Maybe one problem with American education is that some teachers (not all) don’t know enough academic material themselves. Thus, they are more comfortable with silly projects and crafts, because real academic work would be too much of a stretch for their own abilities.

Check out an amazing article by Montclair’s Patricia Clark Kenschaft, titled, “Racial Equality Requires Teaching Elementary School Teachers More Mathematics.” In it, she describes her experiences conducting math workshops for elementary teachers — and finding that they are “appallingly ignorant,”and that they can’t even calculate the area of a rectangle or understand what an average is.

What I’m going to quote is lengthy, but well worth reading.

Like most Americans, I found it difficult to believe how poorly prepared mathematically they are. They are well chosen. They are kind, diligent, and smart, qualities that nobody can teach. They have been failed mathematically by our system. They need to be taught. I have found them eager and quick to learn—and

appallingly ignorant of the most basic mathematics. . . .The teachers are eager and able to learn. I vividly remember one summer class when I taught why the multiplication algorithm works for two-digit numbers using base ten blocks. I have no difficulty doing this with third graders, but this particular class was all elementary school teachers. At the end of the half hour, one third-grade teacher raised her hand. “Why wasn’t I told this secret before?” she demanded. It was one of those rare speechless moments for Pat Kenschaft. In the quiet that ensued, the teacher stood up.

“Did you know this secret before?” she asked the person nearest her. She shook her head. “Did you know this secret before?” the inquirer persisted, walking around the class. “Did you know this secret before?” she kept asking. Everyone shook her or his head. She whirled around and looked at me with fury in her eyes. “Why wasn’t I taught this before? I’ve been teaching third grade for thirty years. If I had been taught this thirty years ago, I could have been such a better teacher!!!”

. . .

The understanding of the area of a rectangle and its relationship to multiplication underlies an understanding not only of the multiplication algorithm but also of the commutative law of multiplication, the distributive law, and the many more complicated area formulas.

Yet in my first visit in 1986 to a K-6 elementary school, I discovered thatnot a single teacherknew how to find the area of a rectangle.In those innocent days, I thought that I thought that the teachers might be interested in the geometric interpretation of (x + y)^2. I drew a square with (x + y) on a side and showed the squares of size x^2 and y^2. Then I pointed to one of the remaining rectangles. “What is the area of a rectangle that is x high and y wide?” I asked. There was no response, so I asked the question again. “What is the area of a rectangle that is x by y?”

The teachers were very friendly people, and they know how frustrating it can be when no student answers a question. “x plus y?” said two in the front simultaneously.

“What?!!!” I said, horrified.

Then all fifty of them shouted together, “x plus y.” Apparently my nonverbal reaction had not been a sufficient clue that the original answer was wrong.

How can children in such a school attain a profound understanding of fundamental mathematics? I am now convinced, after visiting many schools, that this one was not unusual. . . .A year later I won one of the first K-3 grants from Exxon Education Foundation. This enabled me to spend twelve days on campus in the summer with five teachers from that school and to visit the school two mornings a week during the following school year. I spent those mornings teaching math to one to three first-grade classes, one to three third-grade classes, and one fifth-grade class.

During my first class teaching elementary school children, a fifth grader raised his hand and asked, “What is that word you keep using instead of take away?” Enter “minus”—for fifth graders!

The best first-grade teacher told me she never bothered to teach subtraction during the first half of the year because the children couldn’t learn everything at once. I started visiting the school in October, and it seemed to me natural to teach addition and subtraction together. She told me she would not reinforce my teaching of subtraction between my weekly visits, and I said that was no problem.

. . .

The following year I led a team that won an Eisenhower grant and began working in an urban-suburban coalition, going to both all-black schools and all-white schools. My first time in a fifth grade in one of New Jersey’s most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, “Near three, isn’t it?” The children, however, soon figured out the correct answer; they came from homes where such things were discussed.

Flitting back and forth from the richest to the poorest districts in the state convinced me that the mathematical knowledge of the teachers was pathetic in both.It appears that the higher scores in the affluent districts are not due to superior teaching in school but to the supplementary informal “home schooling” of children.

Later she writes of conducting supplementary math workships in a poor African-American elementary school:

Later that spring the Iowa scores were revealed for the three third grade-classes with whom I had been working intermittently for three years. Two classes had median scores at the 60th percentile, a great increase from the 25th percentile only three years earlier. The third class had a median at the 70th percentile, with only one child below the 50

^{th}and that child in the 40s.This dramatic increase in Iowa scores was accomplished by the same teachers and a mathematician with no elementary school background whatsoever.I did have high school certification; I had one year of high school teaching experience and the background of having raised two children of my own, but no official professional preparation except a doctorate in mathematics with a specialty in functional analysis. The teachers and I shared a concern for the students.

They were good teachers, and I had access to the national materials of the late 1980s. We talked with each other. I certainly was not teaching “to” the Iowas. I was trying to share my understanding of fundamental mathematics—and it seems that that was what was needed for the children to do well on the highly computational old-fashioned standardized tests.

Finally, even more examples of teacher ignorance:

A couple of years ago I discovered that the problems are even more basic than I had realized earlier; teachers’ understanding of addition is murky.

Montclair State certifies teachers without providing a special course in either mathematics or mathematics education for them, so they are scattered in our general education courses. I had one pleasant, diligent young woman in such a class who intends to be an elementary school teacher. On the last day of her formal mathematics education she responded to my offer to answer questions before the exam by saying there was something wrong with exercise 11 on page 69 of the text (my book Mathematics for Human Survival).

“In 1999 U.S. cars achieved an average of 28.11 mpg, but light trucks were rated a mere 20.3 mpg. Their mileage was 23.8 mpg altogether. What proportion of American vehicles were light trucks in 1999?”

“What’s wrong?” I asked.

“‘Altogether’ means add, so the mileage altogether must be 48.41 miles per gallon.”

I tried to explain but to no avail. Some of the other students gave fine explanations. She is a cooperative person and realized she was outvoted, but it was clear she did not understand.

Wow.