When I was a kid, my life was mathematics. When I was little, my mom would hand me sheets of addition and subtraction problems that did not involve carrying or borrowing, and I would fill them out for her, and she would give me more. That’s my earliest memory. Second earliest is the procession for President Kennedy after his death. Third earliest might be my Dad changing my brother’s diaper. My mother once said, “You must have remembered it because of uniqueness.”
Anyway, when I was a kid, before I went into first grade, my mom taught me multiplication and division. My Dad taught me some heuristic rules around percentages.
So when I came into first grade at age six, I was shocked that no one else could do math. I was good enough with math and reading that the special education teacher took note of me, and helped give me my unusual education 1967-1969. Though I was in a normal classroom, I was a “group of one.” I had my own special reading and math books. The teachers pushed a variety of enrichment materials to me, but it was like subpar homeschooling. I was on my own, and no one would correct my work. For a little kid, I was pretty motivated, but it would have helped a lot to have more adult interaction.
SRA helped on the reading side, and there were later SRA attempts at math though I don’t think that did much.
In third grade, they gave me a programmed instruction curriculum in math, in addition to the ordinary class. At some points, I acted as a tutor to other students. The programmed instruction was modeled after the “new math” fad. I could get it, but at the time, I realized that I was so different from my peers, that I knew that if I could get it, that did not mean that others my age could get it.
Then in fourth grade, they mainstreamed me. I spent time playing around with how to do square and cube roots by hand. Tedious, but not that hard to do.
In fifth grade, my Father brought me an algebra book that used programmed instruction. I puzzled over it, and didn’t get it until I talked to an older friend about it who told that “x’ is a number that we do not know, but are trying to calculate.
There’s more to this story, but I will drop it, lest I bore you…
What I experienced as a child affected me. I could see the abstraction of math while young, and it amazed me. But now look through my eyes as I find out that I am unusual. There is a normal track for math, and a normal way to teach it. As I tried to tutor my classmates, I realized math was not intuitive for almost everyone else as it was for me.
I became a good math tutor. Parents would hire me , and ask me what my rates were. I don’t know how I thought of this, but I said, “Five dollars per sitting. A sitting could be five minutes or two hours. If I lose their attention span the sitting ends.” That motivated the parents to motivate the kid. After one short sitting, future sittings got longer.
As an adult, I married a math teacher. She admits that I am the better with math, and that I often come up with creative ways to teach concepts that she could not. She is still quite good with teaching math such that all of our biological and adopted children made progress in their percentile scores in math and other topics as they grew.
I think that I know math, and how to teach math. I have done it while young, and older with my older children. I have never taken a course in education, and thus my views of pedagogy have never been sabotaged by what is taught in most colleges regarding teaching children.
Some may think this assessment too harsh, but remember, we had the “New Math” in the Sixties. It was a disaster. For me, a math prodigy, thinking about math through the lens of set theory, it was challenging and interesting. To most students, it was deadening.
So now we have the evil “Common Core Math Standards.” [CCMS] When I was a kid we joked about Communist plots to destroy America. Well, I think the Communists are pretty weak in general — they don’t understand the nature of man. But here we are trying to make kids try to make adult judgments regarding math. That’s just plain stupid, because it doesn’t get the way children develop.
The Holy Grail of Critical Thinking
I don’t think critical thinking can be taught. If you are smart enough, you will think critically. If not, no.
I say this as one where my wife and I have homeschooled our eight children for 18 years, and as my children get older they disagree with us to varying degrees. We taught them well. Sadly, some disagree with our premises. But, they are all smart and the seven that have gone through standardized testing have all shown significant progress, moving 25% or more in the percentile rankings from elementary school to high school. My wife teaches very well, and I support her. Please also note that five of the children were adopted, and the same effects happened with them.
But the CCMS flips things on its head. Children need to learn facts. They can absorb facts because they are easy for the young to absorb. Drill on math facts is a very good thing because it eliminates a hurdle to learning more in math. Once you know the basics, the mind is capable of absorbing more abstract reasoning.
It is the opposite of what the experts say. Math should focus on the concrete with young children, and as they get wiser, on to things that are more abstract. They should not begin with abstraction, and try to move to the concrete.
Think about it for a moment: would you rather hire a guy who understood the basics of your business, or hire a guy who had a theory about your business, but did not understand the basics? You would hire the former if you were smart.
Understanding the basics is important, and sadly, we have gotten away from it in the last 60+ years in math. We did much better in the past, and we paid teachers less in real terms back then. The colleges that teach teachers should be dismantled, and teacher accreditation should be eliminated, because there is