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Emma, 10, at Reid Park

NAEP

Let's face it. We teach math badly, starting in the 1st grade!

In the real world, pouring billions into programs without substantial evidence, because they are such good ideas, does not work. Indeed, reducing class size and upgrading tech in the classroom (both, very expensive), and so on, have not boosted achievement or made America into a math superpower. We have been going in the wrong direction. We are told that our kids are doing well in math when, in fact, compared to international achievement, they are not. Top-performing nations are years ahead. Common Core, state standards, and state testing have not been the answer. Math is taught as a version of NCTM reform math that failed in the past. Furthermore, Common Core and state standards based on Common Core are not benchmarked to international math standards.  

We should make sure that kids have the calculating skills to do the applications and solve problems. If you cannot calculate it, then you don't know it. Calculating skills should come before applications. We give beginners word problems before they know 5 + 7 = 12. We should first require kids to memorize facts like 5 + 7 = 12 for instant recall before asking them to solve word problems that involve the addition and subtraction facts (i.e., standard algorithms). The same is true for long-division and the other operations with whole numbers, fractions, decimals, and percentages.   

NAEP Changes Won't Solve the Problem
In November 2018, the NAEP committee made name changes: NAEP Basic, NAEP Proficient, and NAEP Advanced. The committee stated that the NAEP Proficient achievement level does not mean grade level, even though the tests are given at the  4th-,  8th-, and 12th-grades. The definitions of these levels were altered somewhat to make them more accessible (equitable), but at what cost? Less achievement? The commission now says that NAEP-BASIC is grade level. "See, our kids are doing okay in math! Most kids reach the NAEP-Basic level." I don't buy the committee's argument. Our kids are lousy at math, and everyone seems to know that. But, for decades, American educators and leaders have glossed over the problem and made excuses. The committee, in my mind, has done the same by making the NAEP-Basic grade level. 


Another red flag is in your community and elsewhere. For example, in the Tucson area, from the nine school districts that feed into a local community college, 74 to 88% of the students who had applied at Pima Community College were placed in remedial math (PCC 2014).

U.S. math is going down (across the spectrum), not rising, and the state standards, which were strongly influenced by Common Core and reform math enthusiasts, add to the decline.

Notes 
1. NAEP (National Assessment of Educational Progress) is The Nation's Report Card. 
2. TIMSS (Trends in International Mathematics and Science Study)

I don't think I am misusing the NAEP results because they indicate major problems with U.S. math instruction. Achievement is not getting better. It is flat. The same is true for international math tests such as TIMSS. 

NAEP 2017 (Nations Report Card): How well we educate children in math can be inferred by examining the Advanced levels of NAEP and TIMSS. For example, only 8% of 4th graders, 10% of 8th graders, and 3% of 12th graders scored at the Advanced level of math in NAEP government tests. 

Latest TIMSS: In TIMSS, only 14% of U.S. 4th graders reached the Advanced Level in math, but Singapore had 50%, Hong-Kong 45%, S. Korea 41%. We are not in the same ball park. The trend continues. Only 10% or U.S. 8th graders reached the Advanced Level of achievement, while 54% of Singapore 8th-grade students reached that level, and so on. The international difference in achievement is stark. 

ACT Math Scores Are at a 20-Year Low.
NAEP fails to mention that the most recent ACT math scores are at a 20-year low and that the most recent NAEP math scores are no better than they were a decade ago. On the other hand, students from Asian nations leave U.S. students in the dust. U.S. state math standards are not world class, and it shows up early when only 8% of 4th graders, 10% of 8th graders, and 3% of 12th graders scored at the Advanced level of math in government tests (NAEP).  

International: Asian countries dominated the 2015 TIMSS Math results. 
U.S. students were not in the same ballpark. The more "rote" East Asian learners, who memorize and drill-to-improve-skill, soared far above U.S. students not only in content knowledge and ability to perform mathematics correctly but also problem-solving at the Advanced levels. WOW!  At the 4th-grade TIMSS level, for example, Singapore's scale score was 618 compared to the U.S. scale score of 539, which, incidentally, is slightly better than Finland's 535. 

Reform Math
Reform math, which focuses on reasoning and multiple strategies often at the expense of learning content knowledge and standard algorithms, permeates math teaching today. Kids are novices, not experts. They need to memorize stuff for mastery. Proficiency on the state test is the goal, not the mastery of essential content.

Kids Should Master Standard Arithmetic
In contrast to reform math, the traditional approach focused on the mastery of content knowledge and basic calculating skills such as the standard algorithms. It worked for the Asian students who are leaps ahead of U.S. students. Common Core and state standards are not benchmarked to international standards. They are not world-class. To do arithmetic well means to know facts and procedures. It requires memorization and practice-practice-practice. Unfortunately, the trend in U.S. education has been to eschew memorization and the practice of standard (traditional) arithmetic fundamentals. 

Kids are not mastering simple arithmetic. 

Learning
If you had learned arithmetic in school but can’t remember it, it means you never really learned it in the first place. Learning something is recalling it from long-term memory. It involves mastery, which, in turn, requires practice-practice-practice. 

To apply mathematics to solve problems, first, you need to know the building blocks. You don't start with an application; you start with basic arithmetic--numbers and how they behave and relate to each other, the single-digit number facts, and the standard algorithms.  These are the building blocks of arithmetic, along with patterns (i.e., rules) such as the Commutative rule.


