😎Observations, Ideas, and Opinions on Math Education by a Contrarian in 2021, a Divergent View.
Happy Birthday, Cindy! 4-4-21
What has happened to Math Education?
Education has been hijacked by extremists who have a utopian vision of the future (Marxism). Traditional arithmetic has been replaced with liberal reform math that puts our kids behind internationally. The curriculum is dumbed-down, "so everyone can pass--but no one can excel." It is feel-good grade inflation (for no good reason) in K-8 and above, sometimes in college. Charles J. Sykes (Dumbing Down Our Kids) explains, "In the name of equity, fairness, inclusiveness, and self-esteem, standards of excellence are being eroded throughout American education."
I often hear the motto that "All Children Can Learn." Really? What does it mean? Calculus, Advanced Physics? Does it mean that all children have the same capabilities to learn what schools teach? Like so many liberal ideas, it sounds great, but it is wrong! 4/7-13/21
"Facts seem to have become irrelevant for all too many people, who rely instead on visions and rhetoric. The entitlement mentality has eroded the once common belief that you earned things, including respect, instead of being given them. One of the trends that can become part of a perfect storm of disasters for American society has been a decades-long dumbing down of education, producing a citizenry poorly equipped to see through the political rhetoric and even more poorly supplied with facts and the ability to analyze opposing arguments." (Quotes: Thomas Sowell)
I see smart kids who don't know much. Memorization and practice for mastery have drifted out of liberal classrooms over the decades. Thus, students don't know much science, history, mathematics, grammar, or vocabulary, and so on. Still, they know how to text, snap selfies, use social media like Instagram and YouTube, and Google. Often, students don't have textbooks to take home to review or study. Parents are left in the dark. What does a "pass" grade mean--a D- or an A+? Who knows? 04-7-8-21
In my opinion, equalizing down and academic rigor are inverses. The more equalization, the less academic progress. (I could be wrong, but I don't think I am.) You can't balance the two. By equalizing down in schooling, you are asking for lower standards. If this is true, then equalizing down is a fallacy of fairness, as Thomas Sowell explains.
"Equal opportunities do not result in equal outcomes."
Nevertheless, opportunities should be available for anyone who wants them. But, schools should not water down the curriculum to accommodate ill-prepared students, let's say for Algebra. Remedial courses might be needed if students stumble over fractions and long division. The K-12 math curriculum needs to be upgraded and grade inflation erased. "Do you want to build a snowman?"
Can the students figure out these equations that I give to 2nd and 3rd-grade students? No calculators!
Success Maker [math] software is built on a framework of fun and games. [Clifford Stoll would have a fit!] The company (Pearson Education) says Success Maker software, which is used to supplement a school’s math program, has 40 years of success, but I never heard of it. Furthermore, Pearson backs its claims with faulty research (time-on-task was not controlled, schools volunteered for the study, etc.), unproved education theory, and anecdotal evidence--not hard facts. Its most recent study ("Pearson Success Maker Math Efficacy Study," published 9/15/2010), which is used to promote the math software as effective, was funded by Pearson.
Moreover, the standardized math test (GMADE) used to make comparisons is from Pearson also. Gee, a Pearson product study (Success Maker), is funded by Pearson. Publishers often fund studies to show that their math programs, textbooks, and software work in the classroom. They are in sales! (Note: I wrote about Success Maker in 2010.)
➜ Testing and accountability are still the national educational strategies and more critical than reforming and upgrading curriculum and instruction for all students. (More important, Really?) Test-based accountability has failed. The teacher functions as a facilitator while children are expected to teach one another in group work and find or use multiple ways to solve a problem without a strong background in factual and procedural knowledge. Crazy? Diane Ravitch explains that "kids must have background knowledge, not just strategies."
