Learning Hierarchy

Learning Hierarchy (Gagne)

Teach Kids Algebra
Starting in the 1st grade, students learned about expressions, equality, equation structure (expression = expression), and letters like x and y that can represent unknown numbers. The students solved x using guess and check. They started with true/false: 3 + 4 = 10 - 2. The statement is false because the left side is 7, and the right side is 8. In short, 7 ≠ 8, therefore it is false. An equation is like a balance. The left side must equal the right side in value. For the equation x - 3 = 8, x must equal 11 to make a true statement: 8 = 8. Unlike science, math is absolute. It is not an opinion. It consists of true statements made from other true statements. For example, if 3 + 4 = 7, then 7 - 3 = 4.  (Note that the equation 7 - 3 = 4 is true because 4 = 4.) 

Many teachers don't know how to explain math to kids. They don’t know how to write behavioral learning objectives (Mager) or construct a hierarchical-based curriculum (Gagne). Often, elementary school teachers, even middle school teachers, are weak in some necessary math skills such as fractions, long division, and algebra. It's not about understanding, which is difficult to quantify; it's about knowledge in long-term memory and applying it. 

When I first started giving algebra lessons to 1st through 3rd grades, I made a topic list. Then, I developed specific behavioral objectives (i.e., performance-based) (Mager) and sample problems (i.e., worked examples) to explain the performance. I often asked the students questions and gave them a couple of problems to work on their own (guided practice). Finally, I handed out the lesson's practice sheet (independent practice), which included both current and problems from previous lessons (review). For the remaining 30 minutes, I walk around the room to give encouragement and corrective feedback to individual students.  I called my algebra program Teach Kids Algebra or TKA. Sessions were weekly for an hour. In short, I constructed my own curriculum.

In the 2018-2019 school year, I gave TKA lessons to two 4th grade classes. No group work. No Common Core. No manipulatives. No calculators. The only crutch was an integer number line. In the Spring, I gave algebra lessons to a class of 2nd-grade students. They received 6 hours of instruction. The 7th lesson was a culminating activity. 

Gagne’s idea of curriculum development was hierarchical and indicated specific prerequisites and background knowledge. See the example below.

Break a problem into smaller problems. 

It is a fundamental idea taught in mathematics, and it can carry over to everyday life. The idea that new knowledge builds on old knowledge is central to learning math. Because math builds in long-term memory, the proper sequencing that creates coherence (a learning hierarchy) in a math curriculum is paramount. Below is a sequencing example from Science--A Process Approach (SAPA), which had used Gagne’s hierarchical approach. 

Part C is 2nd grade in the K-6 SAPA science sequence, but I used it in 1st-grade TKA. Incidentally, 4 of the 6 processes taught in the 1st-grade SAPA lessons (Part B) were arithmetic or math-related: using numbers (arithmetic), communicating (graphing), measuring (metric units: g, cm, m, mL), and using space/time relationships (geometry). 

There are multiple problems in math education today. One is the lack of a coherent learning hierarchy (Gagne). The same is true for science. 

Not only did SAPA upgrade science education, but it also shifted higher level math content down to lower grades. SAPA taught the math kids needed to know to do the science. In other words, the math in SAPA was much more advanced than K-6 students traditionally learned. In first-grade SAPA materials, 4 of the 6 processes were math-related. There has been nothing like it since the 60s.  

In math, experienced teachers know what’s essential and what’s not. They can figure out prerequisites and develop curriculum. Most teachers can’t do this. They can't teach what they don't know well.

When I left K-8 classroom teaching (2000), I started tutoring high school mathematics, especially Algebra-2 and precalculus. Tutoring led me to this idea: If you can’t calculate it, then you don’t know it. 

Teachers must use effective instructional methods, such as explicit teaching, memorization, practice (drill), and continual review so that fundamentals stick in long-term memory, not minimal guidance methods that are ineffective and lack scientific support.

Minimal Guidance = Minimal Learning
(Kirschner, Sweller, and Clark, 2006) 

Lastly, if learning is remembering from long-term memory, then we have not been teaching children to learn and master essential content. Much is taught, but little is learned. 

Also, we should not limit students to so-called grade-level content.

Reading Real Books
I think that much learning can be attained by reading books--history books, science books, math books, geography books, literature books, art books, and so on. 

Some of the responsibility for learning should be placed on the shoulders of students, too, not just the classroom educator. Reading books outside the classroom was significant in my learning. It still is. 

One GATE student expressed to me, "I hate reading screens or one to two-page handouts; I want to read real books, with physical pages I can feel and turn." Where are the books? Students don't have subject matter books in digital classrooms--not real books. Also, it's hard to focus (pay attention) because the GATE classroom is so noisy.

©2019 - 2020 LT/ThinkAlgebra

PISA-2018

American students did poorly in math (PISA, 2018).

Test scores will not change much when we keep administrating the same curriculum--which is not world-class--and applying the same minimal guidance (inquiry/discovery) methods (i.e., via group work), which are ineffective. Then, there is test prep that also limits content. 

American parents, teachers, and students do not take learning arithmetic and algebra seriously, unlike Asian parents and schools.  Moreover, American educationists tend to give excuses for poor performance. The fact is that our students are not mastering the fundamentals of arithmetic and algebra. (Note. The test results for PISA 2018 were released in December of 2019.)

You won't find this South Korean motto in American Schools:
"Study hard enough to become Smart enough!" 

