Although I have taught for many years, it is only within the last 5-10 years that I have begun to realize that part of what I need to do is to think more deeply about what I teach, so that I can "chunk" the algorithms and concepts into such small pieces that I can begin to understand how complex the things I teach are, to my students, and why these things (which seem so simple, to me) are so difficult for them.
For example, we use the Glencoe series (Level 2, for our 7th grade classes). Today, we were looking at a problem in the book. It told us that there were 7.2 million people in the state of GA in 1990, and that the projection for the year 2000 census was a 1/10 increase in poulation. Several students, having struggled this year, to develop a clear concept of the meaning of a fractional part (although we have been working mostly with decimals) volunteered, and I called on one young man, who proudly replied, "The population will be 7.3 million." I was, at first, puzzled by his response, but when I asked him to clarify his reasoning, he quite clearly referred to the "one-block model," that we use extensively, explaining that "1/10 is one out of ten (like one row of squares, in our one-block, which contains ten rows of squares), so we need to add one in the tenths place, the place just past the decimal point. That means that 7.2 million plus 1/10 becomes 7.3 million, because we've added one in the tenths place."
Because of past experiences involving similar situations (with the distributive property, and with an equation like 1/2f = 1/5), I immediately recognized the misinterpretation. I was, then, able to clear up the misunderstanding by differentiating between the 1/10 of the population and the 0.1 million that my student was adding. In fact, I then asked leading questions, in order to guide my class to see that we could use our "one-block model" to represent ONE million. Thus, we needed 7 whole one-blocks and another two rows of the eighth one-block. Then, we needed to take 1/10 of each block, as well as 1/10 of the two rows. When we combined all of those "one-tenths" together in a single "one-block", my students could easily understand that we needed to add 0.72 million to the original 7.2 million, to come up with an answer of 7.92 million, rather than simply adding one to the tenths place.
I firmly believe that, if we wish to encourage more mathematical thinking on the parts of our students, we need to learn to think more deeply about what we teach.