core. No drawings or writings paragraphs. Everything was explained with symbolics (e.g., y = x + x + 2)! Teachers of reform math have vastly underestimated what students can do given proper instruction. Teach Kids Algebra was a reaction against Common Core reform math. 

Of the trio of a good math program (skills, ideas, and uses), I focused on the ideas (i.e., concepts). 

It is true!
The more I know, the more I can learn, the faster I can learn it,
the better I can think and solve problems. 
WOW, the Cognitive Science of Learning is GREAT!

Remote schooling leaves some children sad and angry, writes Hannah Natanson and Laura Meckler at the Washington Post. For example, a 9-year-old student is stuck in Zoom school and has not seen a friend since March. She cries when she gets angry and frustrated with remote schooling and fears she is not learning enough new material to pass the 4th grade. Like many remote students, she often has Internet connection problems. She is right, of course. She is not getting the education she needs to advance. (Note: Some phraseology from Natanson & Meckler)

✓ 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? Remote is stupid. Teachers are caught in the mess. They are told to teach reform math from standards that were not world-class. If your kids are not in school full-time with in-person teachers then blame the teacher unions and the biased media. 

✓ "If you cannot explain difficult mathematics to little kids, then you do not know it well enough." Alternatively, you do not know how children learn math. H. Wu, a mathematician at UC-Berkeley who has been teaching workshops and courses for K-8 teachers for decades, wrote that most teachers do not know enough math to teach Common Core math. Many teachers are just average and have difficulty explaining complicated math to students. Teachers take my course in the summer, says Wu, but they teach the same old reform math with minimal guidance group work methods when they return to the classroom. Learning more content does not always alter the way teachers teach, noted Wu, but it is a step in the right direction. The premise has been that teachers know best how to teach math and reading, but national and international test scores do not support the premise.

Note: Common Core is not world-class because it does not meet international benchmarks in math. By 4th or 5th grade, students are about two years behind their peers from top-achieving nations. The advent of Common Core in 2011 revived NCTM reform math and its minimal guidance methods, which have been promoted by math educators from schools of education. Teachers seem to be teaching pedagogy, such as many different ways to multiply, not content. The standard algorithms are pushed to the side.

✓ The reformers want to stamp out Old School, such as traditional arithmetic, memorization, and practice-practice-practice. They say they know best how to educate your child. Well, I think not. Reform math has been a bust, along with fads such as Common Core, remote and hybrid learning, minimal guidance methods of instruction (e.g., group work, discovery learning, project-based learning, etc.), equalizing downward by lowering those at the top (Thomas Sowell: "a crazy idea taught in schools of education"), mathematical practice standards (from the NCTM people), below world-class math standards, grade inflation, and low expectations, all of which make K-12 math education inferior and mediocre. Teachers misrepresent student progress with grade inflation.  

✓ After eight years of Common-Core-based reform math, a meager 24% of 12th graders were proficient in math (NAEP 2019), an awful statistic. Our K-12 math teaching is horrible. The problem originates in 1st grade with inadequate teaching of basic arithmetic. In my analysis of 2011, the Common Core 1st grade math standards were significantly behind the Singapore 1st-grade syllabus of arithmetic content. Singapore 1st-grade students learned a lot more arithmetic content than American students, such as memorizing arithmetic facts, using formal algorithms for whole numbers, learning multiplication, and writing equations to solve word problems. In short, Common Core math standards were not world-class, so why did states eagerly adopt them? Math standards have failed. 
Remote is a bust. 

✓ The late Richard Feynman wrote, "I would rather have questions that cannot be answered than answers that cannot be questioned." It is why I oppose those in education who think they know best how to teach children math, such as Jo Boaler, a so-called math educator. If they had known how best to teach math, then almost all our kids would learn algebra in the 1st grade. Frankly, when asked how I teach 1st and 2nd graders real algebra, I reply, "I do not know how I explain complex material to children ... However, I do know that if I cannot explain difficult math to students, then I do not know it well enough." Also, I think in terms of prerequisites (Gagne), not stages (Piaget), and write coherent "worked-out examples" that are performance-based (Mager's behavioral objectives). Is there a more straightforward way to get from A to B?

