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

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