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.
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