Thursday, March 28, 2019


In education, everyone talks about understanding, but no one defines it. Advocates of understanding even theorize deep understanding. Frankly, I don't know what that means because the word "understanding" is ambiguous, confusing, and very difficult to quantify.  

I think "knowing something" implies some level of understanding. Still, the assumption is ambiguous and lacks specification. To me, “understanding" is a vague term, hard to define, and means different things to different people in different situations and disciplines. I don't define understanding, and I don't measure it. I avoid the word. Like creativity, understanding is complicated to program, measure, and test, so I use the word knowledge, which I can measure. 

The problem I have with critical thinking is that I don't know of any valid tests that can measure it. So-called critical thinking changes from one discipline to another. In short, "critical thinking is difficult to measure," observes Daniel T. Willingham, a cognitive scientist. Also, the critical thinking learned in math (i.e., problem-solving) doesn't help much in science or other disciplines because thinking is domain-specific. 

The same is true for other ambiguous, hard to measure words tossed around in education, such as collaboration, self-esteem, creativity, analytic ability, innovation, understanding, "enthusiasm, wisdom, or attitudes toward learning." 

On the other hand, content knowledge is easily measured on tests. Educators should stick to performance, not ambiguous ideas that sound great but are confusing and difficult to quantify. 

Gaps in content knowledge hold kids back, starting with math facts and standard procedures for operations. Unfortunately, educators are teaching a combination of reform math and test prep rather than essential factual and procedural knowledge of arithmetic. 

Good calculating skills (i.e., factual and procedural knowledge) are required from problem-solving and learning concepts in math, starting in the 1st grade. In my opinion, if you can't calculate it, then your knowledge of an idea is limited, and your calculating skills are weak. 

I can test the knowledge of arithmetic (ideas, skills, and uses). I tell students: "If you can't calculate it, then you don't know it." The idea resonates with little kids in my algebra lessons, but it had started decades ago when I tutored precalculus at a private school. Being able to calculate something is an essential part of problem-solving. 

I can judge a student’s performance (i.e., knowledge), but, even then, I cannot determine precisely the student’s level of understanding, only to say that the student has some level of understandingIn short, the student’s understanding is sufficient to calculate a solution to the problem, which, I think, is the same as saying that the student has acquired enough knowledge to solve the problem.

Understanding grows slowly over the years as the student gains more experience solving similar problems. Perhaps, this is what Barry Garelick meant when he wrote about the "interplay between procedural fluency and conceptual understanding." 

I  think I am on safe ground when I make these assumptions:
1. Knowledge in long-term memory enables problem-solving in mathematics, which is domain-specific. 
2. Practice unleashes talent or ability.  
3. The differences in school achievement are 60% DNA. (Plomin)
Genetic variation means that children do not have the same abilities to learn.
4. Intelligence is not genetically fixed. Other factors add to intelligence.
5. Unlike Singapore, U.S. educators do not teach standard arithmetic for mastery. The lack of mastery of fundamentals has been a significant problem in math education.
6. Understanding can be implied from a student's performance

The arts, it has been said, improves achievement. Really? Still, there is no credible evidence for this. There is no cause and effect. Kids who excel at piano or violin also tend to excel academically--math science, English, history, etc. And, it is mostly DNA. That said, I think children should be taught music, art, drama, etc., but the arts won't help them master arithmetic, which requires hard work and effort. 

©2019 - 2020 LT/ThinkAlgebra 

Friday, March 8, 2019


Children are unequal, not identical. They vary extensively in academic ability, athletic ability, musical ability, writing ability, science ability, and so on. Still, low achieving students in math are placed with high achieving students, which has been a recipe for mediocrity and a lowering of expectations. Every student gets the same instruction for equity, but, as Thomas Sowell explains, so-called equity is a "fallacy of fairness" that prevents students, including minority students, from excelling. Educators must acknowledge that the differences in school achievement are 60% DNA (Plomin Blueprint). Academic achievement is intrinsically tied to academic ability, which varies widely. 

Little Progress in Academic Achievement
I blame schools of education for not training teachers much better academically, for not sorting out unqualified students, and for inculcating teachers in progressive ideology.

