Against Reforms

There is no substitute for knowledge in long-term memory 

and the practice that gets it there.

"Drill to develop skill" is essential.

My Contrarian Math Page is a response to far-reaching, progressive reforms that for decades have twisted and trashed standard arithmetic into "something" I call "reform math." The reformers say that "drill and practice are always wrong. Real teaching is always inquiry-based, student-centered, and constructivist." These claims and other progressive notions are bunk! 

​[ Note: There are many beliefs in education that lack scientific evidence. Belief is not evidence. Anecdotal claims are not reliable because they are based on "personal accounts rather than facts or proper research." ] 

Consequently, students do not master standard arithmetic and the standard algorithms to perform arithmetic. Moreover, the reformers have branded "Old School" ideas such as "memorization" and "drill to develop skill" as obsolete and bad teaching. Countering the reformists' claims, the Old School ideas worked well for most students. They are essential, not obsolete. Also, progressive reformers refuse to sort kids. It makes no sense to place high achieving students in math with low achieving students in the same math class via so-called inclusion or fairness policies. Thomas Sowell points out that "equalizing downward by lowering those at the top [is] a crazy idea taught in schools of education across the country." The high performing students need a different curriculum taught by an algebra teacher starting no later than the 2nd grade. Differentiated instruction within a classroom has never worked well. Chester Finn, Jr. and Brandon Wright write in EducationNext, "Rare is the teacher who can do right by her ablest pupils at the same time she provides slower learners in her classroom the attention that they need." 
Click: My Contrarian Math Page

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My Contrarian Math Page is a response to progressive pedagogy and its illogical reforms. Charles Payne, University of Chicago (So Much Reform, So Little Change, 2008), points out the Holy Postulates for progressives. Here is one: "The Only Pedagogy is Progressive Pedagogy, and Thou Shalt Have NO Other Pedagogy Before it. Drill and practice are always wrong. Real teaching is always inquiry-based, student-centered, and constructivist." The progressive assumptions are bunk.  (Quote Source: Larry Cuban's blog)

Progressive pedagogy is an ideology, not a science of learning content. Evidence doesn't matter to progressives who trash the standard algorithms and put calculators in the hands of K-12 students for arithmetic and algebra. Memorization, imitation, repetition, review, and "drill to develop skill" are often downgraded or disparaged in progressive pedagogy or should I say ideology.

Examples of progressive pedagogy are reform math and state-mandated standards, which are primarily Common Core. A popular reform math curriculum is Everyday Mathematics. Group work and minimal guidance during instruction such as discovery/inquiry activities are characteristics of reform math. In 2015, U.S. 4th grade students outscored Finland 539 to 535 in the math content section of TIMSS, an international test, but the East Asian nations dominated with Singapore at 618, Hong Kong at 615, and S. Korea 608. How has the resurgence of reform math worked out? The U.S. TIMSS math scores for 2013 were better than the 2015 scores. Furthermore, by the 8th grade, 54% of Singapore students reached the Advanced TIMSS Level compared to 10% of U.S. students, which indicates that we are not teaching math at a world-class level starting in the 1st grade. The teaching of items on a test is a fragmented curriculum and not the same as standard arithmetic and algebra.
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Other illogical reforms include an obsession with technology as a panacea, the use of calculators in K-12 mathematics, the intense concentration on critical thinking without content, and the minimal guidance "constructivist" methods during instruction. Moreover, the popular Piagetian notion that kids learn best (naturally) through a child-centered discovery/inquiry approach without a formal curriculum is bogus and violates the basic tenets of the science of learning content. To learn something is to remember it from long-term memory, which requires memorization, imitation, repetition, review, and hard work. Kids need to "drill to develop skill" to learn arithmetic well, that is, they need to practice-practice-practice.

There is no substitute for knowledge in long-term memory and the practice that gets it there.

At the Brookings' Brown Center Chalkboard blog, I noticed that all the topics were about issues, such as teacher diversity, personalized learning, integrating schools, technology, teacher pay, graduate degrees, equal pay, but nothing about teaching, itself, which is what teachers are supposed to do. 


One reason that most kids grossly underachieve is that educators do not teach the basics of arithmetic for mastery. Yes, it often is that simple. (There are other reasons, too!) The Reform Math frame-of-mind marginalizes the standard algorithms. The standard algorithms are not taught first if they are taught at all. They are not the top priority in reform math, but they should be! Progressive educators trash the standard algorithms saying kids don't need to learn the multiplication table or the mechanics of long-division and fractions because they can use calculators.

More is said than done. It is especially true in education. We say we want students to engage in "higher-level" thinking, yet we don't focus on lower-level thinking (i.e., knowing and applying content) that leads to higher-level thinking. Put simply: our actions do not support our goals. We say one thing, then do another. We say x causes y based on scant or anecdotal evidence when there is no cause-effect. Hence, education is loaded with false claims, junk science, and so-called "exemplary" reform math programs that do not work well. Moreover, kids are seated in small groups facing each other. Consequently, they are easily distracted and off task. Much instructional time is wasted. 