©2018 LT/ThinkAlgebra

such good ideas

In education, we have embraced many beliefs, fads, innovations, trends, reforms, and theory without substantial evidence because they are such good ideas. A lot of good ideas turn out to be duds such as Common Core and personalized learning through tech, blended learning, group work, and so on. According to a recent government report, a whopping 82% of the innovations funded by the U.S. Department of Education did not improve student achievement. In short, many of the so-called innovations often hyped in our schools don't work, even the popular ones, such as using technology and software to tailor instruction to each child. 

Many educators believe educational technology can personalize learning. It sounds like another good idea! But, innovations such as using technology and software to tailor instruction to each child have failed to post gains in achievement. The U.S. Deparment of Education keeps funding innovations, but almost all of them fail.

Still, we keep hearing that the way to improve achievement is only through innovation. It's bunk. I think, the best way to improve math education is to look to the past and resurrect the well-established ideas that worked for almost all students, such as memorizing the math facts and learning the standard algorithms as the primary method of calculating. Kids need to "drill to develop math skills." In short, they need to practice-practice-practice because they are novices. 

Also, students need to learn content knowledge, lots of it, because critical thinking in math (i.e., problem-solving) is empty without specific content in long-term memory. Reform math has left students with weak calculating skills and deficient content knowledge. The ACT Math scores have sunk to a 20-year low. In short, many students don't have strong math skills and the content knowledge that are essential for STEM jobs to power the economy. 

The popular problem-solving approach might work for a lesson now and then provided children have strong, calculating skills and sufficient content knowledge. However, often, this is not the case. If a student cannot calculate it, then they do not know it. We should make sure that kids have the calculating skills to do the applications and solve problems. 

The use of software and tech may not be the best way to teach multiplication.
The teacher can demonstrate this on the chalkboard and ask students questions to clarify that multiplication is a sum. Software is not a substitute for the teacher. Software can't ask follow up questions, explain points, or talk to students.

Multiplication is a 1st-grade level concept, not a third-grade level. Students should use a number line on their desks. Multiplication is a sum and should be taught that way. First graders in Singapore are good at adding, so 4 x 7 means the sum of four sevens: 7 + 7 + 7 + 7= 28. 

To multiply, Zig Engelmann asked disadvantaged, urban pre-1st-grade students to memorize multiples, such as the 7s: 7, 14, 21, 28, and so on, to do multiplication and find areas of rectangles. In other words, students should be able to do the calculations before an application is introduced.    

Robert Holland writes about Common Core and state standards and quotes Mark Bauerlein: “These standards rely on process-and skills-based ideas [thinking skills], which brush aside knowledge of Western and English literary traditions,” remarked one of the Pioneer analysts, Emory University English professor Mark Bauerlein."


Holland writes, "Instead, the new-age standards offer a heaping serving of multiculturalism, plus the preposterous notion of 21st-century mathematics that renders obsolete principles validated centuries ago. This is how the dumbing-down so beloved by social levelers proceeds." Traditional or standard arithmetic has been bumped by reform math and Common Core, which is one-size-fits-all. Indeed, Common Core and state standards based on Common Core are below world-class math standards.

Teaching kids to fish is a waste of time if there are no fish! 

H. D. Hirsch, Jr. writes, "American educators have been pronouncing ideas about all-purpose critical-thinking skills for more than a century. But we now know from cognitive science that the idea is essentially mistaken. Critical thinking does not exist as an independent skill." It is specific to a domain. 

"The domain specificity of skills is one of the most important scientific findings of our era for teachers and parents to know about." Still, it is widely unknown. Critical thinking in math is called problem-solving. You have to know some trig in long-term memory in order to solve trig problems. In short, the "basis of thinking skills is specific domain knowledge." 

Evidence to the contrary has been ignored by the liberal education establishment. Jill Barshay points out, "Experts have long known that the research evidence doesn’t consistently support the notion that smaller classes increase how much students learn." 

For math, no benefits were found in reduced class size. 

Personal Note. In the early 80s, I had 25 students in a self-contained 1st-grade class at a city Title-1 K-3 school. Except for 2, the students learned to read, write, and do arithmetic at or above grade level. The math was above the level found in the Singapore syllabus in that students worked with negative numbers, fractions, and algebra ideas. A few years ago, Singapore lowered the number of students for 1st grade to 30. But, the problem isn't the number of students. Zig Engelmann would say, "It's the teaching in the classroom." 

Kids aren't learning enough factual and procedural knowledge in long-term memory because reform math instruction doesn't stress "mastery." Kids need to drill to develop skill in arithmetic. Also, reform math, with its many cumbersome, alternative, nonstandard algorithms (strategies) to do straightforward arithmetic, induces cognitive overload. Moreover, E. D. Hirsch, Jr. writes, "Teaching strategies [in both math and reading] instead of knowledge has only yielded an enormous waste of school time." 

Furthermore, in arithmetic and algebra, the algorithms should be efficient, not cumbersome or complicated.  

Andreas Schleicher, director of the education and skills unit at the Organization for Economic Cooperation and Development, has long been arguing that the U.S. overemphasizes small classes at the expense of good teaching. In his 2018 Book, “World Class,” Schleicher debunks the “myth” that “smaller classes always mean better results.”

