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Linear measurement is typically the measurement skill that elementary mathematics students study first. For several years I have given college students an assignment to measure the perimeter of their dorm room or their kitchen at home using whatever measurement tools they had available. Very few students come to a residential college with a carpenter's tape measure, so the variety of tools that students have come up with has been interesting! Students have measured their rooms with everything from pencils to notebooks to ceiling tiles to their own shoes. Of course, many students are unable to conceive of measuring their rooms with anything but standard units, and these students often search high and low to find a friend who has a ruler or a yardstick. One creative and very patient student decided to use the 10 cm Cuisenaire Rod that was part of her course manipulative kit to measure her room!
When we discuss this assignment in class there are always some students who are surprised that some of their classmates measured with nonstandard measurement units. This opens an interesting discussion in which we explore what is necessary in order to do a measurement. Is there really a difference between a standard length such as foot and a nonstandard length such as "my mathematics notebook"? The only real difference is that no one but me knows how long my mathematics notebook is, so communication about the results of measurement becomes difficult. The use of nonstandard units allows us to assign a number of units to the attribute of length and width of our room. The use of standard units allows us to communicate our results to others. For most real measurement tasks, communication of the results is an integral part of what we need to do, which is why we almost always measure using standard units.
After students get over their surprise about the use of nonstandard units, they sometimes draw the incorrect conclusion that there are no constraints at all governing the use of measurement units. In order to shed light on this misconception I tell them a story of how I once used my pencil as a measurement tool. After measuring the length of the room I wrote down my results: 108 pencils. Before measuring the width of the room I received a phone call. Following the phone call, I was distracted and graded some papers. After a time I remembered my unfinished business, and I measured the width of the room. My results indicated that the room was 108 pencils wide. This astonished me, because I knew the room wasn't square. My question for my students: Why did my measurements suggest a square room? And how could I correct my results? What I didn't tell the students, what I had not realized myself at the time, and what we figured out together by deduction, is that I must have sharpened the pencil before measuring the width of the room. The conclusion I am looking for is that nonstandard units are fine, as long as they remain the same units throughout the entire measurement process. By sharpening my pencil I had changed the units!
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.