"Students should come to school to learn, not text," write Friedman and Mandelbaum (That Used to Be Us), but today's students don't want to study to master the content; they want to use social media and gadgets (poor attitude toward school). Moreover, the progressive reformers insist that students should dive right into critical thinking or problem-solving before the basics are learned, which, in my opinion, is an inane strategy. It's backward and a fundamental reason that students stumble over simple arithmetic. Math is no longer taught for mastery, and explicit teaching was swapped for minimal guidance teaching; teachers have become facilitators rather than academic leaders in the classroom. Also, memorization and practice (e.g., drills) have fallen out of favor in progressive classrooms. What could have gone wrong? There is a large gap between what K-12 schools say they are teaching and what students are actually learning. 

The best K-12 teachers have always provided students with increased layers of difficulty in math (i.e., deep practice⁶) to stretch their knowledge and extend their thinking. Moreover, they teach math hierarchically (properly sequenced), linking new ideas and skills to knowledge already learned in long-term memory. The problem is that there are very few of these teachers left. They are either leaving the teaching profession for one reason or another or retiring. 

6 The idea of "deep practice" is from The Talent Code by Daniel Coyle, who states that repetition (to automaticity) is the key to learning.

The NAEP, PISA, and TIMSS tests show that essential math content is not taught well or taught for mastery. The way we teach math often contradicts the science of learning and can block a child's future. The U.S. spends a lot of money on education but gets mediocre to poor results. 

In-depth content is missing in many U.S. schools. 

Contrarily, the focus of the BASIS schools is in-depth content. 

We keep saying that our kids are doing okay in math when they are not. Their reform math curriculum is not world-class, and the progressive pedagogy, which consists of mostly minimal guidance instructional methods, is inefficient and not supported by science. The U. S. math programs do not stress enough content knowledge or competency. Instead, they stress progressive pedagogy over knowledge, which has been a significant mistake. 

Knowledge is the basis of critical thinking, creativity, and innovation.

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

Teachers should teach more content and in-depth content, not only in science but also in arithmetic and algebra. That won't happen because many elementary and middle school teachers don’t know enough science or math to teach it well, much less in-depth content. (Dr. H. Wu, a mathematician at UC-Berkeley, wrote that teachers don't know enough math to teach Common Core.) 

Unlike Science A Process Approach (SAPA 1967), math has been absent in many of our K-6 science programs. Nobel-Prize-winning Physicist Richard Feynman became a member of the California State Curriculum Commission, read the elementary school science and math books, and proclaimed they were "UNIVERSALLY LOUSY.” Where’s the math? (In my opinion, not much has changed since then. Note. Read more about Feynman at the end of this page.) 

Bruner rails against Piaget: If the prerequisites are in place, there is no developmentally inappropriate content. The problem has been that our education leaders adopted Piaget's theories, which were wrong or inaccurate. Still, many American teachers often say that "XYZ" is developmentally inappropriate, which is an excuse not to teach some content. Unlike the United States, most Asian nations and many European nations did not adopt Piaget's theories.

Moreover, students should learn to recognize routine problem types, translate word problems into mathematical symbols, and know when to add, subtract, multiply, or divide. Students need excellent calculating skills to solve problems. In my 3rd-grade algebra program, in addition to the perimeter and area formulas, two applications were taught through physics demonstrations--free fall (d = 5t^2) and speed (s = d/t), which comes from distance formula, distance = rate x time (d = rt), and others. (The freefall formula (d = 5t^2) was introduced to my earlier TKA classes, but not in the past several years when basic arithmetic was marginalized by Common Core.)

In 4th grade, I reviewed formulas, including the free fall and distance formulas. I added circumference and area formulas of circles and calculated averages (i.e., arithmetic mean). Students must learn to calculate with decimals and fractions and do long-division in the 3rd and 4th grades to prepare algebra, but this seldom happens. Weak arithmetic skills have hindered my algebra program. Common Core and test prep are to blame. (In my opinion, not much has changed in regular classrooms.)

Also, we have a bunch of unintended consequences in testing. "Students may attain higher test scores [which is the purpose of test prep]--but without having actually learned much about the subject" [such as basic arithmetic, geometry, and algebra], writes Jerry Z. Muller (The Tyranny of Metrics, 2018). At the state level, we can "improve the numbers by lowering the standards [which often happens to appease parents and others] ... [or] we can measure inputs rather than outputs" [which is an ill-advised, negative approach]. Even if we could control the inputs so that they were the same for all students (equity), which I doubt, then the outputs would still be different. You cannot equalize outputs (Sowell). You cannot legislate outputs. The concept of sameness, such as in Common Core or state standards, doesn't work by "equalizing downward (i.e., lowering those at the top)," writes Thomas Sowell, who calls it a "Fallacy of Fairness." 

We know that kids vary in musical ability, athletic ability, and so on. Similarly, students also widely vary in academic ability, yet, with rare exceptions, all students are fed the same reform math curriculum with the same progressive instructional pedagogy. 

My advice isn't popular: 
Don't equalize down. 
Junk sameness, most group work ideas, reform math, testing, and progressive pedagogy. 
Sort students by achievement for each subject (low, average, high). 
Upgrade the curriculum (i.e., content) to world-class. 
Use pedagogies supported by science such as memorization, drill, and explicit teaching with worked examples. 
Require all teachers to learn more math (precalculus), science (college chemistry and physics), and the cognitive science of learning. 

In short, change the way math, science, and reading are taught because what we have been doing has not worked. 

Instead of Common Core, which is not world-class, use the K-8 Core Knowledge scope and sequence as a guide for teaching arithmetic. Core Knowledge gets kids to Algebra 1 in 8th grade, which is a much better curriculum.  

We keep doing the same things repeatedly with different twists and packaging and expect different results. Moreover, we blame disparities or achievement gaps on discrimination rather than "the teaching" governed by progressive pedagogy taught in ed school. 

