Falling Down The Rabbit Hole
Alice exclaims, "I never heard of uglification."
Like Alice, we have fallen down the rabbit hole where kids learn the different "branches of Arithmetic--Ambition, Distraction, Uglification, and Derision" as Mock Turtle points out. Mock Turtle is not a teacher; he is a facilitator, which is an important aspect of progressive pedagogy.
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In contrast, Alice explains, "I never heard of uglification." She's not alone! Down the rabbit hole, Alice encountered an alternative, unorthodox, cumbersome, constructivist form of arithmetic--not the straightforward, standard arithmetic that she had memorized, studied, and learned.
Click: China is at the top.
Reform math is upside down and backward; that is, it grossly misinterprets Bloom's taxonomy by jumping to higher-level thinking skills before the fundamentals had been learned in long-term memory. Indeed, today, math education has become bizarre, cluttered, and counterproductive. Reform math makes little sense. Memorization is considered old school. "The prevailing theory is that students must engage in constructing their own knowledge rather than memorizing facts that will only bore them and that they don't really understand," writes Natalie Wexler (The Knowledge GAP). It's the wrong approach. She observes that "skipping the step of building knowledge doesn't work."
It is common sense that if kids don't learn the fundamentals of arithmetic, starting in the 1st grade, then they are blocked from higher-level math (Engelmann).
Aim for Competency!
It is common sense that if kids don't learn the fundamentals of arithmetic, starting in the 1st grade, then they are blocked from higher-level math (Engelmann).
Aim for Competency!
If we want students to become competent in arithmetic and algebra, then they need to be more like ballet dancers, gymnasts, swimmers, violinists, chess players, etc. That is, students need to practice and review so that the fundamentals of math stick in long-term memory (i.e., for automaticity). We don't do that! In fact, what we often do in the classroom is the opposite of the cognitive science of learning.
The Knowledge Gap
The knowledge gap has been known for decades and disregarded. In math, there is a considerable knowledge gap, too, because children don't memorize times tables or learn the mechanics of the standard algorithms like they used to 50 years ago. Hence, calculating skills are weak, and learning is not coherent, systematic, or hierarchical.
Calculating skills are conveyed to concepts or ideas in math. Even though the ideas in math, themselves, are abstract, such as operations like addition or the concept of perimeter, they are learnable even in 1st grade. I know; I did it!
Math Knowledge means math skills. Having math knowledge is the ability to convey the proper math skills to the appropriate problem type and solve the problem by paper calculating. Getting the correct answer has always been critical in applying math, but it has been marginalized in current math teaching, which is a mistake.
The math fact 3 + 8 = 11 is the solution to a vast number of word problems. Because the math fact is abstract, it can be used again and again. That's the power of abstraction. If the child memorizes the 3 + 8 fact, suddenly she has the solution to hundreds of math problems. It's math power! Even though a student may have some idea of perimeters, she cannot calculate a perimeter unless she knows sums. If you can't calculate it, then you don't know it! The memorization of facts is good for novices.
Calculating skills are intrinsic skills for solving problems in math.
Calculating skills are intrinsic skills for solving problems in math.
Education, today, does not work. Math needs a radical break from the status quo, which has been reform math. Teachers should embrace a knowledge approach that is supported by the cognitive science of learning. Also, in math, knowledge is math skills. There is a lack of content in elementary school math. Common Core is a continuation of a content-free approach. It is not world-class. Schools fail to build knowledge, not only in reading but also in math, science, and other subjects.
In addition to calculating and logic, math skills include knowing vocabulary, applying concepts and mechanics, recognizing patterns (e.g., problem types), and grasping the meaning of structure (e.g., 2n means 2 times the number n; 3^4 means the product of 4 copies of 3, which is 3 x 3 x 3 x 3 = 81; ab = ba, and so on). In other words, students must know math stuff in long-term memory to perform math. Furthermore, knowledge is the basis of problem-solving.
You can't solve trig problems without knowing some trig. You can't translate Latin without knowing some Latin. You can't grasp literature without knowing some textual criticism. All fields of study require the memorization of the basics and experience. It's a process of building knowledge. There is no generalized thinking skill. Thinking is domain-specific.
Study hard enough to become Smart enough!
(South Korean Motto)
©2019 - 2010 LT/ThinkAlgebra