Immanuel Kant wrote that thought (e.g., critical thinking, problem-solving, analysis, etc.) without content is empty. To learn something means remembering it from long-term memory such as the single-digit number facts and standard algorithms in arithmetic. Learning requires effort, memorization, drill to develop skill (practice-practice-practice), and review. Unfortunately, the focus has been on reform-math alternatives rather than the standard fundamentals of arithmetic that prepare students for Algebra-1. 

By 8th grade, American students are at least two years behind their peers from some other nations; only 33% of them are proficient in math (NAEP 2015: The Nation's Report Card).
  
Moreover, teachers are often required to teach "items on the test" and use inferior methods of instruction; consequently, many students never master arithmetic. In effect, the math curriculum is fragmented and below world-class levels. The disparity begins in the 1st grade. 

In Everyday Mathematics (EM), which has been a popular reform math program, "The addition algorithm is probably the best of the U.S. traditional computation algorithms," but "Everyday Mathematics does not focus on it." Why not? Also, Everyday Mathematics shows five algorithms for whole-number subtraction: "trade-first, counting up European, left-to-right, and partial-differences," but not the standard algorithm for whole number subtraction. 

According to Everyday Mathematics, "Learning a single traditional algorithm for each operation, especially at an early stage, may inhibit the development of children's mathematical understanding." Wrong!!! 

As Carl Sagan once said, "Extraordinary claims require extraordinary evidence" And, the supporting evidence just isn't there. The reform math claim is absurd!  (FYI: Today, reform math ideas are prevalent in most K-8 classrooms.) Unfortunately, EM recommends four-function calculators for the early grades (K-3) and a scientific calculator starting in 4th grade. A keystroke sequence on a calculator is not the same as learning basic arithmetic in long-term memory. 


If Reform Math had worked well, then our students would be at the top on international tests such as TIMSS and PISA, but they are not.

Last update: 10-12-17

©2017 LT/ThinkAlgebra

Cognitive Load Theory

Not knowing the single-digit number facts for instant recall or the standard algorithms may create a cognitive load in working memory that interferes with solving problems and learning. John Sweller's cognitive load theory (CLT) is one of the most important ideas in the science of learning, yet most educators have never heard of it. You don't want your working memory's limited space busy with stuff that should have been automated in long-term memory such as the single-digit number facts and standard algorithms. 

First-Grade students should not figure out facts as needed by finger counting. It eats up valuable working memory space and is a bad habit. Instead, they should memorize single-digit number facts to avoid cognitive load when solving problems. Ton de Jong writes, "The basic idea of cognitive load theory is that cognitive capacity in working memory is limited so that if a learning task requires too much capacity, learning will be hampered. The recommended remedy is to design instructional systems that optimize the use of a [very limited] working memory capacity and avoid cognitive overload." (Also, any classroom distraction increases the cognitive load in working memory, so it is important that students focus and pay close attention in class which is much more difficult when students sit in small groups facing each other.) 

Students should not finger count. Instead, they should memorize (automate) single-digit number facts and practice the standard algorithms, so they stick in long-term memory. Beginning in the 1st grade, students should drill to develop skill. A 1st-grade student hasn't learned 5 + 7 = 12 if she cannot remember it instantly. Learning is remembering from long-term memory. 

Cognitive load indicates the amount of mental effort needed in working memory. For example, if a student has automated 5 x 7 = 35 in long-term memory, then, as needed, it pops into working memory effortlessly. But, when the student needs to figure out 5 x 7 each time it is needed in working memory, then the mental effort or cognitive load increases and the cognitive capacity shrinks. Learning is impeded. The goal is to get the important stuff (i.e., the fundamentals) into long-term memory, which requires practice-practice-practice. Students should overlearn the basics of arithmetic and algebra. 

Learning the standard algorithms of arithmetic in the primary grades is a key step. The standard algorithms organize and simplify place value and apply single-digit number facts. These fundamentals must be in long-term memory to solve problems and perform arithmetic well.

If a student counts on his fingers to solve 5 + 7 each time he needs it, then it stays in the working memory and is quickly forgotten. The single-digit addition fact does not move to long-term memory without much practice and review.

In other words, the student hasn't learned 5 + 7 = 12 because he can't remember it. Learning is remembering from long-term memory. You don't want kids to work it out each time, which is the essence of constructivism, discovery learning, and other minimal guidance approaches that are inefficient. 

To reduce cognitive load when solving a problem in math, students should have automated single-digit number facts and standard algorithms as early as possible (Grades 1-3). Working Memory space is very limited. 

Human processing power in working memory is limited, so it is important to decrease cognitive load as much as possible when students deal with the demands of solving a math problem. Essential factual and procedural knowledge should be automated in long-term memory "leaving room to attend to the details of the problem."


Math problems should be stated clearly without extraneous information that increases the cognitive load in working memory. Requiring students to show different ways to find a solution increases the cognitive load in working memory. 

Sweller writes, "Providing unnecessary information can be a major reason for instructional failure."

Don't permit 1st-grade students to count on their fingers. It's a bad habit. After the 1st month of school, the number line and charts should be removed, too. Kids should memorize the single-digit addition facts, not calculate them as needed on their fingers or a number line. 