Kids who lack strong standard computational skills, one of the five priorities in elementary school mathematics (W. Stephen Wilson), are not prepared to succeed in high school and college math courses. Instead, they end up in remedial high-school-level algebra classes at community colleges. Test-prep games the system. Grades are inflated on report cards. What else? Common Core is not world-class. What a mess! Adding to it is remote/hybrid. (Note: By essential computational skills, I mean the standard algorithms. By instruction, I mean explicit teaching with worked examples--not group work, discovery or project-based learning, etc.)
Note: Beliefs, fads, practices, and policies in our schools are often based on anecdotal evidence or ideology rather than scientific evidence. Many so-called "studies" do not account for time-on-task or other variables, which are hard to control. Still, many studies hyped correlation as causation, which it is not. Also, just because an association is "statistically significant" doesn't mean it works in the classroom or is meaningful. Lastly, any evaluation of a class, school, or district "based solely on test scores will present a dangerously inaccurate picture," writes Charles Wheelan. But isn't this what educators and leaders do all the time: rank schools, students, programs, etc.? "School spending" and "parental education" are other variables that are difficult to control. Often a study or research in education cannot be replicated. 4-6-21
➜ Not all students have an equal opportunity to learn arithmetic and algebra, reports William Schmidt ("Equality of Educational Opportunity, Myth or Reality" from American Educator/Winter 2010-2011). Schmidt links a student's social-economic status to the opportunity to learn. This is a correlation, but it should not imply causation. Schmidt points out, "Not only do we have great variability [of math content] across districts but by international standards, our eighth-grade students are exposed to sixth-grade content." He is correct. But, I think Schmidt assumes that children of lower social-economic status have lower IQs. I disagree. He also assumes that all students have the same ability to learn content that schools teach when, clearly, ability widely varies.
Robert Plomin writes, "Performance on tests of school achievement is 60 percent heritable on average." The other 40% is environmental (nurture). Plomin's percentages are not deterministic; they are propensities. He explains, "The most important point about equality of opportunity from a genetic perspective is that equality of opportunity does not translate to equality of outcome." It's DNA that makes us who we are. (Quotes: Robert Plomin: blueprint, 2018, MIT)
Equal does not mean identical. (Thomas Sowell)
I know average Title-1 minority kids, ages 6 to 10, in my algebra program (TKA) who are stellar in algebra. The difference often boils down to attitude, persistence, and effort (nurture), qualities I stress. Some students listen better, work harder, study more, and don't give up as fast as others.
The radical reformists claim that the only way to level the playing field is to lower the bar so that similar outcomes are possible for all children. But equalizing downward is not equity, writes Thomas Sowell. It's hardly fair. It's a crazy idea! Moreover, chasing after "gap closing" should not be an educational goal, writes Sandra Stotsky (The Roots of Low Achievement, 2019).
Note: Most students, if not all, should have the opportunity to take a rigorous pre-algebra course in the 7th grade to prepare for Algebra-1 in 8th grade, but many would fail because they were not taught the prerequisites in K-6. Common-Core-based reform math is not world-class math and doesn't get students ready for prealgebra in 7th grade. Equal opportunity to learn is not the same as "equal learning." In short, as Plomin suggests, equal opportunity doesn't produce equal outcomes.
➜ Math, starting with counting and standard arithmetic, is hierarchical but often not taught that way. Traditional or standard arithmetic has been replaced by a corrupt version of Common-Core-related reform math, an aberration of standard arithmetic. Indeed, math education has been usurped by liberal extremists and a radical agenda of forcing equality of outcomes (i.e., sameness) on all students. It's not equity! Leading the equalizing crusade is reformist Jo Boaler and her cohort.
➜ The reform math idea of "gaining conceptual understanding and learning applications in math [without] mastering procedures" has been promoted through Common Core and influential reformists like Jo Boaler and others. What good is learning about perimeters if you can't calculate them? For example, Boaler brags that she never memoried the multiplication table, yet she acquired a Ph.D. in math education, which is far different from a degree in mathematics. She wants to kick out standard algorithms because they" interfere with the students' progress of discovering the properties of mathematical objects on their own." Memorizing the multiplication table and finding exact answers are not needed, asserts Boaler. Moreover, kids should work in groups, engage in discovery lessons, think critically, and use calculators. Frankly, I am not sure how students can engage in critical thinking when they lack factual and procedural knowledge that enables it.