I think it is okay to have large classes with special tutoring (pull out) for those students who are behind. For example, Singapore places incoming 1st-grade students who have weak number skills with a separate teacher for math class to catch them up within two years. In short, Singapore sorts kids for math class. We don't do that. We mix low-achieving math students with high-achieving math students in K-8 under the pretense of equity and inclusion. U.S. achievement in math has been lackluster, flat, and below expectation. Our policies often hinder achievement.     
  
Robert Slavin, director of Johns Hopkins’ Center for Research and Reform in Education, told Jill Barshay, “But…no studies of high quality have ever found substantial positive effects of reducing class size.” Despite popularity with parents and teachers, review of research finds small benefits to small classes (by Jill Barshay, The Hechinger Report).

The liberal Education Establishment promotes unconfirmed ideas and fads because they are "such good ideas." No need to test, they say. Scientific evidence of effectiveness isn't required. But, too many allegedly "good ideas" have turned out to be duds. They failed in the classroom and hurt the students. It is called reform! Unfortunately, there are many education practices, beliefs, fads, innovations, reforms, and theory that are not supported by evidence. No wonder the U.S. has major education woes. In fact, progressive reformers say there is no need for proof because the ideas and theories are just common sense and accepted by everyone (e.g., self-esteem, learning styles, small classes). Really?  

Teaching kids to fish is empty if there are no fish! 
The way we teach basic arithmetic and reading has produced lackluster results. East Asian nations focus on performance in math procedures (doing and applying math knowledge well) while the U.S. educators stress higher level thinking strategies, fads, and small classes. Our approach has failed.

E. D. Hirsch, Jr. points out that the education establishment is "captive to bad ideas." Thus, over the decades, "learning has declined." Sadly, the progressive education establishment insists that "students should 'learn how to learn' to develop abstract thinking skills that they could apply fruitfully to various facts, events, and conditions." Sounds great! But, it is contrary to the cognitive science of learning and domain specificity of skills. Indeed, a thinking-skills-orientated approach has radically shaped the curriculum. It is entrenched and hard to change. Hirsch explains that it's the wrong approach because "abstract thinking skills falter without a foundation of content supporting them."  Kids need background knowledge--not an "anti-core knowledge curriculum favored by the educational establishment."  Thinking without content is empty. (E. Kant)

We should first emphasize lower level thinking (i.e., knowing and applying) to build a strong foundation for higher level thinking. Kids need background knowledge.  

According to Mark Seidenberg, the U.S. culture of education has produced "chronic underachievement" in both math and reading.

This post is updated frequently. 11-16-18, 12-6-18, 12-8-18

©2018 LT/ThinkAlgebra


 

Deficient Skills

Deficient math skills begin in the 1st grade and rise up the grades and into adulthood.

Only 22% of 12th-grade students are proficient in science, 25% in math, and 37% in reading (NAEP, 2017). Also, the most recent ACT Math scores are at a 20-year low. And, according to an ETS report, U.S. students lack the "necessary skills across the domains of literacy, numeracy and problem-solving." 

Why? 

I have an answer, but you won't like it: It's the teaching. 

Many teachers are handed a substandard curriculum and use inferior instructional methods that are advocated by liberal professors in ed schools. The curriculum in math is not benchmarked to international standards. Unfortunately, teachers often implement minimal guidance methods of instruction (e.g., group work) that are inefficient.

Fundamentals Are Not Taught for Mastery!
The fundamentals, starting with 1st-grade arithmetic, are not taught for mastery, which involves memorization and drill-to-develop-skill. Students are taught reform math, not standard arithmetic. Also, the minimal-guidance teaching methods (group work) are inefficient compared to explicit teaching. Moreover, the math curriculum is not world class. For example, U.S. 1st-grade students learn substantially less than their counterparts in Singapore. Furthermore, teaching to the test items is a lousy method to teach fundamentals. 

Standard arithmetic is straightforward and simple, but the liberal reformers have complicated it and made it complex, difficult, and confusing for students. To make matters worse, there are all the fads, regulations, trends, policies, and extras that are pushed into the classroom by progressive reformers. 

Kids are novices, not little mathematicians. Novices need to memorize stuff. 

ETS Report states that Many Millennials Lack Literacy, Numeracy, and Problem-Solving Skills!
1. Literacy: U.S. millennials scored lower than 15 of the 22 OECD participating countries (OECD is the Organization for Economic Co-operation and Development).
2. Numeracy: U.S millennials ranked last with Italy and Spain.
3. PS-TRE (Problem Solving in technology-rich environments): 
U.S. millennials ranked last with Slovak Republic, Ireland, and Poland. 

The ACT Math Scores have sunk to a 20-year low! Kids don't have strong math skills that are essential for STEM jobs to power the economy. In the classroom, the mastery of fundamental content through memorization and practice-practice-practice has not been the goal. Instead of traditional (i.e., standard) arithmetic, students are taught reform math.

US education has been following the wrong path for decades.  
Test-based reforms, NCLB (now ESSA), Common Core, state standards based on Common Core, state tests, fallacious "fairness" policies, and other progressive fads and trends--such as credit recovery, inclusion, group work, more money, more tech, and so on--have been false starts. 

