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 understanding. In 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
Thursday, March 28, 2019
Friday, March 8, 2019
Teaching
Unequal
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.
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.
https://medium.com/@dr.paideia2020/misusing-blooms-taxonomy-f570c5f56c2a
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."
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 Approach) to 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.
Comments: ThinkAlgebra@cox.net
Last update: 3-13-19, 5-23-19
© 2019 -2020 LT/ThinkAlgebra
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 |
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.
https://medium.com/@dr.paideia2020/misusing-blooms-taxonomy-f570c5f56c2a
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.
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.
Comments: ThinkAlgebra@cox.net
Last update: 3-13-19, 5-23-19
© 2019 -2020 LT/ThinkAlgebra
Monday, February 11, 2019
Random Thoughts Part 2 - 2019
Much of what I write diverges from trendy education policies, widespread classroom practices, prevailing theory, and other issues, such as equity.
For example, the major claims made by the Common Core (CC) people were bogus, such as readiness for college and world-class benchmarks. Beginning in the 1st grade, U.S. students are not learning world-class math. Also, parents may clamor for small class sizes, but the evidence doesn't support it. Remarkably, students don't need to start at the concrete level to get to the abstract level. The 1st-grade equation n + 7 = 12 is already abstract and can be used to solve countless situations that are concrete. Another issue is that equity trumps achievement in the classroom, hurting all children, including children of color. Moreover, costly technology in the school has not transformed education.
Over the years, I have found several precocious children in my algebra classes, children who can deal with abstraction better than others.
1. Abstract
We have been told that children get to the abstract only through concrete experiences and developmental stages (Piaget), so teachers ask kids to use manipulatives such as cubes and images in textbooks to demonstrate math ideas. But, what if we go from abstract to concrete, which is what I do in my elementary school algebra classes?
In 1st grade, I want children to learn the core of a concept, which means that the base of the concept of addition, for example, is not linked to counting cubes or manipulatives, or images in books. Concepts (i.e., ideas) are abstract, so are numbers, operations, equality, rules, and algorithms, too. Students don't need to start at the concrete level to get to the abstract level. Equation 3 + 7 = 10 is already abstract and can be used to solve countless situations that are concrete.
In my Teach Kids Algebra (TKA) program, 1st-grade students used symbols for unknown numbers and ideas and solved simple equations using arithmetic knowledge and reasoning. Ordinary 1st-grade students also built x-y tables and drew pictures of linear functions (i.e., make a graph in Quadrant-1). I created TKA as a reaction against Common Core. I fused algebra to standard arithmetic.
The equations x - 5 = 12 and y = x + x - 2 are abstract. For example, in the second equation, 1st-grade students were asked to find y given several x values, build a table of values, and make a graph of the equation. Find y when x = 1 corresponds to the point (1, 0) on the coordinate plane. Find y when x = 2, which is the point (2, 2). Find y when x = 3, which is the point (3, 4), and so on. Also, if y = 12, then what is x? {using guess-and-check reasoning, x is 7 because 7 + 7 - 2 = 12} Even 1st-grade students follow the algebraic rule for substitution.
2. Not College Prep
What irks me is that many schools offer kids so-called "college-prep" courses in math, but when they apply to a community college, for example, they are placed in noncredit remedial math courses, mostly the equivalents of middle school prealgebra and Algebra-1 and high school Algebra-2. Indeed, beginning in the 1st grade, kids are not learning world-class math. Common Core and state standards based on it are not world-class math. Our kids start behind and stay behind.
3. Not College-Career Readiness
The primary claim of Common Core has been college or career readiness without remediation in college, but there seems little consensus on how to do this, writes Alyson Klein. Klein explains that readiness is judged by what has been done in high school, not after high school. College readiness “will not mean that students enter college, or enter college without remediation,” says Phillip Lovell of the Alliance for Excellent Education. It is a bad idea to extrapolate beyond known data and say students are ready for college.
In Alyson Klein's article ( Education Week), she quotes Allison Timberlake: "You could have fairly high graduation rates but still pretty high remediation rates when kids go onto to postsecondary studies. And that's a problem, right?" said Allison Timberlake, the state's deputy superintendent for assessment and accountability (Georgia Department of Education).
The schools in the Tucson area have high graduation rates, but, depending on the school district, from 74% to 88% of the students applying at a local community college are placed in remedial math.
Okay, then what was the purpose of implementing Common Core (CC) and state testing? Common Core math standards were interpreted as a version of reform math that failed in the past. Reform math emphasizes group work and minimal guidance methods such as discovery learning, not the mastery of the fundamentals in math, i.e., factual and procedural knowledge. So, what was the purpose of CC? Many of the CC writers are cashing in.