➜ Tom Loveless observes that Common Core reform math failed to boost student achievement in math and reading. Math is hierarchical. Each objective has a sequence of prerequisite objectives (Robert Gagne) that must be learned before moving to the next. Gagne (Science--A Process Approach or SAPA, 1967) identified the sequences and applied performance-based learning objectives (i.e., behavioral objectives) with measurable or performance verbs of action. In short, learning objectives must be quantifiable. Gagne did not use standards. He wrote tangible, clear, and specific objectives that were observable and measurable (i.e., performance-based). Note: Understanding, creativity, appreciation, problem-solving, and other vague terms cannot be behavioral objectives because they are difficult to measure accurately. What does it mean to say that a student has acquired problem-solving skills? How do you quantify that? It's not objective; it's subjective. It's a guess! On the other hand, the fundamentals of arithmetic are observable and measurable. The arithmetic objectives follow a sequence of prerequisites and are clear, specific, and quantifiable. 3-31-21
Of the six science processes taught in 1st-grade SAPA, four were math or math-related: Using Numbers (arithmetic), Measuring (metric system), Communicating (graphing), and Using Space/Time Relationships (geometry). SAPA emphasized the importance of math in science. The lessons were thoroughly field-tested and revised often before becoming commercial-ready to schools through Xerox. 3-30-21
➜ Below: Science--A Process Approach (SAPA K-6, 1967): Part C is 2nd-grade SAPA. I used a similar hierarchical sequence for integers in my 1st-grade algebra program: Teach Kids Algebra, TKA, 2011. It worked well. 3-30-21
➜ LT: If you can't calculate it, then you don't know it. "The teacher's least favorite thing to hear from a student is 'I get the concept, but I couldn't do the problems.' The student doesn't know that this is shorthand for 'I don't get the concept.' The ideas of mathematics can sound abstract, but they make sense only in reference to concrete computations," writes Jordan Ellenberg (The Power of Mathematical Thinking, 2014). When I tutored Algebra2 and precalc, some students would say they understand the idea (concept) but had difficulty calculating the answers. To which I would say, "If you can't calculate it, you don't know it. Let's review the steps." Ellenberg points out, "It is pretty hard to understand mathematics without doing some mathematics." G. Polya (How to Solve It, 1956) explains, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics." The stress is on the doing, the performance! You won't understand perimeters if you can't do sums to calculate them. 3-30-21
➜ Common Core reform math, often called today's New Math, "is a nightmare for those of us who know full well how to solve most basic math problems, just not the crazy-making, convoluted way it's being taught today," writes Jennifer Jolly (USA Today). Parents are upset because they cannot help their K-8 children with math. Reform math is absurd. Its alternative algorithms taught for equity and understanding at the expense of mastering standard algorithms, which are necessary for problem-solving in math, are a radical approach that doesn't work. The progressive extremists say that math has changed. No, it hasn't. Indeed, 7 + 5 still equals 12, a basic fact that should be memorized along with many others in 1st grade. The standard addition algorithm is organized around place value and supported by single-digit sums. The single-digit sums like 6 + 8 = 14 are easily understood on a simple 0-20 number line. Indeed, the single-digit sums (+) and products (x) should be sanctified, not marginalized.
Indeed, a fact such as 4 + 7 = 11 is valid and useful in hundreds of concrete situations and allows us to solve real problems. Also, if 67 + 85 = 152, is true, then 152 - 67 = 85 and 152 - 85 = 67 are also true by the basic rerlatiionship between addition and subtraction. Using smaller numbers, if 4 + 7 = 11 is valid and true, then 11 - 7 = 4 and 11 - 4 = 7 are also valid and true. 3-28-21
The Standard Algorithm Is Primary!