Introduction: What we have been doing in the classroom has not worked well. Achievement is flat in reading, math, science, and other academic domains. Teachers should focus on the mastery of fundamentals like Singapore, not learning for a state test!​ So, why don't they? Some think that knowledge is lower-order thinking and not nearly as important as higher-order thinking. So, the progressive curriculum is based on higher-level thinking such as analysis, synthesis, and evaluation, which are at the top the Bloom's pyramid. The approach is wrong. Educators are misusing Bloom's Taxonomy. They should start at the bottom, not the top.

Bloom Modified for TIMSS
It is inconceivable that elementary school reform math programs, which claim they are strong in problem-solving (i.e., higher-level thinking), are deficient in fundamentals or basic skills, such as computational fluency, which was stressed in the 2008 report from the National Mathematics Advisory Panel (NMAP). 

Unfortunately, the favored reform math programs, which had evolved from the National Council of Teachers of Mathematics (NCTM) standards, seem to reverse Bloom’s cognitive schema by emphasizing higher-order thinking (Problem-Solving in math) and by going against key cognitive science findings that emphasize background knowledge (both factual and procedural) in learning arithmetic well. The reform math programs do not work because students lack sufficient knowledge in long-term memory. The reformers are almost all from the radical left. 

Adding to the muddle, progressive education leaders keep dumping stuff (e.g., reforms, policies, innovations, etc.) into the classroom, such as personalized learning, blended learning, project-based learning, minimal guidance instruction such as discovery learning; so-called "mathematical practices," reform math that failed in the past, early use of calculators (NCTM), "content-indeterminate standards"* that are not world-class, social-emotional learning (SEL), mindfulness, tech use, test prep, yearly testing, "nonexistent all-purpose skills,"* learning styles, feelings (self-esteem), innovations/reforms that don't increase achievement, group work (i.e., collaboration), higher-order thinking (critical thinking or problem-solving in math), and equity policies that are "fallacies of fairness." It makes no sense! 
* The two phrases are from E. D. Hirsch Jr., Why Knowledge Matters: Rescuing Our Children from Failed Educational Theories, page 104.

​It seems that every year, teachers need to assimilate and implement a new reform, trend, fad, program, or innovation, when, in fact, 82% of the reforms, programs, trends, fads, and innovations funded with grants from the U.S. Department of Education failed to improve achievement in math or reading. WOW!

Students need to know basic arithmetic and algebra so they can move forward to higher math. They do not learn arithmetic to be collaborative or creative problem solvers. Later, they do not sign up for calculus to become more creative. 

In short, I am not interested in students being collaborative and creative problem-solvers when they don't know the fundamentals and stumble over routine math problems. Students are novices ​and need to build a storehouse of essential factual and procedural knowledge in long-term memory for use in problem-solving.  

Our kids are not doing okay or just fine. For example, in the Tucson area, depending on the school district (9 of them), from 74% to 88% of incoming students at the local community college are placed in remedial math. Starting in the 1st grade, kids are learning reform math, which clutters the curriculum with a bunch of extras, not standard arithmetic. The reform math curricula are insufficient, and the progressive methods of instruction are inefficient.
(Note: Some of the phrases in quotes are from E. D. Hirsch, Jr. and Thomas Sowell; Data from Pima Community College)

Misusing Bloom's Taxonomy
Dr. Paideia writes in Medium that educators are misusing Bloom's Taxonomy.  He explains, "Bloom’s Taxonomy is often used to structure students’ learning objectives. Because Bloom stated that Evaluation and Synthesis are “higher-order thinking,” while knowledge is “lower-order thinking,” the people who make up the curricula have made the mistake of thinking that “higher-order” means “the only things that are important.” As a result, teachers are required always to ask “higher-order questions,” no matter the age or degree of knowledge a student has." But, critical thinking without knowing content is empty (Kant). 

Paideia continues, "Bloom’s Taxonomy is structured like a pyramid because the easier “lower-order thinking” levels are absolutely necessary to master before you can move up to the next level. No pyramid exists with only a top or a middle. There has to be a foundation, and that foundation is knowledge. What that knowledge is going to look like will vary according to grade level, and the more complex the knowledge, the more of the higher-order thinking levels can be involved." 