Reference: Alyson Klein, Education Week. https://www.edweek.org/ew/articles/2019/01/23/states-using-post-high-school-readiness-as-essa.html
4. Not Internationally Benchmarked
The second main claim was that the math standards of CC were benchmarked internationally. But, as I pointed out in 2011, it is not the case. The 1st-grade students in Singapore learn much more standard arithmetic than CC 1st-grade standards.
For example, 1st-grade students in Singapore start multiplication, memorize addition facts, learn rules, and practice standard algorithms. By 4th or 5th grade, American students are at least two years behind their peers from top-performing nations.
5. Class Size
Robert Plomin (Blueprint) asserts that class size makes little difference in achievement. "It is widely assumed that children learn more in classrooms containing fewer children." But the effect size is small, only 1% of the variance is class size, says Plomin. In contrast, 60% of the differences in school achievement is genetics. Only 1% is class size.
6. Equity
"An analysis of state ESSA plans finds that many states are prioritizing equity over performance," writes Dr. Elizabeth V. Primas (From The 74). Inclusiveness, for the sake of equity, has created major problems in the classroom. There is no reason to mix low achieving math students with high-achieving math students in the same math class. Also, the one-size-fits-all concept of Common Core and state standards is not fair to most students, including students of color. Furthermore, it is anathema to sort kids for math class according to their achievement.
Indeed, equity often has dominated over achievement in school policy. Thomas Sowell says it is false equity ("the fallacy of fairness"). He writes, "Equalizing downward by lowering those at the top ... is a crazy idea taught in schools of education across the country." Equity overachievement is a liberal ideology in which "achievement is equated to privilege," says Sowell, a black economist.
Note. I volunteer at a K-8 Title-1 school where 85% of the students are of color. Over the years, I have found many children of color who excel in my weekly algebra lessons. This school year, I have two 4th-grade classes. In late February (2019), I started to give algebra lessons to a 2nd-grade class.
7. Reform & Innovation
Most of the reforms today are linked to a business or organization trying to sell its materials and ideas to the schools. It is lucrative for the sellers, but the students do not get any better. Unfortunately, some of the Common Core writers are cashing in, too. Most reforms fail.
Incidentally, almost all the innovations funded by the U.S. Department of Education (a whopping 82%) have failed to improve achievement. Furthermore, costly technology and software in the classroom have not transformed education and, often, is more of a distraction than a help. Larry Cuban, (Stanford Graduate School of Education) writes, "The fact is that many teachers continue to struggle in integrating devices and software into their lessons." Teachers, I think, waste valuable class time and energy attempting to fit technology into lessons as dictated by the school district. Sadly, they do not have much choice in the matter.
Unfortunately, time-on-task for math, reading, history, and science and the mastery of content through practice are less important in today's classrooms to make room for tech use, doubtful reforms, and social-emotional learning (SEL), which is self-esteem, revisited. The idea is that students need high levels of self-esteem before they can learn. It's nonsense, of course.
©2019 - 2020 LT/ThinkAlgebra
Last update: 2-11-19, 2-12-19
For example, the major claims made by the Common Core (CC) people were bogus, such as readiness for college and world-class benchmarks. Beginning in the 1st grade, U.S. students are not learning world-class math. Also, parents may clamor for small class sizes, but the evidence doesn't support it. Remarkably, students don't need to start at the concrete level to get to the abstract level. The 1st-grade equation n + 7 = 12 is already abstract and can be used to solve countless situations that are concrete. Another issue is that equity trumps achievement in the classroom, hurting all children, including children of color. Moreover, costly technology in the school has not transformed education.
Over the years, I have found several precocious children in my algebra classes, children who can deal with abstraction better than others.
1. Abstract
We have been told that children get to the abstract only through concrete experiences and developmental stages (Piaget), so teachers ask kids to use manipulatives such as cubes and images in textbooks to demonstrate math ideas. But, what if we go from abstract to concrete, which is what I do in my elementary school algebra classes?
In 1st grade, I want children to learn the core of a concept, which means that the base of the concept of addition, for example, is not linked to counting cubes or manipulatives, or images in books. Concepts (i.e., ideas) are abstract, so are numbers, operations, equality, rules, and algorithms, too. Students don't need to start at the concrete level to get to the abstract level. Equation 3 + 7 = 10 is already abstract and can be used to solve countless situations that are concrete.
In my Teach Kids Algebra (TKA) program, 1st-grade students used symbols for unknown numbers and ideas and solved simple equations using arithmetic knowledge and reasoning. Ordinary 1st-grade students also built x-y tables and drew pictures of linear functions (i.e., make a graph in Quadrant-1). I created TKA as a reaction against Common Core. I fused algebra to standard arithmetic.
The equations x - 5 = 12 and y = x + x - 2 are abstract. For example, in the second equation, 1st-grade students were asked to find y given several x values, build a table of values, and make a graph of the equation. Find y when x = 1 corresponds to the point (1, 0) on the coordinate plane. Find y when x = 2, which is the point (2, 2). Find y when x = 3, which is the point (3, 4), and so on. Also, if y = 12, then what is x? {using guess-and-check reasoning, x is 7 because 7 + 7 - 2 = 12} Even 1st-grade students follow the algebraic rule for substitution.
