This article is excerpted from “Math Education in the U.S.” (Chapter 10) I went to school in the 50’s and 60’s when students first learned how to add and subtract in second grade. After spending some time memorizing the basic addition and subtraction facts and learning how to add and subtract single digit numbers, I was excited to hear my teacher announce one day that we would now learn how to solve problems like 43 + 52 and 95 – 64. In teaching the method, the teacher explained how the procedure relied on place value–what the ones place and tens place meant. I became bored with the explanation, began to daydream and missed the description of the procedure. The teacher then announced that we would now take a test on what we had just learned.

Your point that an exercise repeated until mastered will lead to understanding/ comprehension is excellent. This is a basic method to learning many things. We need some exposure to a thing, process or method before we can begin to see how it connects in other ways.

Thank you. I just modified one of the paragraphs in this article. I have included this sentence: "This idea that one hundred percent of all students must understand one hundred percent of all the concepts behind one hundred percent of all procedures is one hundred percent unrealistic, misguided and damaging."

Learning to ride a bike helps a physics student understand center of gravity, and force vectors.

Your point that an exercise repeated until mastered will lead to understanding/ comprehension is excellent. This is a basic method to learning many things. We need some exposure to a thing, process or method before we can begin to see how it connects in other ways.

Thank you. I just modified one of the paragraphs in this article. I have included this sentence: "This idea that one hundred percent of all students must understand one hundred percent of all the concepts behind one hundred percent of all procedures is one hundred percent unrealistic, misguided and damaging."

Yes, testing and evaluation were basic to helping a student move forward. The ability to consistently get the right answers proved understanding.