In short, teachers should focus on lower-order thinking, which is knowledge and the demonstration of that knowledge in solving routine math problems. Factual and procedural knowledge stored in long-term memory enables problem-solving. You don't start at the top. You start at the bottom of the pyramid, which is Knowledge.

Part I: Myths, Fact-Inference
Kids and adults are terrible at separating facts from an opinion on the Internet. Also, there has been much emphasis on using devices for engagement, but engagement is not the same as learning content. Kids today know less content, and without substantial background knowledge, they can't identify what's fact or fake on media, social media, and the Internet. Moreover, kids and adults "rush to judgment on scant information," observes Leslie Valiant (Probably Approximately Correct). The media is packed with opinions. It blends fact with opinion, which passes as "journalism" today, so it is difficult for the student to sort the facts from the inferences (opinions).  Students are told to trust their feelings. They don't learn facts. Knowing something isn't essential. "I can just google it." Really? 

Gary Stix (Scientific American, August 2011) writes, "Some widely held ideas about the way children learn can lead educators and parents to adopt faulty teaching principles."

Here are common myths about the brain.
1. Humans use only 10% of their brains.
2. Left brain and right brain people differ. 
"Humans use both hemispheres of the brain for all cognitive functions."
3. Each child has a particular learning style. Really?
4. Brains of boys and girls differ in ways that dictate learning abilities. 

Stix points out, "Practicing a musical instrument appears to improve attention, working memory, and self-control ... Listening to [Mozart] alone is not sufficient." We are not sure "exactly what type of [musical] practice enhances executive function." But, we do know that practicing a musical instrument, such as the violin, "trains the entire brain" to become "better listeners."

Model Credit: Caitlyn

Education is loaded with unproven beliefs, widespread classroom conventions that lack evidence, claims that overreach, and multiple reforms and innovations that don't work as expected in the classroom.

Today, teachers have to contend with so-called mathematical practices, mixed groups (low ability mixed with high ability), group work, engagement, using manipulatives, minimal guidance instructional methods (e.g., discovery learning), and poorly designed curricula (e.g., reform math). Toss in some social-emotional learning (SEL), feelings, mindfulness, tech (gadgets or screens), discipline problems, critical thinking without content (Kant), learning styles, and much more. You end up with a huge mess.

Part II Teachers can't teach what they don't know well, which became evident to me in the early 70s when regular classroom educators were asked to explain SAPA (Science--A Process Approachto elementary school students. Many of the teachers had taken SAPA courses, summer workshops, and Professional Development, but very few K-6 classroom teachers were capable of teaching the SAPA science curriculum or math in it. SAPA is loaded with math ahead of the grade-level curriculum. In short, teachers didn't have the proper background knowledge in science and math. In college, future teachers weren't required to take harder academic courses in math and science.

Fast forward to 2019. 
Not much has changed. Today, most K-6 teachers major in education, not in a regular academic discipline. Moreover, many are weak in math and science. Still, they are expected to teach these disciplines in self-contained classrooms with mixed ability groups of children. Also, reading scores in national and international tests are flat, too. I blame schools of education for not training teachers much better academically, for not sorting out unqualified students, and for inculcating teachers in progressive education, a failed ideology.

According to a study from the Organization for Economic Cooperation and Development (OECD), only smart people make good teachers, which is nonsense. How teacher candidates are selected, trained, or taught is the difference. It doesn't matter how smart you are unless you stop and think, says Thomas Sowell. One student said, "I am smart; I am just not good at academics." Really? 

In the U.S., future teachers shy away from rigorous college courses in science and mathematics, that is, most don't sign up for chemistry and physics or precalculus and calculus. As a group, they also have weaker SAT scores than students in other majors, such as English, finance, history, physics, computer science, engineering, business, etc. And, with the dictum of "smaller class sizes," more teachers are needed. The result has been that "smartness" within the teacher pool has been diluted over the years. 

Furthermore, the best college students don't want to become teachers--poor working conditions, little respect, no career ladder, less pay, and so on. Many veteran teachers are leaving, too. 

Last update: 3-13-19, 5-23-19

© 2019 -2020 LT/ThinkAlgebra