Kids stumble over simple arithmetic. 
Teaching reform math rather than traditional or standard arithmetic in the early grades has flunked. The national and international math tests show that our students are behind the best-performing nations in math by a wide margin. Furthermore, the high rates of remedial math (up to 88% of incoming students at community colleges) are another red flag.

In my opinion, the progressive-left or liberal narrative has damaged public education with radical ideas such as credit-recovery schemes and "fallacious fairness policies" as Sowell has pointed out. In mathematics, for example, the mastery of content knowledge and skills is not stressed in progressive (liberal-left) classrooms, starting with 1st-grade arithmetic. 

Grade Inflation Kills
Children do not live in Lake Wobegon where all are above average. They are unequal. Regrettably, merit and achievement do not count for much in many schools today. Still, parents think their schools are doing great when they are not. In math, for example, teachers often give As and Bs for substandard performance. Grade inflation is entrenched in school policies. It has been a pretense for equality, inclusivity, diversity, and social justice. 

In the real world, getting a passing grade on the state test should not imply that the student has mastered grade-level content and skills in math. Many parents do not realize how poorly their child is doing in math because the math curriculum is not world-class beginning in the 1st grade. Also, teachers inflate grades and pass students to the next grade level. A student can get As in class and fail the state test in math. It's crazy! Students are passed to 4th grade without mastering the times table in 3rd grade, and so on. Indeed, by the 4th or 5th-grade, math students are at least two years behind their peers from other nations. One cannot make the unequal, equal. Note. Parents want their child with grade-level kids, not in a class fitting their instructional needs, so teachers pass them on. 

Mathematics Professor H. Wu, UC-Berkeley, discussed the problems of implementing Common Core state standards. He wrote parts of it (e.g., fractions). Wu acknowledged that Common Core would likely fail because K-8 teachers do not know enough math content to teach it well. No amount of professional development will change this, he said.

Mark Pulliam writes, "Diversity means singling out certain races for special treatment." It's called equity, but it is hardly fair. According to the progressive-left narrative, opinion about anything is as valid as the next. It's nonsense.

Education, today, is not about learning and applying content; it is about diversity, inclusivity, equality, and social justice. You are on the wrong side if you don't buy into the progressive-left narrative that equality in everything is everything. Really?

Individuals vary widely. 
We are unequal. 
You can't legislate equity via social justice. You can't make the unequal, equal. Merit and achievement are not considered in the progressive narrative. Everyone is the same. But, we are clearly not. 

Against the status quo of the progressive-left, kids who excel in math achievement should not be placed with low achieving students for math class. They should be pulled out and taught by an algebra teacher for math class. On the other end of the spectrum, Singapore pulls out students with weak numeracy skills at the start of 1st grade to catch them up.   

Many Millennials Lack Literacy, Numeracy, and Problem-Solving Skills!
"The rationale for the focus on millennials is simple: This generation of American workers and citizens will largely determine the shape of the American economic and social landscape of the future." The deficient skills of U.S. Millenials should alarm us, but it has not. (Quote: Education Testing Service (ETS) report America's Skills Challenge: Millennials and the Future 2015)

"The youngest segment of the U.S millennial cohort (16- to 24-year-olds), who could be in the labor force for the next 50 years, ranked last in numeracy and among the bottom countries in PS-TRE. In literacy, U.S. millennials scored low, too. Note. Data is from the  Education Testing Service (ETS) report America's Skills Challenge: Millennials and the Future, 2015)

The ETS report states that even our best millennials don't measure up to their peers from other nations. It does not surprise me. Starting in the first grade,  U.S. students are not learning basic mathematics well enough to compete in a world driven by technology engineering, and science. The millennials have been in progressive schools that stressed understanding and problem-solving. What's wrong? The "teaching" was wrong! The mastery of domain knowledge and skills in long-term memory was not stressed. 

Today, students lack the knowledge to do critical thinking (aka problem-solving), not only in math but also in science. Knowledge in long-term memory enables problem-solving, which is domain-specific. 

Only 22% of 12th-grade students are proficient in science, 25% in math, and 37% in reading (NAEP, 2017). Not good!

In short, U.S. students lack the necessary "skills across the domains of literacy, numeracy and problem-solving." 

Also, "The scores of U.S. millennials whose highest level of educational attainment was either less than high school or high school are lower than those of their counterparts in almost every other participating country."  It does not surprise me because government schools have used credit recovery schemes, grade inflation, and watered-down courses to increase their graduation rates.

First-Draft Form. Expect Changes. 
10-27-18, 11-6-18 


©2018 LT/ThinkAlgebra

Sowell - Skewed Distributions

The causes of skewed distributions of outcomes are many.

"In recent times, virtually any disparity of outcomes is almost automatically blamed on discrimination, despite the incredible range of other reasons." 