So, why not return to ideas and methods that worked well in the past, such as traditional arithmetic content and old-school instruction methods? 

Teachers should focus on content, not progressive pedagogy. They should worry less about understanding, the so-called mathematical practices, and self-esteem or social-emotional stuff, and more about the student's competency or performance to build confidence. But, I am not sure teachers know how to teach content. 

Sowell: Not all disparities are from discrimination.

Teachers and schools cannot close the achievement gap. It is beyond their reach, so stop blaming teachers and schools for disparities. A black scholar, Thomas Sowell, reminds us that disparities don't automatically mean discrimination. Sowell goes against the entrenched progressive pedagogy that asserts that all disparities are discrimination. Well, as usual, the progressives are dead wrong! Also, not all individuals or groups of people value education in the same way, explains Sowell in Discrimination and Disparities. There are many reasons for disparities, but the progressives ignore the facts. 

Do K-5 teachers know enough math to teach arithmetic well? 
Apparently not! Dr. Hung-Hsi Wu, a UC-Berkeley mathematician who has worked with K-8 teachers for decades, concludes that K-8 teachers don't know enough math to teach Common Core mathematics. He should know!

1. Dr. Wu writes, "The truth is that skills and understanding are completely intertwined in mathematics. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding." 
2. "Conceptual advances are invariably built on the bedrock of technique. For example, the familiar long division of one number by another provides the key ingredient to understanding why fractions are repeating decimals." 

We should teach kids techniques first, such as standard algorithms' mechanics, starting in 1st grade with addition and subtraction. The standard addition algorithm is the best model for place value. It also requires the auto recall of single-digit facts. Math skills (technique), such as long division, calculating percentages, or applying the quadratic formula, are the knowledge from where understanding comes from. 

Ian Stewart (Letters to a Young Mathematician) writes that "mathematics requires a lot of knowledge and technique." In short, you must have a "solid grasp of the basics" and be able to calculate on paper (i.e., "technique"). 

Sadly, for decades, the prevalent reform math view in elementary school arithmetic has been this: "Children can have conceptual understanding without learning algorithms." Really? What a dumb idea! Thus, according to Wu, reformers want to delay or marginalize standard algorithms, which is nonsense, who points out that a "deep understanding of mathematics lies within the skills." Teach math skills, which are fundamental knowledge in math. He also explains that it is a terrible idea to "skip the standard algorithms by asking children to invent their own algorithms," which is another stupid idea from progressives. 

Teach the mechanics of the standard algorithms first with an explanation much later. The mechanics require the auto recall of single-digit math facts.

The problem starts in the early grades. 

In grades 1-5, U.S. schools value strategies and alternative algorithms (reform math) over math facts and standard algorithms (traditional arithmetic), reading-comprehension skills over knowledge. The result has been gross deficiencies in achievement. Students are not exposed to a lot of content, much less in-depth content. 

Memorization and practice (drill) have fallen out of favor in today's schools that stress progressive pedagogy over knowledge (content). In short, the reading and mathematics instructional programs used by many districts and teachers are not backed by cognitive science. Also, I am not sure how young children will learn to read the more complex texts if their textbooks and readers are written at grade level or below grade level. 

Furthermore, many students don't have math, science, and history textbooks, but they have Google, Instagram, and other social media. What's wrong with this picture? In contrast, "the BASIS schools teach their students as Europeans and Asians do: in-depth content. Now they beat them (Asian students) on international tests." At BASIS, students take Latin and Algebra in the 5th grade.  

First-grade algebra (TKA, 2011)

FYI: I taught basic algebra (e.g., y = x + x - 2) to 40 1st-grade students and 50 2nd-grade students in the Spring of 2011, for a total of 7 instructional hours at each grade level. I shifted algebra down to 1st grade and fused it with basic arithmetic. It would be accessible to novices who were memorizing single-digit math facts and learning the standard algorithm mechanics. 

(Click Basic Algebra)

According to the National Council of Teachers of Mathematics (NCTM), children are no longer expected to master paper-pencil arithmetic, which opens the door for calculator use as early as kindergarten, writes Charles Sykes, Dumbing Down Our Kids. Moreover, the progressive reformers insist that students should dive right into problem-solving before the basics are taught, which, in my opinion, is an inane strategy. It's backward! Also, the so-called math educators—straight from schools of education—insist that young students will pick up the arithmetic along the way and invent their own math by discussing math problems in small groups. (Sure, and the moon really is cheese.)

How will students learn in-depth content unless an expert teaches the subject with another teacher who makes sure they pay attention and stay on task? In addition to PE and lunch recess, students need time to play several times a day, built into the academic schedule.

There is no social promotion at BASIS. Students must pass all the comprehensive content exams to go to the next grade level. In contrast, social promotion in public schools has been rampant. Many kids are below grade level when passed to the next grade level.  

Old School Worked: In the 1950s, teachers focused on content and competency.
Photo: 46 students at a Catholic parish school in 1950. Students sat down, got quiet, and paid attention. No Common Core. No federal regulations. No test prep. No unions. No bureaucracy. No reform math. No group work. No manipulatives. No calculators. No cell phones, and so on. Nuns were the teachers and disciplinarians. They used real textbooks and read to the class in the afternoons. Children studied vocabulary and read books. At the time, many public schools had similar seating and ideas, such as drills for learning and memorizing facts that were good for novices. It's called Old School! It worked then and works now, but it is hard to find as memorization and drill have fallen out of favor in progressive schools. 

Working hard is Old School. So are generosity, loyalty, and honesty.

Desks in a row may be old fashioned, but it was highly effective. 
Kids learned and listened to the teacher. 