2. Not College Prep
What irks me is that many schools offer kids so-called "college-prep" courses in math, but when they apply to a community college, for example, they are placed in noncredit remedial math courses, mostly the equivalents of middle school prealgebra and Algebra-1 and high school Algebra-2. Indeed, beginning in the 1st grade, kids are not learning world-class math. Common Core and state standards based on it are not world-class math. Our kids start behind and stay behind.
3. Not College-Career Readiness
The primary claim of Common Core has been college or career readiness without remediation in college, but there seems little consensus on how to do this, writes Alyson Klein. Klein explains that readiness is judged by what has been done in high school, not after high school. College readiness “will not mean that students enter college, or enter college without remediation,” says Phillip Lovell of the Alliance for Excellent Education. It is a bad idea to extrapolate beyond known data and say students are ready for college.
In Alyson Klein's article ( Education Week), she quotes Allison Timberlake: "You could have fairly high graduation rates but still pretty high remediation rates when kids go onto to postsecondary studies. And that's a problem, right?" said Allison Timberlake, the state's deputy superintendent for assessment and accountability (Georgia Department of Education).
The schools in the Tucson area have high graduation rates, but, depending on the school district, from 74% to 88% of the students applying at a local community college are placed in remedial math.
Okay, then what was the purpose of implementing Common Core (CC) and state testing? Common Core math standards were interpreted as a version of reform math that failed in the past. Reform math emphasizes group work and minimal guidance methods such as discovery learning, not the mastery of the fundamentals in math, i.e., factual and procedural knowledge. So, what was the purpose of CC? Many of the CC writers are cashing in.
Reference: Alyson Klein, Education Week. https://www.edweek.org/ew/articles/2019/01/23/states-using-post-high-school-readiness-as-essa.html
4. Not Internationally Benchmarked
The second main claim was that the math standards of CC were benchmarked internationally. But, as I pointed out in 2011, it is not the case. The 1st-grade students in Singapore learn much more standard arithmetic than CC 1st-grade standards.
For example, 1st-grade students in Singapore start multiplication, memorize addition facts, learn rules, and practice standard algorithms. By 4th or 5th grade, American students are at least two years behind their peers from top-performing nations.
5. Class Size
Robert Plomin (Blueprint) asserts that class size makes little difference in achievement. "It is widely assumed that children learn more in classrooms containing fewer children." But the effect size is small, only 1% of the variance is class size, says Plomin. In contrast, 60% of the differences in school achievement is genetics. Only 1% is class size.
6. Equity
"An analysis of state ESSA plans finds that many states are prioritizing equity over performance," writes Dr. Elizabeth V. Primas (From The 74). Inclusiveness, for the sake of equity, has created major problems in the classroom. There is no reason to mix low achieving math students with high-achieving math students in the same math class. Also, the one-size-fits-all concept of Common Core and state standards is not fair to most students, including students of color. Furthermore, it is anathema to sort kids for math class according to their achievement.
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| Thomas Sowell |
Indeed, equity often has dominated over achievement in school policy. Thomas Sowell says it is false equity ("the fallacy of fairness"). He writes, "Equalizing downward by lowering those at the top ... is a crazy idea taught in schools of education across the country." Equity overachievement is a liberal ideology in which "achievement is equated to privilege," says Sowell, a black economist.
Note. I volunteer at a K-8 Title-1 school where 85% of the students are of color. Over the years, I have found many children of color who excel in my weekly algebra lessons. This school year, I have two 4th-grade classes. In late February (2019), I started to give algebra lessons to a 2nd-grade class.
7. Reform & Innovation
Most of the reforms today are linked to a business or organization trying to sell its materials and ideas to the schools. It is lucrative for the sellers, but the students do not get any better. Unfortunately, some of the Common Core writers are cashing in, too. Most reforms fail.
Incidentally, almost all the innovations funded by the U.S. Department of Education (a whopping 82%) have failed to improve achievement. Furthermore, costly technology and software in the classroom have not transformed education and, often, is more of a distraction than a help. Larry Cuban, (Stanford Graduate School of Education) writes, "The fact is that many teachers continue to struggle in integrating devices and software into their lessons." Teachers, I think, waste valuable class time and energy attempting to fit technology into lessons as dictated by the school district. Sadly, they do not have much choice in the matter.
Unfortunately, time-on-task for math, reading, history, and science and the mastery of content through practice are less important in today's classrooms to make room for tech use, doubtful reforms, and social-emotional learning (SEL), which is self-esteem, revisited. The idea is that students need high levels of self-esteem before they can learn. It's nonsense, of course.
©2019 - 2020 LT/ThinkAlgebra
Last update: 2-11-19, 2-12-19