It is common sense that if kids don't learn the fundamentals of arithmetic, then they are blocked from higher-level math (Engelmann). The fundamentals start in 1st grade with the meaning of numbers by place value (e.g., 13 is 10 + 3), rules: add zero (3 + 0 = 3), add one (5 + 1 = 6), and commutativity: 3 + 4 = 4 + 3), memorizing the number facts, and learning the mechanics of the standard algorithms, first. Incidentally, the standard addition algorithm is the best model for place value and should be taught in the 1st-marking period of 1st grade.
Content-free is the wrong approach!
Teachers are instructed to teach higher-level thinking skills (i.e., critical thinking or problem-solving) before kids had mastered the fundamentals that support content thinking. For example, learning basic arithmetic content starts with memorizing the number facts, place value, and practicing the mechanics of the standard algorithms for automaticity. Students should practice and review the basics to make them stick in long-term memory for use in problem-solving. Applying content knowledge is the next significant step. Recognizing problem types is critical in arithmetic and algebra.
Robert Pondiscio (Fordham Institute) writes, "Hirsch, myself, and many others have long lamented the content-free, skills-driven, curriculum-agnostic brand of schooling that has come to dominate American primary education. This state of affairs is due in part to mistaken notions about how children learn." I do not blame teachers; they are doing what they had been trained to do, but I question "the teaching" itself. Critical thinking (aka problem-solving) without content is empty. Thought is domain-specific. You cannot solve a trig problem without knowing some trig or translate Latin without knowing some Latin. In math, efficient calculating skills are an intrinsic part of problem-solving. Passing from one grade to the next does not mean the student is competent at grade-level arithmetic. More likely than not, most students are below grade level (NAEP math). It is common sense that if kids don't learn the fundamentals of arithmetic, then they are blocked from higher-level math (Engelmann). So what has happened to common sense?
Thomas Sowell (Discrimination and Disparities, 2019) points out, "Education is an area in which differences in values and behavior play havoc with policies based on an assumption of sameness. There is no reason whatever to assume that education is valued equally by all individuals or groups."
Some children do not value education or study as much as others. In contrast to Asian families, education is not the highest priority in some American families.
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| Focus on Content Knowledge |
Knowledge has always been the best preparation for the future. You can't apply something that you do not know well in long-term memory. Thought without content knowledge is empty. Indeed, strong academic skills, a work ethic, persistence, postsecondary education, and some "chance" are needed to prepare students for future jobs. Many of today's careers use math.
Don't Underestimate the Role of Chance
Much of what happens is random. Opportunities can arise suddenly. Thus, in many cases, being at the right place at the right time with the right set of skills can convey opportunities that others may not have or value. It's persistence and chance. We cannot equalize opportunities or balance outcomes. The real world is not Lake Wobegon, in which all the children are above average.
Leonard Mlodinow (How Randomness Rules Our Lives) writes, "It might seem daunting to think that effort and chance, as much as innate talent, are what count. Our degree of effort [persistence] is up to us." Mlodinow points out that we underestimate the effects of randomness for "successes and failures" in life. Furthermore, he writes, "Ability does not guarantee achievement, nor is achievement proportional to ability."
Kids today, have opportunities in government schools that I never had when I was a student. But, many do not value learning!
Observation. Some students coming into the 7th grade still don't know the times tables for instant recall and long division, which are skills I used to teach in the 3rd grade for mastery. The primary reason kids don't know arithmetic well enough is the teaching. For decades, kids have been taught reform math, not conventional arithmetic, and it shows on national and international tests. Recently, a 2nd-grade teacher complained to me that kids coming into 2nd grade know absolutely nothing.
Ashley Berner (Johns Hopkins) points out, "Numerous recent studies suggest that switching from a low- to a high-quality textbook can boost student achievement more than other, more popular, interventions, such as expanding pre-school programs, decreasing class sizes, or offering merit pay to teachers. It is also cost-effective."
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| Will our kids learn enough math to get into the STEM and math-related fields? Probably Not! |
For decades, American kids have stumbled over simple arithmetic, which is the foundation for algebra and higher-level mathematics. The widespread progressive math reforms do not work. The curriculum is not world-class, and the progressive methods of teaching are ineffective (inferior). If the children aren't learning, then there is something wrong with the teaching, that is, with the curriculum and the instructional methods.
We should prepare more students for a solid precalculus course in high school. Also, Algebra-1 is a middle school course for typical students who are prepared. Likewise, calculus is a high school course for average students who are prepared. But, the reform math curriculum and progressive instructional methods, including teaching the state test, have driven underachievement, not preparedness. Being proficient on the state test does not mean your child is college-ready or knows basic arithmetic.