Note: Thomas Sowell ​taught economics at Cornell, UCLA, Amherst and other academic institutions. Currently, Sowell is a scholar in residence as a Senior Fellow at the Hoover Institution, Stanford University. I first noticed him when I read Dismantling America in 2010. Also, he wrote Wealth, Poverty and Politics 2016. His most recent book is Discrimination and Disparities 2018. Gerald P. O'Driscoll, Jr., Senior Fellow, Cato Institute, writes, “In this provocative book, Thomas Sowell turns the table on those who automatically link disparate outcomes to discrimination. He begins by focusing instead on the myriad of factors that need to come together for success." Read the book.   ​

Thomas Sowell (Discrimination and Disparities) says that many policy fixes in education have been counterproductive, such as immediately blaming gaps in educational outcomes on discrimination. Too often, skewed distributions of educational outcomes were wrongly blamed on "genes or discrimination," observed Sowell. The playing field has never been equal and cannot be made level by government "equity crusades," which are "fallacies of fairness." In short, the one-size-fits-all equity fix of the Common Core and other efforts will not erase the skewed distributions. Sowell writes, "The idea that it would be a level playing field if it were not for either genes or discrimination, is a precondition​ in defiance of both logic and facts." ​

What irks me as a teacher is that divergence in education is now called discrimination, and that achievement is now called privilege. It is typical liberal rhetoric that is offensive to all hard-working students who achieve, including students of color. Sowell points out, regrettably, that "in recent times, virtually any disparity of outcomes is almost automatically blamed on discrimination, despite the incredible range of other reasons." (Sowell calls it the invincible fallacy. I call it  a sham when scholarship and achievement are criticized with empty rhetoric.)

The social vision and rhetoric of progressives "treat beliefs as sacred dogmas beyond the reach of evidence or logic." Facts don't matter much! "Facts seem to have become irrelevant, for all too many people, who rely instead on visions and rhetoric."​ 

Also, the idea that inputs do not influence outputs is another issue. ​Of course, they do! Sowell explains, "To admit that inputs affect outputs, whether in education in the economy or other areas, would be to undermine the [social] vision and agenda" [of progressive ideology]. According to Sowell, the invincible fallacy is that "outcomes in human endeavors would be equal, or at least comparable or random, if there were no biased interventions, on the one hand, nor genetic deficiencies, on the other." The invincible fallacy assumes "an even or random distribution of outcomes in the absence of such complicating causes as genes or discrimination."​ The assumption, of course, is wrong! 

The causes of skewed distributions are many. Indeed, life, itself, is unfair, yet "fallacies of fairness" abound in our classrooms. All students get the same instruction under Common Core and state standards regardless of their math achievement, which should drive outcomes to be the same. Not Really! It is wishful thinking. Outcomes would not be the same. The crux is that even if the inputs were the same, which I doubt would be possible, the outputs would still not be the same! As Sowell says, "Equal inputs does not produce equal outputs."

Incidentally, one of the many consequences of the invincible fallacy has been the "dumbing down of education."

Comments: ThinkAlgebra@cox.net

©2018 LT/ThinkAlgebra

​

STEM

What does STEM mean?
Science - Technology - Engineering - Math

This post is a work in progress. Expect frequent changes. 
The quotes highlighted yellow with a 🔻 triangle symbol are from Nick Arnold, STEM QUEST.

Model Credit: McKaylaS

Click: Visit my New Science Page!

Only 22% of 12th-grade students are proficient in science (NAEP). The Next Generation science standards limit core content in science, especially in chemistry and physics, which are lumped together. Also, many believe that the only way to learn science is through hands-on, inquiry stuff. That's not true! Most science is learned by reading and studying textbooks, in coursework, and from lectures by knowledgeable teachers. Students do not need to reinvent the wheel, not in science or math. Lab experiments are done to reinforce the content taught.  

Below, MIT Professor Catherine Drennan lectures chemistry to over 300 freshmen students packed into a huge lecture hall. Students had textbook reading assignments to study, problem sets to complete, help sessions to attend, labs to do, and exams to pass. MIT Course: Principles of Chemical Science (5-111sc).


In contrast to the Next Gen Framework, "It is never too early to introduce small children to big ideas," observes Chris Ferrie, a physicist, and mathematician, who wrote Quantum Physics for Babies and similar picture books (Baby University).

In teaching complex material in math or science to little kids, I think Jerome Bruner, not Piaget. LT



We should teach the mathematics needed to do and understand the science such as was done in Science--A Process Approach (1967). For example, four of the six processes taught in SAPA's 1st grade were math or math-related. The science teacher taught math, which was often a year or two ahead of the regular curriculum. Today, there seems to be math anxiety in STEM. University of Chicago researchers Sian L. Beilock and Erin A. Maloney contend, "Our students cannot enter into STEM majors if they have a fear of mathematics." In science, teachers should teach STEM Math Skills. K-6 teachers don't have a coherent science curriculum such as SAPA, which stressed the math link in science. Furthermore, elementary school teachers have weak math and science skills. 


🔻"In science, you investigate the world around you." 
Science starts with atoms, atomic structure, and how the universe works, including living things, the Earth, our solar system, the galaxies, and stars. "A is for Atom, B is for Black Hole, C is for Charge," writes Chris Ferrie (ABCs of Physics) for young children. 

"The rules that describe nature seem to be mathematical. Why nature is mathematical is again a mystery. Science is a method of finding things out. The method is based on the principle that observation is the judge of whether something is so or not." Also, according to Richard P. Feynman, basically, scientists guess an idea or rule, then test it through experiment and interpret the observations. "If the rule does not agree with experiment, then the rule is wrong." We don't start with a bunch of observations, then infer a rule. "It does not work that way." In science, we guess a rule, then experiment and examine observations (mostly measurements). The experiment and its results must be repeatable by others. The mathematics must be right. An idea you can't test is speculation, not science. "The purpose of theory is to describe nature, not to explain nature." (Source: Richard P. Feynman, The Meaning of It All)

Note: Technology and Engineering are not science; they are products of science. Mathematics is a tool used by scientists to describe nature.