Ten-year-olds are shown at a public school: Old School! They are reading a lesson from books in the 4th grade. No distractions, talking, tech, or group work! Kids sit in rows facing the teacher. They memorized stuff and pushed content knowledge into long-term memory, which enabled thinking. Thinking is domain-specific.

(Photo from Instagram)

Today

Most U.S. Students Are Behind in Math! 
"Despite billions of dollars in increased funding, American students were still outperformed by Chinese students in a test of reading, math, and science skills. But, perhaps even more alarming, only 14 percent of American students were able to reliably distinguish fact from opinion in reading tests." (The Big Think) The U.S. education machine ignores the science of learning. It disregards facts when they disagree with progressive pedagogy. 

"Study hard enough to become Smart enough!"

(China: From The Big Think) China is doing what we used to do in the 1950s, but much better. They are not chained to Piaget or backward views. Because they memorize and know stuff, they are much better at problem-solving than American students. 

"Compared to other OECD member nations, American students performed especially poorly in math." Yes, the U.S. is near the bottom in math. China is at the top. Other OECD nations think that today's "performance of students predicts future economic potential," but not the United States. We keep importing talent, not developing it. The well of imported talent will run dry, so many tech companies have set up in other nations where there is an abundance of potential talent to develop and a different attitude toward schooling (Study hard enough to become Smart enough!). 

The Asian nations drill basics (so-called "rote learning") and are way ahead of the U.S. in problem-solving. Even though the U.S. claims to teach problem-solving, it doesn't matter when it denigrates knowledge. The students in Asian nations know the content, a lot of deep content in math and science, in long-term memory. Typical American students do not. You can't solve math problems without specific knowledge in long-term memory. Knowledge enables thought (i.e., critical thinking, problem-solving, etc.). 

OECD - Organisation for Economic Co-operation and Development
"The quality of their schools today will feed into the strength of their economies tomorrow.” (The OECD and The Big Think)
Chinese Provinces -  Beijing, Shanghai, Jiangsu, and Zhejiang
PISA Mathematics (15-year-olds)  

From The Big Think

Even the neediest students in China outperformed the OECD average. On average, Chinese households earn "three times less than the OCED average of $30,500 a year," according to the OECD. 

We have a reading problem, too. Only 14% of students can tell the difference between fact and opinion. Adults have the same problem.

Basic mathematics and reading skills are taught poorly in many K-12 schools. Popular reading programs diverge from the cognitive science of learning: math, the same.   

1. DNA dominates school achievement. 
2. "Children differ in their ability to learn the things that schools teach." See #1.
3. In education, we need to recognize genetic variation; We don't. Everyone gets the same. How did that happen? Think, Common Core. See #1.
4. "Intelligence is the capacity for abstraction."
5. Critical thinking is domain-specific. There is no generalized thinking skill or strategy that is independent of content.
6. Thinking comes from knowledge, not thin air. 
7. Singapore students start multiplication in 1st grade.

If you put laptops in the classroom, then they will be more of a "distraction from learning than a tool for learning." No kidding!

Primary References: 
1. "U.S. students lag behind" (The Big Think).
2. "BASIS School Network" by Kate Stringer, the 74.
3. "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education," by Professor H. Wu (Mathematician, UC-Berkeley)
4. "Surely You're Joking, Mr. Feynman," by Richard Feynman, Physicist  
5. Science--A Process Approach, 1st Grade Lessons (1967)

Other sources include Plomin (Blueprint), Hirsch (Why Knowledge Matters), Murray (Real Education), Kant (Critique of Pure Reason, 1781), Willingham (Why Students Don't Like School and When You Can Trust the Experts), Stotsky (The Roots of Low Achievement), Wexler (The Knowledge Gap), Sowell (Dismantling America and Discrimination and Disparities), Muller (The Tyranny of Metrics), Friedman and Mandelbaum (That Used to Be Us), Singapore Math Syllabus, PISA 2018 results, J. Bruner, OECD, TIMSS, NAEP. 

Feynman

“I am looking at all these books, and none of them has said anything about using arithmetic in science," explains Feynman. In stark contrast, 4 of the 6 processes taught in 1st-grade Science--A Process Approach (SAPA, 1967) were math or math-related: using numbers (arithmetic), communicating (graphing), measuring (metric units), and using space/time relationships (geometry). Math is essential in science, but it is not taught in hands-on science programs. (So-called hands-on activities or labs should reinforce the content students are currently studying in class, but it seldom happens.) Also, SAPA taught arithmetic ahead of the grade-level arithmetic curriculum of that time, which was much more advanced than the reform math kids learn today. 

For example, SAPA taught negative numbers in the 1st grade. (In my Teach Kids Algebra algebra program, I taught negative numbers in the 2nd grade, 2019. Also, I taught integers in my self-contained 1st grade in the early 80s.) 

"Everything [e.g., elementary school science textbooks] was written by somebody who didn’t know what he ... was talking about. How anybody can learn science from these books, I don’t know, because it’s not science,” concludes Feynman. Also, Feynman had harsh words for the math textbooks, too: "no applications and not enough word problems."  

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LT
12-6-19, 12-11-19, 12-14-19, 12-16-19, 12-20-19, 12-21-19, 12-24-19, 12-26-19, 12-29-19, 1-4-2020, 10-23-20

©2019 - 2020  LT/ThinkAlgebra

Rabbit Hole Math

Falling Down The Rabbit Hole 

Alice exclaims, "I never heard of uglification." 