Note: The discovery and problem-solving approaches, which are commonplace, are not a good learning stratagem compared to explicit teaching and worked examples. 

Also, there are ways to move working memory information to long-term memory via repetition, imitation, and practice. Students must drill to develop skill.

In summary, not knowing the single-digit number facts for instant recall or the standard algorithms, for example, may create a cognitive load that often interferes with solving problems and learning.

To Be Revised

Last update: 10-1-17, 10-4-17



© 2017 LT/ThinkAlgebra

Why Learn Arithmetic & Algebra?

Why should students master Arithmetic and Algebra?
So, they don't live paycheck to paycheck and beyond their means.
So, they save and invest to become millionaires when they retire.
So, they are frugal, debt-free, and have a cash flow from investments to build wealth. (Kids need to learn Algebra-2 to get into college.)

Knowing math and having financial savvy can greatly improve your quality of life. Briefly, learning math and science opens the door to opportunities. Isn't this what we want for students? 

Why are you in debt and living paycheck to paycheck? That's stupid!
Arithmetic was deemed important because it was practical, but how many of us perform arithmetic when everything is calculated by machines? Most of us do not balance our checkbooks. We don't have a practical budget to track where the money goes; consequently, 78% of us live paycheck to paycheck and above our means, which is stupid. Most of us don't understand the science of money such as the miracle of compounding, which is exponential growth, not linear change. In short, most of us don't know enough arithmetic or simple ideas of algebra and cannot apply them even with a calculator. The concepts are not difficult. 

The difference is between those who know math and those who don't, between those who save and invest and those who spend and spend beyond their means. Income does not determine wealth. As your income increases so do your expenses. Cash flow from other sources (i.e., investments) determines your wealth. 

"Income is not wealth. Wealth is cash flow from other sources," write Tracy & Strutzel. Spend less than you earn and invest the difference in a good index mutual fund to start. 

Tracy & Strutzel point out that "successful people do not gamble" or buy new cars. They state, "The reason people don't retire financially independent is that they spend everything that they earn." 

The average American does not have cash. Nearly 70% do not have an emergency fund of at least $1000, much less 6 months of expenses should something happen. 


Unfortunately, most of us don't know how percentages work because of poor teaching. If we did, then we would never take out a seven-year car loan, pay the minimum on credit card purchases, or use "payday" or "title" loans, etc. We have been taught that debt is the normal way of life, and it builds a credit score, so we spend everything we make and then some. It is a dumb idea! 

In contrast, financially savvy people don't buy anything on credit or use credit cards unless they have the money in the bank to pay for it.  If you don't have the money in the bank, then don't make the purchase. They don't spend everything they earn. They budget and invest the difference.  

Financially savvy people buy an affordable car with cash. It's not a new car. Financially savvy people know math. For example, they know probability and don't buy lottery tickets. 

Student loans are another problem for young people. I had student loans, but I also had a job with enough income to pay them off.

You can become a millionaire by investing about $100 a month over an extended period. The investment would have to grow by a factor of 10000 over time. It would take 45.67 years of investing at least $100 a month in an index mutual fund that has an average annual return of 9.69%, which was the 50-year annual return in the stock market from 1966 to 2015. Even if your take-home pay is $1,666 a month to start, $100 a month is only .06 or 6% of your monthly take-home pay. You can become a millionaire only if you think that way. 

Also, as your income increases, you should increase the monthly investment to 10% to 20% or more of your monthly take-home pay. You can access the American dream, but not by living paycheck to paycheck and going into debt with credit cards and car loans, etc. If your take-home pay is $2500 a month, then 10% to 20% would be $250 to $500 a month. If you do this (10%) for 36 years, then you will have one million dollars, but you may need much more because nasty inflation kills off buying power over time. The only debt you might carry during this process is a reasonable mortgage, but after 30 years, it should have been paid in full. As your income increases, you should pay off the house mortgage within 15 years by making extra payments. Also, if you can't put at least 20% down, then don't buy the house.

We need to teach kids key financial lessons from 1st grade on up. One is that debt is not normal. If you want something, then save for it and pay cash. It is where arithmetic and algebra come into play. Easy credit with high-interest rates can kill your future. 

Extras

Wobegon does not exist.
Our educators and schools are loaded with wishful, Utopian thinking and illusions of fairness, most of which is absurd. "Fairness as the equal treatment does not produce fairness as equal outcomes," writes Thomas Sowell.  

Many teachers don't know how to teach content because, they, themselves, have not mastered the content. Teachers who advocate flipped classrooms, personalized learning, tech as a panacea, and other schemes and fads don't know how to teach. 

The idea that children don't need to memorize or drill to improve math skills (i.e., practice) shows ignorance of cognitive science and of how children learn, which is remembering from long-term memory.

Many have the idea that new is better, so they must have the latest tech, newest car, etc. Why would you want a car payment, mortgage, or credit card debt the rest of your life? The "new is better" attitude has resulted in more lackluster achievement. Reformers preach "out with the old," even if it worked well, and "in with the new," even if it has no basis in evidence. Reformers seem to ignore the evidence. 

To Be Revised as needed.

©2017 LT/ThinkAlgebra