Contrary to the NextGen science standards, elementary school students should study atoms, which are the building blocks of matter, and important molecules like DNA, along with atomic theory or structure (electrons, protons, and neutrons), quantum physics, chemical reactions, motion (Newton's Laws), electromagnetism, the roles of famous scientists, and other topics. 

In classical mechanics (classical physics), at certain distances, opposite charges are attracted and like charges are repelled. It is simple to show this to 1st and 2nd-grade students with N-S magnets. But, when students learn about atoms, classical ideas don't work.

The negatively charged electron should smash into the very tiny, positively charged nucleus according to classical mechanics, but it doesn't. The electron (-) and nucleus (+) stay apart. Why? Also, positively charged protons in the nucleus of atoms stay put. They don't fly apart. Why? 

The classical mechanics of Coulomb's Force Law and Newton's 2nd Law do not work at tiny, atomic distances. A new way of thinking was needed: quantum mechanics (i.e., Quantum Physics). In fact, Chris Ferrie wrote a picture book called Quantum Physics for Babies. 


Atoms and molecules exist, move, and bounce off each other. 

(Einstein, 1905)

Chris Ferrie writes, "All electrons have energy. An electron can take energy to jump up an energy level and give up energy to fall down an energy level. A quantum is the smallest unit of energy." There is much more to this, of course, but it is a start.  

🔻"In technology, you develop products and gadgets to improve our world." 
The cell phone is a good example of technology.


🔻"In engineering, you solve problems to create extraordinary structures and machines." Engineering often starts with simple machines. In short, you need to learn some physics. Yet, the simple machines are absent from NextGen science standards. Students can be introduced to engineering by building a bridge or a robot. They can build circuits and design and build a structure with Lego. 

Students can use gumdrops and toothpicks to design and build a stable structure such as a bridge. Shapes are important. 

The Six Simple Machines
Lever
Wheel and axle
Pulley
Inclined plane
Wedge
Screw

🔻"In math, you explore numbers, measurements, and shapes." 
Beginning in 1st grade, it is essential that students get used to symbols that represent numbers and unknowns. Math starts with numbers and equations and how they work. For example, x + x + x = 10 - 4 is a 1st-grade level equation. The symbol x must be 2, which is the only solution to the equation: 2 + 2 + 2 is 6. Students must follow the algebraic rule for substituting. The right side (10 - 4) is 6; therefore, the left side must equal 6. Thus, 6 = 6. 

Math is a major tool of science. An equation can be represented by symbols, pictures (i.e., graphs) or data tables.

Students should study math, especially equations and equation solving. But, to move forward to algebra, they need to master the basics of arithmetic such as whole numbers, integers, fractions, percentages, and ratios.


Learning how numbers work, calculating with numbers, and solving equations are important mathematics in early elementary school. 

In the first lesson for the 4th-grade classes (8-8-18), I introduced a system of three equations, which requires logical reasoning, knowledge of algebraic substitution, and some basic arithmetic. I meet with kids once a week for an hour in my Teach Kids Algebra program.  

z + x = 13
y + y = 14
z + y = 12
x =
y =
z =

==============
A quadratic equation defines an up-side-down parabola at the water fountain.



©2018 LT/ThinkAlgebra

Math Teaching 2

Running in Place & Fallacy of Fairness
The abstract goes to Concrete
The How always preceded The Why

No matter how hard Alice ran, she stayed in the same place, so has U.S. math achievement. "Faster! Faster! Now here you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!?" (The Queen to Alice, Through the Looking-Glass) 

Running in Place
Math Achievement Stagnation

For many years, students seem to be running in place in math achievement. It's stagnation. In short, our kids are not getting better at math. They are staying in the same place. Kids are not learning much because the fundamentals are not taught for mastery. Indeed, most teachers do not want to admit that they teach math poorly. Teachers get good evaluations when they teach math the Common Core reform math way. It's group work and test prep. Teachers focus on test-based proficiency not the mastery of content. Even if a student scores at the proficient level on the state test, it should not imply that the student has mastered grade-level content.

The constructivist approach of reform math hasn't worked. Scores on national and international tests have been low. Students are not required to practice arithmetic basics for mastery. 

The only way to judge whether or not the instruction is working is through performance testing. Can the student do standard arithmetic efficiently?  Zig Engelmann writes, "How successful is your version of the constructivist approach?"  

Petrilli: Lost Decade of Educational Progress: 2007-2017

After all the reforms and billions and billions ($) spent over the years, our 4th-grade and 8th-grade students are no better in mathematics than they were 10 years ago. Not only are the math scores below expectations; they are also flat. 

Michael J. Petrilli says the NAEP scores in math from 2007 to 2017 indicate a "lost decade of educational progress." Also, Bill Gates spent "hundreds of millions to improve teaching," but the initiative "failed to improve student achievement." 

What is wrong with education? More money, various reforms, so-called innovative programs, and professional development haven't worked. We have been unable to improve the "teaching" in the classroom, that is, the teaching that substantially boosts real student achievement. 