Why make learning arithmetic harder than it is? It is another reform math radical idea that makes no sense. Learning arithmetic fundamentals should not be confusing, hard, or complicated.

✓ I agree with Daniel Willingham that "children are more alike than different in terms of how they think and learn." Learning styles are not supported, that Jane learns better in one way that would be bad for Bill. No, both Jane and Bill think and learn in the same way through practice that makes memory long-lasting.

Note: This website counters reform math, discovery learning, project-based learning, and other minimal guidance, constructivist methods (via group work). It also opposes the status quo, including teacher unions that tell teachers not to teach as they demand more money. Schools are loaded with beliefs, fads, and policies, such as remote and hybrid learning, which have scant evidence supporting them. This website voices my opinion. It includes Radical Ideas and who endorses them. Sameness is a basic tenet of socialism, which has gripped our schools for decades. The latest strategy is Common Core reform math that started in 2011. After eight years of Common-Core-based reform math, only 24% of 12th graders were proficient in math (NAEP 2019), an awful statistic. Common Core and state standards have failed to deliver career and college readiness for 3/4 of the math students.

The math curriculum is not world-class, according to my 1st-grade analysis of 2011. By 4th grade, many of our students have fallen two years behind in math achievement. Moreover, grade inflation dominates the landscape of our public schools. Kids get good grades for no good reason to boost their self-esteem, so we are told by education leaders. It is mostly nonsense, in my opinion. Now, it is augmented by so-called social-emotional lessons. Really? Asian kids learn more math because they are in school longer. 

Radical Ideas 

Let's Equalize Outcomes, Inflate Grades, Teach Less Content, Dump the SAT/ACT for Diversity & Equity, Change OBE (Outcome-Based Education) from academic goals to "feelings" goals, Dump traditional arithmetic (Old School) for reform math that is to world-class, Dump explicit teaching for ineffective Minimal Teacher Guidance During Instruction methods, Dump paper-pencil calculating for calculators, Dump memorization and repetition for content-free critical thinking, etc. The influential leader of these and other radical ideas is Jo Boaler, a "math educator" at Stanford University, not a mathematician or scientist. 

Adding It Up is the blueprint for reform math. 

Jo Boaler presented reform math to the Committee on Mathematics Learning, established in 1998 by the National Research Council. The committee produced a report in 2001 called Adding It Up, which was a blueprint of Boaler's brand of reform math, a radical idea. Many of the committee members were from schools of education. Others had degrees in psychology and sociology, not mathematics, except two. Most of the Committee on Mathematics Learning consisted of people who did not know mathematics well. It was a diverse group! 

Furthermore, the comments of the 15 reviewers were not part of the report and "remain confidential." The reviewers "were not asked to endorse the conclusions or recommendations, nor did they see the final draft of the report before its release." Thus, their input was in name only and marginalized. (Incidentally, one of the reviewers was Richard A. Askey.)

Notes from Daniel Willingham, Richard Feynman, Amy Chua, Stanislas Dehaene, and others.
  • 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 does not 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 with flashcards. 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." Furthermore, "No matter what, you do not talk back to your parents, teachers, elders." (Amy Chua, Battle Hymn of the Tiger Mother, 2011).
The use of worked-out examples can substitute for problem-solving in learning Algebra (Sweller). Over many decades, I found that explaining carefully selected worked examples on the board can boost K-5 arithmetic learning (i.e., skills, ideas, and uses) and middle school algebra, especially when joined with flashcards for retrieval practice from long-term memory (Dehaene). Flashcards have many learning applications: math facts/definitions/procedures, spelling, definitions, vocabulary, procedures, language, and so on. Why do progressive teachers ignore flashcards? (Note: ZOOM is a fad that will fade, but flashcards are not. They help children learn.)

Problem-solving is a function of knowing stuff. Try solving a trig problem without knowing trig or translating Ovid's Metamorphoses without knowing Latin vocabulary and structure.
There is no substitute for knowledge in long-term memory and the practice that gets it there. It is the cognitive science of learning. Is ZOOM fatigue spreading?

