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Problem introduction
Halfway though the year, a teacher introduces a kindergarten class to a pretend measuring problem. In the problem, Big Bill, a tall man with very large feet, and Little Larry, a short man with very tiny feet, came to the classroom to measure the length of the classroom for a new rug. Unfortunately, their answers were quite different. Little Larry’s answer of twenty feet was twice as long as Big Bill’s measurement of ten feet. The teacher asked the children to offer suggestions about the rug’s length as well as why differences existed in the measurements. Later, the teacher asked them to actually measure the room using whatever tools the required.

Observed investigations
Children’s ideas were humorous and generally indicated only a surface understanding of length measurement. Most children attributed the differences to counting errors and “not good” measuring: “Big Bill just forgot how to count his numbers past ten. I do sometimes!” “I think Little Larry was counting too fast and skipped numbers.” “Big Bill forgot what to measure and so he went up the wall instead of long-ways!”

The five-year-olds measured the length of the classroom by using a variety of methods with some commonalities. Surmising that length had a beginning and an ending point, all the children began counting from one end of the room and ended their counting at some point on the other side. In addition, they used a type of measuring unit such as their own foot, a measuring tape, a ruler, a piece of yarn, their finger, or their hand span. The differences in their measuring processes, however, provided evidence of their partial notions of measuring. Children counted feet by walking toe to heel “like a tightrope walker.” They walked sideways by opening and closing their legs, counting the number of leg openings. Children in one partnership even measured by jumping while keeping their feet together, a process that they said was “really, really hard!” Sometimes, children counted their footsteps while holding a ruler “ ‘cause you’re supposed to,” they said.  Others spread tape measurers across the room, looked at the last number on the tape measure, and reported either 1 or 150 as the measurement, depending on the orientation of the tape  Still others ran out of rulers and decided to use pencils for measuring tools because “they kinda fit.”

Problem conversations
After the measuring experiences, children shared their results. To their surprise, the answers were all different. Children explained the difference by stating that some people “did it more better” or that “we need to do it more to find out the real answer.” Perhaps the most interesting conversation occurred when children were prompted by the teacher’s “confusion” about Big Bill and Little Larry. She stated, “Big Bill is big and he has big feet, so he should have a big number. Little Larry is little and he has little feet, so he should have a little number. But that is not how it worked. Big Bill has a little number and Little Larry has a big number. I wonder why.” An animated conversation ensued as children explained to the teacher how big feet “take up lots of room” and how they “spread out more.” They explained Larry’s measurement by saying that Larry takes “baby steps so they are real close together.” The conversation continued even after the class was dismissed, and children suggested many solutions over the next few days.

Follow-up activities
Children did not want to leave this problem. They continued to persist until they recorded all their answers using both standard and nonstandard units as well as a variety of tools. This investigation prompted even more interest among some children when school district personnel were measuring the building for new tiling. Several conversations about the size of the men’s feet and their measuring methods were overheard, and the investigation continued.

Connection to the Standards
This activity connects to the Problem Solving Standard as well as the Measurement Standard expectations referring to the application of “appropriate techniques, tools…to determine measurements” (NCTM 2000, p. 102). When the teacher expressed confusion over the “big feet, small number” dilemma, she created yet another opportunity for peer discussion and a beginning understanding of the processes of measurement.