In schools of education, teachers are taught inferior (minimal guidance) instructional methods, unreliable theory, flawed equity and diversity guidelines, and a reform math curriculum that is not world-class. Moreover, K-8 teachers are weak in math and science content because they majored in education, which is not an academic subject. How can they teach subjects they don't know well? I am convinced that some elementary school teachers, today, do not know how to teach math or reading effectively. 

The progressive scheme dumbs down math for equity. High achievers and low achievers are mixed in the same classroom or in the same math class (inclusion for fairness). Furthermore, all students get the same instruction (e.g., Common Core or its state rebrand) regardless of achievement level or ability (equalizing for fairness). Indeed, "equalizing downward by lowering those at the top," says Thomas Sowell, is "a crazy idea," It is a "fallacy of fairness" that hurts all children, especially children of color. In American schools, it has been taboo to sort kids by achievement for math. In contrast, Singapore sorts kids in math in the 1st. grade. All kids should be given opportunities in schooling but not always the same opportunities or the same instruction. Kids who are advanced in arithmetic, for example, need an entirely different math curriculum from other kids. Instead of advanced math, they are given grade-level material. It is called equity, but it is false equity. 

Also, the priority should be given to learning the compact, standard algorithms first, but this is not the case in the reform math as taught in our schools. Starting in 1st grade, we don't stress the mastery of fundamentals. Memorization and drill-to-develop-skill have fallen out of favor in modern classrooms. Group work, discovery learning, and other fads are much more important than automating the fundamentals of arithmetic, progressives (aka liberals) say. They are wrong!

Help

We fell down the Rabbit Hole decades ago! 
Let's say we needed to calculate 6 x 1584. In Mock Turtle's 3rd-grade "Uglification" class, students were taught a wide range of alternative methods or strategies to multiply: lattice, repeated addition, array, area, partial products, make a drawing, calculator, distributive. (Which method should I use?) But, learning all these alternative methods clutters the curriculum and the minds of children (cognitive load). Who multiplies using the area model? The standard algorithm has been pushed aside. Some kids never learn it. Progressive education strikes again. We spend a lot of money, time, and effort on progressive ideas and theories that don't work well. Why make arithmetic harder than it is? 

The Real World: To move forward, students must know the compact, standard algorithms, not complicated alternatives.


Part I Abstraction
Why isn't math taught this way?
We think concrete goes to the abstract because that's what everyone says. But, what if the "experts" were wrong? The idea is that children develop number concepts only from concrete representations, then, later, as abstract ideas (Piaget's theory), but it doesn't work that way. Indeed, 5 is 5 and can be represented in hundreds of different ways such as 5 people, 5 pencils, 5 cubes, 5 dots, etc., but the number 5, itself, is an abstract idea. It is a concept in our mind. You don't need to go from concrete to abstract to understand the meaning of 5. The number 5 is already abstract, which is the starting point. 

The number line helps novices visualize what 5 is. It helps beginners with the idea of magnitude. The number 5 is a point on the number line that is one more than 4 and two less than 7, and so on. 

The meaning of 5 is derived from its connections with other numbers. 

What about 3 + 4? You don't need to show 3 + 4 is 7 in several concrete ways, or make a drawing, or write an explanation. If need be, the number line can verify that 3 + 4 is 7, which is obvious. But the number line is not a proof or a why. Also, showing different ways to represent 3 + 4 is not a verification of "deep understanding" or why 3 and 4 add to 7.

The number line is essential arithmetic, but it is seldom used in the primary grades, which is a huge mistake as children try to grasp magnitude and how numbers behave. 

The Number Line Is Important Arithmetic

Numbers are invented in our minds. They are abstract. Children start with the whole numbers as concepts and should learn connections between numbers on a number line, such as "add 1" (4 + 1 = 5), or 3 + 4 = 7, or 7 - 3 = 4, or 3 x 4 as the sum of three fours: 4 + 4 + 4 = 12. 

So, why do we ask novices to explain the obvious several different ways? We are told that it shows understanding. Really? I don't buy it. If needed, the number line verifies it. That said, 3 + 4 = 7 should be memorized early in the 1st grade, but, unfortunately, the memorization of number facts has fallen out of favor in progressive classrooms. 

In short, arithmetic is invented in our minds. In every word problem, no matter how simple or complicated, the student needs to pull out the numbers and figure out what to do with them (execute an operation) to find an answer. You see, arithmetic is about numbers that are abstract ideas. It's not about 5 pencils. It's about the 5 and how it relates to other numbers. We use straightforward arithmetic to solve word problems. 

If arithmetic is abstract, then why isn't it taught that way?

Indeed, the equation 3 + 4 = 7 is the solution to hundreds of word problems. In 2nd grade, I would write an equation on the board and ask students to make up a story (word) problem. Some students surprised me by turning 3 + 4 = 7 into a missing addend problem (a subtraction problem) such as, "I have 7 apples and give 3 of them to Bill. How many apples do I have left? Or, 7 - 3 = 4. The problem isn't about apples; it is about the numbers and how they are related. The basic relationship between addition and subtraction is abstract.

Part II
The HOW always preceded the WHY.
Why isn't math taught this way?
Tobias Dantzig (Number: The Language of Science), a book highly praised by Albert Einstein, writes, "In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy. This is particularly true of arithmetic. The strength of arithmetic lies in its absolute generality such as a + b = b + a. Its rules admit of no exceptions: they apply to all numbers. Every number has a successor [add one]. There is an infinity of numbers." Teaching novices arithmetic should be the "how," not the "why." Kids are novices, not little mathematicians. 