Like Alice, we have fallen down the rabbit hole where kids learn the different "branches of Arithmetic--Ambition, Distraction, Uglification, and Derision" as Mock Turtle points out. Mock Turtle is not a teacher; he is a facilitator, which is an important aspect of progressive pedagogy. 

HELP!

In contrast, Alice explains, "I never heard of uglification." She's not alone! Down the rabbit hole, Alice encountered an alternative, unorthodox, cumbersome, constructivist form of arithmetic--not the straightforward, standard arithmetic that she had memorized, studied, and learned. 

Note. In math, American high school students are at the bottom (PISA, 2018).
Click: China is at the top.

Reform math is upside down and backward; that is, it grossly misinterprets Bloom's taxonomy by jumping to higher-level thinking skills before the fundamentals had been learned in long-term memory. Indeed, today, math education has become bizarre, cluttered, and counterproductive. Reform math makes little sense. Memorization is considered old school. "The prevailing theory is that students must engage in constructing their own knowledge rather than memorizing facts that will only bore them and that they don't really understand," writes Natalie Wexler (The Knowledge GAP). It's the wrong approach. She observes that "skipping the step of building knowledge doesn't work."















It is common sense that if kids don't learn the fundamentals of arithmetic, starting in the 1st grade, then they are blocked from higher-level math (Engelmann). 

Aim for Competency!

If we want students to become competent in arithmetic and algebra, then they need to be more like ballet dancers, gymnasts, swimmers, violinists, chess players, etc. That is, students need to practice and review so that the fundamentals of math stick in long-term memory (i.e., for automaticity). We don't do that! In fact, what we often do in the classroom is the opposite of the cognitive science of learning. 

The Knowledge Gap

The knowledge gap has been known for decades and disregarded. In math, there is a considerable knowledge gap, too, because children don't memorize times tables or learn the mechanics of the standard algorithms like they used to 50 years ago. Hence, calculating skills are weak, and learning is not coherent, systematic, or hierarchical.  

Calculating skills are conveyed to concepts or ideas in math. Even though the ideas in math, themselves, are abstract, such as operations like addition or the concept of perimeter, they are learnable even in 1st grade. I know; I did it! 

Math Knowledge means math skills. Having math knowledge is the ability to convey the proper math skills to the appropriate problem type and solve the problem by paper calculating. Getting the correct answer has always been critical in applying math, but it has been marginalized in current math teaching, which is a mistake. 

To advance, kids must know math facts and standard mechanics in long-term memory.

  

The math fact 3 + 8 = 11 is the solution to a vast number of word problems. Because the math fact is abstract, it can be used again and again. That's the power of abstraction. If the child memorizes the 3 + 8 fact, suddenly she has the solution to hundreds of math problems. It's math power! Even though a student may have some idea of perimeters, she cannot calculate a perimeter unless she knows sums. If you can't calculate it, then you don't know it!  The memorization of facts is good for novices.

Calculating skills are intrinsic skills for solving problems in math. 

Education, today, does not work. Math needs a radical break from the status quo, which has been reform math. Teachers should embrace a knowledge approach that is supported by the cognitive science of learning. Also, in math, knowledge is math skills. There is a lack of content in elementary school math. Common Core is a continuation of a content-free approach. It is not world-class. Schools fail to build knowledge, not only in reading but also in math, science, and other subjects. 

In addition to calculating and logic, math skills include knowing vocabulary, applying concepts and mechanics, recognizing patterns (e.g., problem types), and grasping the meaning of structure (e.g., 2n means 2 times the number n; 3^4 means the product of 4 copies of 3, which is 3 x 3 x 3 x 3 = 81; ab = ba, and so on). In other words, students must know math stuff in long-term memory to perform math. Furthermore, knowledge is the basis of problem-solving. 

You can't solve trig problems without knowing some trig. You can't translate Latin without knowing some Latin. You can't grasp literature without knowing some textual criticism. All fields of study require the memorization of the basics and experience. It's a process of building knowledge. There is no generalized thinking skill. Thinking is domain-specific. 

Study hard enough to become Smart enough! 

(South Korean Motto)

©2019 - 2010 LT/ThinkAlgebra 

Cognitive Science

Cognitive Science

The late Zig Engelmann said that if you want to know if an idea works, then go into a classroom, teach it, and see what happens, which is what I did in the winter of 2011 as a volunteer guest algebra teacher for two 1st grades, two-second grades, and one-third grade.  It was clear to me that we underestimate the content children can learn.  

Eric A. Nelson ("Cognitive Science and the Common Core Mathematics Standards" 2017) writes, "Between 1995 and 2010, most U.S. states adopted K‐12 math standards which discouraged memorization of math facts and procedures [NCTM]. Since 2010, most states have revised standards to align with the K-12 Common Core Mathematics Standards (CCMS). The CC does not ask students to memorize facts and procedures for some key topics and delay work with memorized fundamentals in others." Because of the limited space in the working memory (WM), Nelson observes, "When solving math problems of any complexity, due to WM limits, students must rely almost entirely on well‐memorized facts and algorithms." It is what the science of learning is all about. Kids are novices, not experts. They need to memorize essential stuff in long-term memory to solve math problems in the Working Memory. They need to learn content.  

The late Zig Engelmann observed: "You learn only through mastery!"

Paying attention is important in learning. If you are distracted by something, then you diminish your working memory's thinking capacity. Daniel Kahneman (Thinking Fast and Slow, 2011) writes, "Anything that occupies your working memory reduces your ability to think," which is not good for learning and problem-solving. For example, sitting in groups is a distraction for some kids. 

Attention in the classroom is often inconsistent, so is learning.

It takes effort to pay attention in class!