Progressive reformers hide behind the concept of sameness (i.e., everyone gets the same instruction, regardless of achievement level), which is another inane idea. Sameness is an illusion. Thomas Sowell (Discrimination and Disparities, 2019) points out, "Education is an area in which differences in values and behavior play havoc with policies based on an assumption of sameness. There is no reason whatever to assume that education is valued equally by all individuals or groups."
Progressive ideas litter the education playground.
For example, an "education" professor claims that teaching kids math discriminates against children of color. How stupid!
Comment: "Why would anyone think that minorities would be less able to do math than anyone else?"
Rochelle Gutierrez, an education professor, not a mathematician, claims that teaching kids algebra and geometry discriminates against students of color and perpetuates white "unearned privilege." She is dead wrong! Gutierrez is one of a host of left-leaning-ed-radicals who assert that achievement is "privilege." It's not! Her message is clear: Don't Achieve. If you achieve, it is unearned, which is a toxic message for both minorities and whites who are trying to better themselves.
Contrary to Gutierrez and others like her, achievement in math is earned through hard work, practice/review, and study. Indeed, students of all colors need higher-level math courses to expand their career choices later on.
Note. How do we prepare children for the jobs of the future?
We prepare students the same way we have always trained students for the future. Kids need strong academic skills (math, science, reading/vocabulary, writing/language) and a work ethic to get a good job now and in the future. Most students will need some form of postsecondary education or training. Moreover, students should take a lot of math and science in schools, such as precalculus and algebra-based physics.
For decades, K-5 schools have been weak in both math and science. It carries over through middle school and high school. Unlike Asian nations, we don't push kids into math, and it shows: 54% of Singapore 8th graders scored at the Advanced Math Level compared to only 10% of American 8th graders (TIMSS). If our kids are not good at math, then we made them that way. Moreover, many of our students are not linking the learning of math to future careers and employment. There is a multitude of jobs that use math.
The bottom line is that all students need to upgrade their math skills to move forward.
The progressive educationists do not take learning math seriously enough. Math education has been beset with problems for decades. For example, State Math standards, which are based primarily on the Common Core, are significantly below world-class standards. So, why were they adopted? Consequently, by the time American kids reach the 4th grade or 5th grade, they are about two years behind their peers from top-performing nations.
The math gap starts in the 1st grade and grows through the grade levels. For example, Singapore 1st-grade students learn much more basic arithmetic than American children, including multiplication and formal algorithms to add and subtract (i.e., standard algorithms). Also, Singapore 1st-grade students memorize math facts and drill for developing skill.
With some pivotal changes, we could do the same. We could teach for the mastery of fundamentals using explicit teaching and a world-class curriculum, starting in the 1st grade, but we don't. Parents should take the initiative and teach basic arithmetic to their children at home, but will they? Also, the policy of mixing low-achieving math students with high-achieving math students in the same math class has been a recipe for mediocrity. Thomas Sowell explains that "equalizing downward by lowering those at the top is a fallacy of fairness."
What many teachers don't get is that mathematics is cumulative, starting with arithmetic: one idea builds on another. You can't teach math like you teach social studies.
In math, the learning of future lessons depends on the mastering of previous lessons: the prerequisites (Gagne). Learning is what students remember later on, not just for a test. Children are not learning basic arithmetic because it is not being taught for mastery. It's the teaching, as the late Zig Engelmann had said, repeatedly. The widespread math reforms have not worked the children aren't learning, then there is something wrong with the teaching, that is, the curriculum and the instructional methods. Some valuable content isn't taught because it is not on the state test.
But, progressive educationists don't see it that way. They blame permissive parenting, societal ills (poverty, drugs), and insufficient funding. I heard the same arguments 50 years ago. Nothing has changed! Also, educationists claim that higher pay, smaller class size, more group work, and technology-technology-technology (laptops for all, etc.) would magically fix the problem.
Click Use Math
"Everyone has asked themselves: When will I use math? Believe it or not, hundreds of careers use skills learned in high school math on a daily basis." But, learning high school math well (through precalculus) depends on mastering K-8 arithmetic, geometry, and algebra, starting with 1st-grade arithmetic. Students must know the content, but many do not.
19th-Century? How many middle school students, high school students, or adults?
19th-Century 4th-Grade Basic Arithmetic in America
1. Find the interest of $60 for 4 months, at 5 percent.
2. If 12 peaches are worth 84 apples, and 8 apples are worth 24 plums, how many plums shall I have for 5 peaches?
(Source: Ray's New Intellectual Arithmetic, 1877, which combined 3rd+4th-grade arithmetic into one compact 140-page book.)
Sometimes, learning arithmetic, such as the multiplication table, is not much fun. Children with weak math skills have limited career opportunities later on.
Also, read the Future.
Knowledge has always been the best preparation for the future, no matter the epoch.
Last update: 7-22-19, 7-29-19, 7-31-19, 8-11-19
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