Welcome to the mad, mad world of math education. 😎
We shall miss you, Rush, 70, 2-17-21. God-speed!
It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness... Charles Dickens, A Tale of Two Cities.
Reflections2 is continued on Inkings1.
1877: First Grade in 1877 was far different from 1st Grade today. In 1877, Kids learned to read and master the classic curriculum of arithmetic. "In America's one-room schoolhouses, Ray's Arithmetic was used alongside the McGuffey Readers." 2-19-21
Ray's Primary Arithmetic (© 1877) for Grades 1 and 2 was a tiny 4.5" x 7.25" book that had 94 pages and 89 Lessons. Primary Arithmetic "covers all four basic functions: addition, subtraction, multiplication, & division in single digits with word and money problems." Below is a typical addition word problem from Lesson XXIII. Students had to be able to read the word problems and use arithmetic to answer the questions. Also, there are many drills in the 1st-2nd grade compact book. Ray's Primary Arithmetic was arithmetic, not reform math. 2-19-21
1st Grade: Ray's Primary Arithmetic, Lesson XXIII
Scale Up (?): I do not think my algebra program for little kids can scale up to other schools. It is more complicated than that. Algebra has deep roots in basic arithmetic, not reform math. Progressive teachers in K-5 find it challenging to change from reform math ideology to prioritizing basic arithmetic, which involves memorizing math facts, learning rules, practicing standard algorithms, and writing and solving equations, all for retention in long-term memory via drills and flashcards. In short, just because something works in one school doesn't mean it will work in other schools. Teachers are different. Still, educators tend to chase after any fad that comes along. 2-18-21
Many schools are adopting so-called "Culturally Responsive Teaching and Leading Standards." I guess teaching children the fundamentals is no longer the goal. How will they get a job later if they can't read, write, or do arithmetic well?
Woke schools claim that getting the "right answer" or "showing your work" to get the right answer is "evidence of white supremacy." How stupid! The algebra program I present to 1st-grade through 4th-grade at a Title-1 urban school, which is 90% minorities, requires students to get the right answer and show how they got the correct answer. Mathematics is based on facts, not woke opinion. I think most woke teachers hate math. Learning arithmetic well is hard work. Memorization and practice have fallen out of favor in woke classrooms. 2-18-21
Progressive educators are wrong! Critical thinking skills are not independent of domain knowledge. Students who study math should develop a toolkit of facts, procedures, rules, and cases in long-term memory. These fundamentals are the knowledge required for critical thinking in math (aka problem-solving). There are no shortcuts. Moreover, in math, problem-solving (deductive, based on factual statements), no one disputes 0 + 1 = 1 or 1 + 1 = 2, is far different from problem-solving in science (observation/inference process). The interpretation of measurements can change over time with new observations. Today, all matter is made of elementary particles: electrons, photons, quarks, and gluons. This idea of fundamental particles is different from saying that all matter is made of atoms.
Frank Wilczek writes, "The basic laws of physics are universal. They hold everywhere and for all times." I wish that were true in education. In my opinion, the fundamentals of reading, writing, and arithmetic are not universal in our schools. After 8-years of Common Core reform math, only 24% of 12th graders are proficient in math (2019 NAEP), not a favorable statistic. Based on national and international tests, I have concluded that learning the basics has not been a top priority in progressive schooling. If it was, then scores would match the math scores of top-performing nations. Given the amount of money spent on K-12 education in the United States, we are not the best or close to being the best. For decades, we have imported talent from the Asian nations because not enough had developed here. Furthermore, many students who enroll in community colleges are typically placed in remedial math (high school algebra). Students who have trouble with algebra are the same students who had difficulty with basic arithmetic, especially fractions, ratio/proportion, and percentages. Unfortunately, many have not mastered the x-table and have been weaned on calculators since elementary school. 2-18-21
Remote and its variants have been costly ($$$$$) and plagued with tech problems. Remote is inferior to in-person classroom teaching. Does remote learning hurt children's health? Many pediatricians think so. Also, in-person safety measures for opening schools are "ludicrous," says Dr. Scott Atlas. "We are off the rails." Atlas explains, "We have children, young children, wearing masks, being separated, thinking they are an infection vector for everyone and that everyone is a danger to them ... It is almost insane." 2-16-21
It is an age of foolishness. (Charles Dickens) What are we doing to our children? Also, there is "gap chasing" and a "fallacy of fairness." Still, I am optimistic! Things will get better.
The achievement gap hasn't closed. Are we chasing after the wrong goal? Sandra Stotsky thinks so because reforms have not worked. (Stotsky, The Roots of Low Achievement, 2019) For example, achievement in math has stagnated ever since states adopted Common Core in 2011. Stotsky observes, "They [the teachers] may teach less content to higher achievers to equalize achievement between higher and low achievers or ignore what college teaching faculty say college readiness means." Thomas Sowell points out that "equalizing ... by lowering those at the top ... is a fallacy of fairness." Lowering the bar has not been an answer.
In the real world, academic ability widely varies, so all students can't possibly attain an identical level of achievement. Even with the same inputs, the outcomes will differ. Also, educators have not figured out how to change low achievers to high achievers. It must be magic! Today, the idea of equity is to keep all students at the same level of achievement, which is wrong. No student gets ahead. It is low expectations in the guise of equity! But, improving the curriculum for all students is a start in the right direction, as Stotsky points out.
The content that high achievers learn should be far different from the content low achievers learn. Schools should accelerate the best students, not hold them back, but acceleration seldom happens at the K-8 level. Why are elementary school children in the talented and gifted programs learning the same grade-level math as the regular kids? (Equity?) In teaching algebra to K-4, I found many minority students who achieved at high levels. We spend too much money on the bottom students who have the limited academic ability and not nearly enough on the other students. We need to level the funds. 2-15-21
|Kids stumble over simple arithmetic. |
Multiplication facts and standard algorithms should be practiced and learned no later than 3rd grade to prepare for higher-level math. (Model: Alyssa)
Learning Math is Hard Work
(It's not always fun until you get good at it through practice-practice-practice and review-review-review).
In math, flashcards work because they force you to recall, which takes cognitive effort. "The least-fun part of effective learning is that it's hard." You have to force yourself to recall a fact or procedure in arithmetic or algebra. Your mind is lazy and doesn't want to think! You must force yourself to think and remember. "To learn something is to remember it," writes Mark McDaniel (Make It Stick, 2014). Also, if nothing has changed in long-term memory, then you haven't learned anything. Kirschner, Sweller & Clark ("Why Minimal Guidance During Instruction Does Not Work") observe, "Learning is defined as a change in long-term memory." Students should quiz themselves at home to learn. Parents should help younger children. Likewise, teachers should frequently quiz students on new content and basic factual and procedural knowledge needed to do the math. If a student cannot recall how to apply the Pythagorean theorem, solve a proportion using the cross-product property of proportions, or use the distributive property, the student hasn't learned it. If a student cannot instantly recall 8 x 7 = 56, they haven't learned it. Indeed, good math students are a product of training at school and home.
In contrast, one parent expressed to me that the AP courses were a bunch of junk for her child. Also, the quality of AP teaching often varies. AP is overhyped and pushed by the College Board. Still, there is a substantial difference between AP calculus in high school and the calculus at the University of Texas for STEM students. Richard Rusczyk, the founder of the Art of Problem Solving (AoPS), writes that AP calculus is for average high school students who are prepared. However, the "AoPS Calculus text is written to challenge students at a much deeper level than a traditional high school or first-year college calculus course." In short, AoPS Calculus is for truly advanced high school students, not average students.