Children can learn much more math content than taught. The problem begins in K-5 with standards that do not meet international benchmarks, a reform math curriculum that marginalizes basic arithmetic and standard algorithms, and minimal guidance teaching methods that are inefficient (group work). It is "the teaching." Math achievement has stalled. For at least a decade, our students have not been getting any better in math. They were not very good, to begin with. Students often fall behind, and remote and hybrid teaching exacerbates the problem. The reformed math curriculum, which is based on Common Core or Common-Core-like standards, is not world-class.

Students are not prepared. Like math, reading scores have stagnated over the past decade, too. Only 37 percent of 12th-graders were proficient in reading, 24 percent in math. The results of the 2019 NAEP tests are bad news for the reformers. It means most 12th graders are not prepared to pass entry-level college courses. However, the deficiencies start in K-5. The fundamentals of arithmetic and reading have been taught poorly.

Alice Crary and W. Stephen Wilson in the New York Times (2013) point out that reform math programs have killed traditional math. They write, "The standard algorithms are either de-emphasize to students or withheld from them [students] entirely." Moreover, "The staunchest supporters of reform math are math teachers and faculty at schools of education," where teachers are trained. Now you know why reform ideas persist in our classrooms. Many of the reform ideas taught in schools of education are rooted in NCTM standards, dating back to December 1989. Backing up the NCTM claims was a report called Adding It Up, PreK-8, 2001. It stated that even students in high-achieving Asian nations could not meet the report's Five-Stranded definition of proficiency. The report promoted the same old reform math stuff, such as the inventing algorithms, early calculator use, multiple representations, teacher as facilitator, and much more. In reform math, "reasoning" is much more critical than learning content knowledge. However, it is not. "Many educational questions cannot be answered by research, including the curriculum and methods of instruction," the report concludes. So, educators often use "professional judgment and reasoned argument" based on assumptions such as "math should be fun." to fill the gaps that fall out of the domain and research. This is how beliefs and fads creep into instruction and stay there for decades, rather than facts and the science of learning.
There is no substitute for knowledge in long-term memory and the practice that gets it there. Problem-solving is a function of knowing stuff, not thin air. To perform arithmetic and algebra well, students must know facts, procedural knowledge, and problem types in long-term memory. Nevertheless, reformers think that facts and techniques do not matter much. Really? All the reasoning (i.e., critical thinking) in the world will not help you solve a trig problem unless you know trig. Incidentally, the latest surge of reform math correlates with Common Core or Common-Core-like state standards.

Link1: How I Shoot Photos
Link2: Gifted Programs
Our best math students often go unchallenged and fend for themselves in mixed classrooms. Some students need advanced-level content and fast-paced math classes because they are a couple of years ahead of their peers in math achievement, but their academic needs are seldom met in elementary and middle school.
Link3: Teach Kids Algebra (TKA)
STEM Math for Grades 1-5
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 support the standard algorithms. It starts in 1st grade.
Link4: PISA 2018
American students did poorly in math. Gee, I wonder why?
Link5: Cognitive Science
Students learn new ideas linked to old ideas they already know. Readiness is not age-dependent; it is determined by the student's mastery of the prerequisites.
Link5: Science
Science does not prove things right. It is a method that eliminates wrong ideas. Correlation should never imply causation. 
Link6: Early Math Early math is just as important as early literacy but is often neglected. The crux is that kids need explicit teaching from K-5 teachers who know math, but many elementary school teachers are weak in math.

Below: Maria,  as a high-school Junior in Texas, took a college physics course at a community college for high school dual credit. She was astounded that many of the college students stumbled over simple algebra needed to solve physics problems.

Maria was a determined 12-year old who later majored in computer science at UT. The University of Texas does not accept AP Calculus toward a STEM degree, so she needed to take the university's calculus courses. STEM students need to be advanced in math. MariaB at age 12 on 7-31-2006

Update: 12-19-2020
More Coming

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