Some basic ideas in the cognitive science of learning:

Note. Some of the content in this section has been taken directly or rephrased from The Science of Learning, 2015.

1. Students learn new ideas that are linked to old ideas they already know.

2. To learn, students must transfer information from working memory (where it is consciously processed) to long-term memory (where it can be stored and later retrieved). Learning is a change in long-term memory. Students have limited working memory space that can easily be overloaded. Worked examples reduce the cognitive burden. A worked example is a step-by-step demonstration of how to perform a task or solve a problem.

3. Grasping new ideas can be impeded if students are confronted with too much information at once.

4. Cognitive development does not progress through a fixed sequence of age-related stages [as Piaget had thought]. The mastery of new concepts happens in fits and starts. Readiness is not age-dependent; it is determined by the student's mastery of the prerequisites. (A college student who has not mastered algebra will have difficulty with calculus, while a 10-year old who has mastered algebra/trig (yes, they do exist) will have little trouble with calculus. Readiness is a function of mastering prerequisites.) 

5. Each subject area has some set of facts that, if committed to long-term memory, aids problem-solving by freeing working memory resources and illuminating contexts in which existing knowledge and skills can be applied. Memory [of math facts, for example], is much more reliable than calculation.

---------------------- End of Section

Note. Inferences are built on observations, which are measurements. An interpretation of the measurements is called an inference. The idea of observation/inference was a fundamental part of a K-6 program called Science--A Process Approach (SAPA, 1967). Today, there is nothing like SAPA. An inference is not a fact, yet textbook writers and journalists often treat inferences and correlations as facts. 

All scientific theories are tentative; they are subject to change or correction with additional data. Almost all studies in education have not been replicated for one reason or another. Repeatability is a basic idea in scientific research. If an experiment is repeated and gives different results, then there is something wrong. If a study has not been repeated, which is often the case in education, it cannot be trusted because its claims have not been verified. Also, it is not surprising that 82% of the government grants for classroom innovations (experiments) issued by the U.S. Department of Education failed to improve reading and math achievement. Unfortunately, there is a lot of junk in education with no basis in science. If you cannot measure something, then it is not science. It's speculation, and we have a lot of that going around in education. 

Some of the difficult to measure ideas include understanding, creativity, innovation, collaboration, critical thinking, self-esteem, analytic ability, enthusiasm, persistence, and so on. 

In progressive education schools, many teachers have been taught idealistic, unrealistic learning theories, such as constructivism, rather than the cognitive science of learning, which is practical. Often, teachers are trained to be facilitators of learning, not academic leaders. They frequently implement test prep and minimal guidance instructional methods, which are not supported by evidence as being effective. These are among the least effective methods. Is it any wonder that our kids aren't learning arithmetic like students in other countries?  Teachers are caught in the middle of a mess they did not create. Bad ideas and fads are often imposed on teachers who have little say. 

I think teachers realize that a one-size-fits-all approach (Common Core or state standards) is fundamentally flawed and counterproductive. We need to stop blaming teachers for our education shortcomings and start blaming the schools of education, a progressive reform agenda, a plethora of untested fads and unproven reforms, and the numerous state and federal policies, mandates, and laws that interfere with and impede the efficient education of our children.

I reject the progressive utopian ideology and agenda of sameness (equalizing downward), displacing essential content for thinking skills, well-intentioned fads (evidence doesn't matter), and a "radical constructivist view of mathematics learning," which are embodied in the “reform math movement [then NCTM, now Common Core] that stresses an undefined conceptual understanding and student-created algorithms," along with early calculator use. The notion in reform math is that pedagogy and group work are much more important than "mathematical substance." The teacher doesn't teach; the teacher facilitates. Long-term memory knowledge is not that important, say the reformists. 

In contrast, I agree with Sandra Stotsky and the 2008 National Mathematics Advisory Panel that endorse an "academically stronger mathematics curriculum as well as for fluency in students’ computational skills with whole numbers and fractions" using standard algorithms. In other words, the Panel argues for explicit teaching of substantive "standard arithmetic" to prepare more kids for algebra. Indeed, one of the Panel's recommendations was that all schools should offer a valid Algebra-1 course no later than 8th grade, which implies that schools need to prepare more elementary students for success in the course by ensuring they know standard arithmetic well. (Quotes from Sandra Stotsky, National Mathematics Advisory Panel)

Teaching in the 21st Century has become unnecessarily difficult, complicated, and perplexing, especially with the extra burdens of Common Core, government mandates and policies, unproven reforms (fads), and pervasive progressive ideology of "sameness" and of "thinking skills displacing essential content." 

One problem is that kids widely vary in academic ability, yet they are grouped together in the same classroom (inclusion policies). Another is that thinking skills, without substantial knowledge supporting them, are at best superficial. Improving inference-making skills implies substantially improving the student's knowledge and experience on a particular topic or subject. Having content knowledge in long-term memory is the key to profound and intelligent thinking. Googling and reading a few paragraphs and bits and pieces do not cut it. Kids are novices and need accurate knowledge in long-term memory to think well, not vice versa. For example, to solve math problems, students need basic knowledge, both factual and procedural, in long-term memory. The knowledge also has to be specific; you can't solve trig problems without knowing and applying trig. In summary, the quality of your thinking is a function of the quality of your knowledge.      

Teaching children "math practices" or "science processes" is not the same as teaching essential standard arithmetic or science content. Children are not pint-sized mathematicians or junior scientists. They are novices who need to master content (facts and efficient procedures) from the get-go. When asked why Asian nations do a better job teaching math and science, Arthur Levine, former president of Columbia University's Teachers College, says they "start earlier, work longer, and work better." He explains, "Kids are capable of learning about mathematics much early than we thought." We have a weak curriculum, even under Common Core, which is not world-class.

