Linear Measurement Compared with Area Measurement
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Pencils, rulers, tape measures, and yardsticks may be the tools of choice for measuring the distance around a room. Questions about the room's perimeter can be answered with the results we get from this measurement. I may be able to buy a wallpaper border to place around the room, or I may be able to buy molding to make the perimeter of the room look nicer. But what if I want to buy carpeting or flooring? How do I measure the area of the floor of my room?
My fifth-grade teacher always marked my answers wrong when I gave an area answer without specifying that the units were square inches or square feet. It seemed to me to be such a trivial detail, but it was one that she would not let me get away with. My answers were marked wrong if I left out that little word square, even if I did the math correctly. Why was that? What was the difference between an answer of 6 inches and an answer of 6 square inches? Was this simply one of those rules that adults made up and that kids had to follow because they were kids? That's certainly the way it seemed to me at the time.
Later in my life, as a teacher of grade five, I found myself on the other side of the issue. When my students gave an answer to an area question as "6 inches" rather than as "6 square inches," what did that signify? I came to realize that it often signified a lack of understanding of the nature of area measure compared with the nature of linear measure. Rather than approach the problem by simply requiring my students to write the correct labels, I began to try to help my students understand the difference between linear measurement and area measurement. I began to realize that in most textbooks and planned mathematics curricula, the topic of area measurement came after the topic of linear measurement, particularly the measurement of perimeter. I began to wonder if the students' prior learning of linear measurement was causing them difficulty understanding what they were measuring when they were measuring area. To make matters worse, the typical area measurement tasks began by having students measure the length and width of a rectangular area. The children were then told to multiply the length and the width (linear measurements) to find the area. I began to wonder what would happen if we reversed the sequence of tasks. What if we began our study of area measurement with a realistic problem that gave us a need to find an area? What if the original question itself had in it a clear depiction of square units? What if I gave the students some actual square units that they could use to cover an area in question in order to directly determine how many of those squares were needed? Our classroom ceiling was conveniently tiled with large, square ceiling tiles. I posed the question, "How many tiles should I order if I want to replace our white classroom ceiling with bright orange ceiling tiles?" Naturally, my students began to count the tiles in order to give me the answer to my question. I interrupted their counting. I raised the stakes. I offered a prize to the first person who could give me a correct answer. To one or two of my students, counting every tile now seemed like a waste of precious time. They realized that they could count one row, count the number of rows, and multiply. They had their correct answer long before their classmates could finish a direct count.
Elementary Mathematics: Pedagogical Content Knowledge, by James E. Schwartz, is designed to sharpen pre-service and in-service teachers' mathematics pedagogical content knowledge. The five "powerful ideas" (composition, decomposition, relationships, representation, and context) provide an organizing framework and highlight the interconnections between mathematics topics. In addition, the text thoroughly integrates discussion of the five NCTM process strands.