Ian Stewart (Letters to a Young Mathematician) clarifies, "One of the most significant differences between school math and university math is proof. At school we learn how to solve equations or find areas of a triangle; at university, we learn why those methods work and prove that they do." We need to stop teaching kids as if they are little mathematicians, which they are not. 

American teachers are hung up on the "why" of everything, but in arithmetic, it is proper to learn the "how" first and the "why" later, which is what happens in Asian nations. Learn the technique first and get it right (i,e., automate it). Learn the "how" first and don't worry about why it works.

Note: Standard arithmetic is simple and compact, but it is not easy to learn without the memorization of single-digit number facts and practice-practice-practice. 

National Mathematics Advisory Panel (2008):
"Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division." I do not recall teaching arithmetic without some level of student understanding. Understanding develops slowly over the years. Also, automating the mechanics of arithmetic for fast access frees working memory space for problem-solving. Arithmetic is a tool for solving problems.  

First-Grade Subtraction: 
Go Vertical! Let's Borrow! Missing Addend Equations!
The standard algorithm of subtraction is a key math skill that can be taught in the 1st grade. The vertical format is a place value system. Start with problems that don't involve borrowing, such as 47 - 25, but use the vertical format to stress place value.

Teach borrowing. Borrowing shifts a ten back into the ones place, that is, one ten equals 10 ones. The 16 can mean 16 ones. It is simple place value and reverse engineering. Students can compose missing addend equations in their heads and recall memorized addition facts to solve them. What number plus 9 is 16? {7 + 9 = 16} Or, x + 9 = 16 . It is a missing addend equation. Children can use memorized addition facts to do subtraction: 
addend1 + addend2 = sum. It is important to relate subtraction to addition.

Like it or not, Arithmetic is based on Rules
I often hear teachers say that they don't want kids memorizing a bunch of rules. But, arithmetic and algebra are governed by rules, and students must know the rules and be able to apply them to do correct mathematics. For example, the commutative rule should be learned in the first full week of 1st grade: 2 + 3 and 3 + 2 give the same answer. Arithmetic is based on rules.  There are not that many, but they should be learned as early as possible, such as the "zero rule" for addition: 3 + 0 = 3. The properties of numbers that children learn are rules. They are also mini procedures.

Learn the procedure first (i.e., the "how," such the procedure for solving 1/2 of 1/2, which is a multiplication of fractions 1/2 x 1/2 = 1/4.)

The stress on understanding "why" something works is misplaced. For example, Isaac Newton invented a calculating method (calculus). He knew it worked because the results of his calculus agreed with the experimental results, but he didn't know why his calculating method worked. The "why" wouldn't come for another 200 years with "limits." But, that didn't dissuade Newton from using his calculating method to figure out physics. It always worked. He worked out the how, but not the why. Still, he was a brilliant mathematician and physicist. 

So, when we teach kids (novices) an operation, such as addition in 1st grade, we should first make sure they can do the procedure well (the "how"), which requires practice-practice-practice. Focus on the how not the why. 

Teachers should teach the procedures (aka algorithms) that are the most efficient and compact and work all the time. These are the standard algorithms. 

Understanding comes out of doing. Why are parents concerned about their children's math education? Kids can't do simple arithmetic. 

Students Learn Math Inductively
Mathematics uses deductive reasoning and logic to create new math. But, students--who are novices, not experts--don't learn math that way. They learn math inductively. That is, the teacher explains several examples and then tells students that the method always works. It is impossible to test every possible case or present a formal proof. The teacher should also explain the counterexample.

Students don't need to draw pictures or count objects to understand math.
enVision Problem Solving
Kara found 5 shells. Then she found 3 more. How many shells did Kara find? Draw a picture. Write the number. 


The problem-solving idea of enVision Math, a popular reform math program, is for the student to draw little pictures then count the objects depicted. The memorizing of critical single-digit number facts is not part of the curriculum in 1st-grade state standards. In my opinion, 5 + 3 = 8 should be memorized in 1st grade. It can be verified on a number line if need be. Moreover, the answer to the question is "Kara found 8 shells," not the number 8.

Teachers teach math for test-based proficiency, not mastery, and that is what is wrong with math instruction today. Proficiency in state math tests should not imply that the student has mastered grade-level math content. Under Common Core, state standards, test-based accountability, and "teach to the test items" mentality, our curriculum has become fragmented, and our instructional methods have been inferior.

Let me repeat. Educators focus on low achievers. It is called "equalizing downward by lowering those at the top," which is a "fallacy of fairness," says, Thomas Sowell. Instead, teachers should focus on equalizing upward to challenge high achievers by tracking them in at least one of their best subjects, such as mathematics, but not all their subjects. It is tracking-by-subject, but there should always be some flexibility in grouping along with way. 

To Be Revised
Updated: 7-7-18, 7-10-18, 7-11-18, 7-13-18, 7-16-18, 7-17-18, 8-31-18
Model Credit: Gabby, Hannah, Shayna
The NAEP chart is from an article by Michael J. Petrilli (Fordham)

©2018 LT/ThinkAlgebra.org