Furthermore, most US teachers do not know or use the "cognitive science of learning" in their teaching. For example, starting in 1st grade, children are not required to memorize math facts and practice standard algorithms for automation in long-term memory under reform math. They seldom master the basic rules that govern the behavior and meaning of numbers and operations.

Essential facts and standard procedures (some of the fundamentals of arithmetic) need to be automated in long-term memory (a vast storehouse) for direct use in problem-solving that occurs in the limited mental space of working memory. Daniel Willingham also states, "The brain is not designed for thinking ... Humans don't think very often (i.e., solving problems, reasoning, reading something complex, or doing any mental work that requires some effort) because our brains are designed not for thought but the avoidance of thought." 

Thinking requires focused attention and effort. In the classroom, "factual knowledge is important," says Willingham, and "practice is necessary." These ideas are key fundamentals in the cognitive science of learning.   

  

Regrettably, reform math via Common Core pays little attention to the science of learning and often exposes novices to cognitive load problems and frustration. "Teaching strategies [such as nonstandard algorithms in Common Core reform math] instead of knowledge has only yielded an enormous waste of school time," writes E. D. Hirsch Jr. [Aside. In Common Core reform math, strategies refer to many different, complicated, inefficient, nonstandard, or invented algorithms to do simple arithmetic. Why not teach kids efficient, standard algorithms to begin with?] 

Each subject (e.g., math, science, history, literature, the arts, etc.) has its unique background knowledge, language, and way of thinking, so generalized skills do not transfer easily, which is why Hirsch says that generalized skills are a waste of classroom time. 

Furthermore, the thinking skills required for solving an equation in algebra, a physics problem, or the exegesis of a literature text are not always the same. (Deductive reasoning from well-established rules (facts) is used in math, but a counterexample invalidates the rule unless conditions are defined, such as a ÷ b, b ≠ 0, etc. Math is based on facts, logic, and true statements. 

In contrast, inductive reasoning is used in science, but a few conflicting measurements (data) do not invalidate a relationship. For example, linear regression analysis can calculate a correlation coefficient and build a linear equation that models the relationship between two variables. Incidentally, regression analysis is a statistical tool used in many different disciplines.

Moreover, a high correlation coefficient between variables should not imply causation. Also, you should not extrapolate beyond the actual data, which is why mutual fund companies caution investors that past performance is no guarantee of future performance.  That said, the ability to use formal logic, that is, logical reasoning is critically important in math, science, and other academic disciplines. 

Our personal decisions on national issues, buying a car, investing, or selecting a school are seldom based on logical reasoning because we don't bother to optimize our choices. Still, neither do the so-called policymakers and decision-makers at the local, state, and federal government levels. For example, the content of the Common Core standards, the costs of their implementation, or their impact on teachers and students were not thoroughly examined, analyzed, or critically reviewed by policymakers and the powers that be, that is, the decision-makers, who hurriedly and blindly adopted Common Core long before the final draft was written. In short, the people-in-charge did not bother to optimize their decisions when making critical choices that have affected every child and teacher in the public schools. 

Common Core was plunged down our throats by higher-ups. The effort and time we spend on making a decision should be proportional to its importance, says Nobel Prize-winning economist Herbert Simon. Hurried or hasty decisions in education are often counterproductive and have unintended consequences. It seems that our policymakers and powers that be have not followed Simon's principle in K-12 education. Moreover, as it turns out, Common Core math standards are often implemented as a repackaged flavor of NCTM reform math. Reform math, based on discredited constructivist learning theory and inefficient minimal teacher guidance for instruction, disregards some of the science of learning's fundamental principles. 

We often make assumptions (inferences) or favor innovations without scientific evidence (e.g., Common Core standards will improve test scores and make all kids ready for college/career, or the latest technology will rescue our lagging math and reading achievement, etc.). We often make decisions on insufficient information, anecdotal evidence, intuition, expediency, trendiness, feelings, opinions of friends, celebrities, politicians, so-called experts, social media, etc. We sometimes believe dubious or far-fetched claims made by alarmists. The effort and time we spend on making a decision should be proportional to its importance, says Herbert Simon, but that's not how we have in the real world. 

Some of the most popular practices and programs in education are not evidence-based and don't work well because they go against learning science. Furthermore, having a steady diet of test prep is not education. Keith Devlin writes, "Decision-making [something computers do well] and thinking [something humans do well] aren't the same, and we shouldn't confuse them." We humans don't bother to optimize our decisions. We are not very good at making good decisions. 

Teaching children "math practices" or "science processes" is not the same as teaching actual arithmetic or physical science. Children are not pint-sized mathematicians or junior scientists. They are novices who need content to think well! Furthermore, content should be taught explicitly or directly, that is, through strong teacher guidance. The content children learn from manipulatives or discovery activities in group work is trivial compared to the content children learn when taught explicitly. Students should not be required to make drawings (visuals) or use nonstandard algorithms to do simple arithmetic. Students should practice and use standard algorithms from the get-go.

In school, the primary objective should be "fill the mind with specific content" to enable smart thinking. Indeed, "training the mind to think" is essential, but smart thinking depends on gaining knowledge in long-term memory via a content-rich curriculum. Furthermore, the content should be a broad-based, liberal arts curriculum, including the arts, humanities, math, and science.

The quality or merit of a student's problem solving, critical thinking, or inference-making skills is a function of substantial knowledge of the subject at hand (Science of Learning, 2015). This fundamental idea of the cognitive science of learning is not new. For example, Immanuel Kant (1724-1804) once remarked, "Thoughts [critical thinking] without content [knowledge] are empty." 