The standard algorithm is built on place value and simplifies regrouping. The idea behind the algorithm is to add ones to ones and tens to tens, etc., because this is how the number system works. The grid above shows that when ones are added to ones, there are 12ones, which is immediately broken into 1ten 2ones. (See the 12.) Record the 2ones in the ones column and "carry" the 1ten over to the tens column (carry mark). Then, add tens to tens for a total of 3tens. The answer is 32, which means 3tens + 2 ones. The standard algorithm is economical and helps automate fundamentals. It is what we want first graders to do. Students are novices; they do not need a perfect understanding of regrouping or place value to use the grid. Understanding grows slowly. At first, only a functional understanding is necessary. The standard algorithm is a focused goal and should be presented at the beginning of the first grade. 3-30-21
➜ In my view, if you can't do it (i.e., calculate it), then you don't know it. You cannot apply something you don't know well. Perimeter is an example. If a student cannot do sums, then the student cannot calculate the perimeters of polygons. Knowing the idea of perimeter and calculating perimeters correctly are not the same. Finding the perimeters of squares, rectangles, and other shapes is a 1st-grade application of sums. The addends must be the same units, such as centimeters.
|1st-Grade Plotting Points and Finding Perimeters of Shapes|
Teach Kids Algebra (TKA) Lesson 7, May 2011
Title-1 City K-8 school.
➜ The problem has been deeply rooted in our curriculum and instructional methods for decades. It will be hard for teachers to change from discovery/inquiry or project-based group work to knowledge-based schooling and explicit explaining of carefully selected worked examples. Ian Stewart, a mathematician, points out, "Mathematics happens to require rather a lot of basic knowledge and technique." Marginalizing the standard procedures (i.e., technique) has been the wrong approach. And implementing constructivist "minimal guidance methods" of instruction has paved the way for minimal learning, explain Kirschner, Sweller, and Clark ("Why Minimal Guidance During Instruction Does Not Work...", 2006). 3-23-21
➜ Minimal Guidance = Minimal Learning!
My algebra program for grades 1 to 4 is called
- Understanding does not produce mastery; practice does!
- You cannot apply something you don't know well.
- If you can't calculate it, then you don't know it.
- You know nothing until you have practiced.
2nd or 3rd-grade TKA
12 + 4 + ◻ = 12
What is box to make a true equation?
-4 (negative 4)
4 and -4 are opposites (inverses) and add to zero, thus 12 + 0 = 12.
Note: Adding negative 4 is the same as subtracting 4. To subtract 4, add its opposite. The idea of inverse is fundamental in algebra.
0 and 1 are identities for addition and multiplication, respectively.
12 + 0 = 12 and
12 x 1 = 12.
In 1st Grade Arithmetic
Add 1 to get the next natural number. (Building Numbers)
6 + 1 = 7
7 + 1 = 8
11 + 1 = 12
Add 0 (Inverse in Addition)
6 + 0 = 6
7 + 0 = 7
11 + 0 = 11
Can the students figure out these equations that I give to 2nd and 3rd-grade students? No calculators!
Answers to the equations:
-4, 1/4, 1, 13, 165/28, 1.75 miles
STEM Math for Elementary School Students by LT: Teach Kids Algebra Program
- Addition: It must be the integer -4 to make 0, the identity.
- Multiplication: It must be a fraction 1/4 to make 1, the identity.
- Any expression to the zero power is 1 by definition.
- The exponent 1/2 means the positive square root of 169 or 13.
- The fraction problem works out to be 165/28, which is in the lowest terms. Students must have lots of fraction practice and know that division is multiplying by the reciprocal.
- Units must be consistent, so 30 minutes is 1/2 of an hour, thus d = rt or d = 3.5 mph x 30/60 or 1/2 of an hour. The multiplication is simple: half of 3.5 is 1.75 miles as hours cancel.
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