Here’s a great example of how language can affect conceptual understanding. The following is from a third grade teacher in China, quoted in Liping Ma’s book Knowing and Teaching Elementary Mathematics: Teacher’s Understanding of Fundamental Mathematics in China and the United States:

        Some of my students may have learned from their parents that you “borrow one unit from the tens and regard it as 10 ones.” I will explain to them that we are not borrowing a 10, but decomposing a 10. “Borrowing” can’t explain why you can take a 10 to the ones place. But “decomposing” can. When you say decomposing, it implies that the digits in higher places are actually composed of those at lower places. They are exchangeable. The term “borrowing” does not mean the composing-decomposing process at all. “Borrowing one unit and turning it into a 10” sounds arbitrary. My students may ask me how can we borrow from the tens? If we borrow something we should return it later on. How and what are we going to return? Moreover, when borrowing we should get a person who would like to lend to us. How about if the tens place does not want to lend to the ones place? You will not be able to answer these questions that students may ask.

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