To base a curriculum on problem-solving or critical thinking without acquiring sufficient content knowledge in long-term memory, let's say in mathematics, has little merit. 

"The reality is that you can think critically about a subject only to the extent that you are knowledgeable about the subject," writes Paul Bruno (Edutopia Blog). In short, problem-solving, critical thinking, or inference-making skills depend heavily on your knowledge of and experience with a particular topic, subject, or domain.

In contrast to Common Core, you cannot do quality critical thinking well by reading a few paragraphs. Bruno explains, "This is in stark contrast to the common desire among educators and policymakers to teach so-called [generalized] thinking skills that can be applied in any situation." Thinking skills are unique to a subject or domain. Improving inference-making skills implies improving students' knowledge base and experience in a particular area, subject, or field.

The so-called strategies in Common Core reform math, that is, nonstandard or invented algorithms, should not replace, delay, or hinder the memorization of basic number facts or standard algorithms' learning through practice in first grade or any grade. Memorization and practice to gain factual and procedural knowledge in long-term memory do not squelch creativity and learning, as some erroneously think. Knowledge gaining engages and enables problem-solving, creativity, and innovation.

Instruction

Many widespread, favored "classroom practices" are not supported by scientific evidence and are among the least effective. Kirschner, Sweller, & Clark (2006) point out that minimal guidance methods, in which the teacher is a facilitator or coach of learning, are typically the least effective classroom practices. These constructivist-based minimum guidance practices have many names, including such favorites as inquiry-based, discovery learning, problem-based, etc. Some teachers are convinced that inquiry- or discovery-based learning, which favors group work, is the best way for kids to learn math, but it is not true. In short, reforms should be based on cognitive science or learning, not popularity, intuition, or ideology. Current cognitive research supports direct [explicit] teacher guidance. "Direct instruction involving considerable guidance, including examples, resulted in vastly more learning than discovery," writes Kirschner, Sweller, & Clark.

Kirschner, Sweller, & Clark explain (long quote), "After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to intermediate learners. Direct instructional guidance is defined as providing information that fully explains the concepts and procedures that students are required to learn as well as learning strategy support that is compatible with human cognitive architecture. Learning, in turn, is defined as a change in long-term memory." 

Kids are novices, not experts, so they need straightforward teacher guidance and encouragement, especially with new, more complex content. Furthermore, Kids don't need to work in groups. Kevin Ashton (To Fly A Horse) points out, "Working individually is more productive than working in groups."  

More principles of the science of learning. 

1. Memory is more reliable than calculating.

Calculating single-digit facts clutter a student's limited working memory and leave less space for thinking. For example, add 6 + 9 using make ten. The student adds 1 to 9 to make 10, then subtracts 1 from 6 to make 5 (to compensate, we are told), then adds the two results: 10 and 5 to get 15. That's a total of three calculations the student must do and hold in working memory. A few students can do this, but many students got confused and mixed up. To avoid the clutter and cognitive overload, simply memorize 6 + 9 = 15 (for instant retrieval from long-term memory). When the situation of 6 + 9 or 9 + 6 arises in a question or word problem, the student instantly retrieves 15 without thinking or calculating. It is automatic. Thus, one of the basic precepts of the science of learning is the relationship between working memory (limited) and long-term memory (vast) and why storing arithmetic essentials in long-term memory frees mental space for thinking, that is, "memory is more reliable than calculating."

2. Another principle of learning is explaining a worked example on the whiteboard step-by-step-by-step. Explaining something in small steps lessens the chance of cognitive overload in the student's working memory.

3. Another principle is that kids are novices, not experts or adults. Kids need straightforward, explicit instruction, not group work or collaboration. They need to learn [absorb] in-depth content in long-term memory to learn to think well. Furthermore, kids don't think like adults, and they are not pint-sized mathematicians or little scientists. Let me repeat, kids are novices and need to transfer content into long-term memory to think well. The transfer of material from working memory to long-term memory storage requires memorization, repetition, practice, and study. 

Many poor decisions (with good intentions) have been made in education over the decades, and the progressive reformists continue making them. The policymakers (those in charge) wrongly decided to "push our existing system harder for incremental improvements and rely on policies calling for curriculum homogeneity, more standardized testing, and teacher accountability tied to student test score performance," write Wagner & Dintersmith (Most Likely to Succeed), to chase after higher test scores. I agree with Wagner & Dintersmith that the mastery of core academic content is essential. Still, I reject the solution of minimal teacher guidance methods (project-based learning) for accomplishing this. Knowledge is not all that relevant. It has always been true that children need to memorize and practice to master the fundamentals of math in long-term memory. 

Frankly, I don't think kids who know bits and pieces of a reduced curriculum will be successful because skillful thinking in a subject without substantial knowledge of that subject is empty. Maybe, one day, the science of learning and an excellent math curriculum taught by teachers who know math will be implemented in our schools, but not under Common Core, its burdensome baggage, and goal to score higher on tests. In short, our schools under Common Core and state standards are test-driven. In ed schools, most teachers have been taught idealistic, unrealistic learning theories rather than the science of learning, which is practical. Like Sandra Stotsky, I reject the "radical constructivist view of mathematics learning," which is a modern “reform math movement [that stresses] an undefined conceptual understanding and student-created algorithms." (Quotes from Sandra Stotsky, National Mathematics Advisory Panel)

  

Note. This page is a work in progress. It still is. It is not an essay. It is far from complete. Please excuse typos and other errors. October 21, 2015, November 1, 2015, November 12, 2015, 11-19-19, 10-17-20

Draft 1.
Models: Hannah

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