Measurement of Length: How Can We Teach It Better? easurement of length is taught repeatedly in the South (Kamii and Clark 1997). Each child starting in kindergarten and continuing in was given a sheet of paper (11 by 17 inches) with an Mgrades 1, 2, and beyond. However, during inverted T photocopied on it (see fig. 1). Although the past twenty-five years, according to the National the vertical line appeared longer—the result of a Assessment of Educational Progress (NAEP), the perceptual illusion—both lines were 8 inches long. outcome of this instruction has been disappoint- This task was designed to find out how the child ing. As can be seen in table 1, only 14 percent of went about comparing two lines that could not be the third graders and half (49 percent) of the sev- compared directly. Because the lines could not be enth graders gave the correct answer—5 cm—to moved and placed side by side, the child had to use a question on the 1985–1986 NAEP (Lindquist an object such as a ruler or a strip of paper to make and Kouba 1989). Similar items included in other an indirect comparison. NAEPs before and after this one have produced The interview procedure consisted of the follow- similar findings. ing four steps: What is so hard about measurement of length? 1. Perceptual judgment. Presenting the child Table 1 is informative because it reveals the incor- with the figure of the inverted T (fig. 1), the inter- rect answers the students gave. Thirty-seven per- viewer asked, “Do you think this line (line A) is as cent of the seventh graders gave the answer 6 cm, long as this line (line B), or is this one (A) longer, evidently determined by counting the numerals 3, or is this one (B) longer?” The purpose of these 4, 5, 6, 7, and 8. This answer indicates that more questions was to spur the child’s involvement in the than a third of the seventh graders did not know task and give him or her reasons for answering the what a unit of length was, especially the segment subsequent questions. measuring 1 cm. About a third (31 percent) of the 2. Transitivity (first attempt). With a tagboard strip third graders made the same error, and another third (12 by 0.5 inches) in hand, the interviewer asked, (30 percent) chose the answer 8 cm, obviously pay- “Can you use this to prove (or show) that this line (A) ing attention only to the end of the line. is longer than the other line [or whatever the child The purpose of this article is to explain, on the had just said]?” This question was asked to find out basis of research, why instruction has been inef- if the child could demonstrate transitive reasoning by fective and to suggest a better approach to teach- using the strip. Transitive reasoning refers to the abil- ing. Following the model of Piaget, Inhelder, and ity to reason logically that if A is equal to the length Szeminska (1960), I individually interviewed and indicated on the strip and the length indicated on the videotaped 383 children in grades 1–5 in two public strip is equal to B, then A and B can be inferred to be schools in a low-to-middle-income neighborhood equal. Piaget et al. (1960) had shown that children are not able to make this logical inference before the By Constance Kamii age of seven or eight; younger children said that the Constance Kamii, [email protected], is professor of early childhood education at the University only way to compare the two lengths was to put them of Alabama at Birmingham. She studied under Jean Piaget off and on for fifteen years and side by side for direct comparison. has been using his theory to improve mathematics education in kindergarten and the early 3. Unit iteration. Offering a small block (1.75 grades. by 0.88 by 0.25 inches) to the child, the interviewer 154 Teaching Children Mathematics / October 2006 Copyright © 2006 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. asked, “Can you use this to prove (or show) that Table 1 this line (A) is as long as the other one [or what- ever the child had just said]?” The purpose of this Percentages of Students in Grades 3 and 7 Responding to an Item of question was to find out if the child was able to the National Assessment of Educational Progress compare the two lengths by using the small block Table 5.3 as a unit to iterate. (The child had to make a mark Rulers on the paper to indicate the top of the block and Percent Responding a then move the block up without any gap or overlap Item between the units. The 8-inch lines were about 4.5 blocks long.) The strip was used to find out if the child could compare whole lengths, but the block cm 1 2 3 4 5 6 7 8 9 10 11 12 was used to determine if the child could think about the 1.75-inch length as a part to iterate within the How long is this line segment? b whole length. 3 cm 4 1 5 cm * 14 49 4. Transitivity (second attempt). This question 6 cm 31 37 was posed only to those children who were unsuc- 8 cm 30 9 cessful with the strip used in the second step. Four 11 cm 6 2 additional blocks of the same size were given to I don’t know. 15 2 the child, and he or she was asked, “Can you use a The response rate was .80 for grade 3 and .97 for grade 7. b An actual centimeter ruler was pictured. these to prove (or show) that this line (A) is as long * Indicates correct response. as the other one [or whatever the child had just said]?” This question was asked to find out if the child could demonstrate transitive reasoning when Table 2 offered a second chance. Table 2 shows the findings from these interviews. Percentages of Students Who Demonstrated Transitive Reasoning The percentages of students who demonstrated and Unit Iteration transitive reasoning can be seen in the third, fourth, Transitive reasoning and fifth columns, labeled “Transitive reasoning.” Unit Grade n With With strip The third column (titled “With strip”) presents the With strip iteration percentages who demonstrated transitive reasoning blocks or blocks with the strip, and the fourth column (titled “With 1 78 21 8 29 10 blocks”) shows those who demonstrated it with five 2 79 56 16 72 33 blocks (but not with the strip). The total percent- 3 75 77 8 85 55 ages who demonstrated transitive reasoning can 4 75 83 1 84 76 be seen in the fifth column, labeled “With strip or 5 76 89 3 92 78 blocks.” The percentages in this column indicate that most children (72 percent) construct transitive reasoning by second grade. These findings support Because unit iteration was not demonstrated by the age reported by Piaget et al. (1960). the majority of students before fourth grade (see The last column of table 2 shows that unit table 2), it is not surprising that, as shown in table iteration develops gradually and that most children 1, only 14 percent of the third graders got the cor- (76 percent) construct it by fourth grade. Each rect answer—5 cm—because it involved units. interview was reviewed to find out if anyone dem- However, the answer 8 cm, which 30 percent of onstrated unit iteration before transitive reasoning, the third graders gave, cannot be explained by the and no such case was found. Our research thus data in table 2. Because 72 percent of the second supported Piaget’s statement that children construct graders demonstrated transitive reasoning, most of unit iteration out of transitive reasoning. the third graders should have been able to relate the whole length of the line to the corresponding part of the ruler. The low-level response given by so Why Has Instruction Been many third graders must be attributed to ineffective So Ineffective? teaching. In seventh grade, the major problem was Children’s development of logic can be used to that 37 percent did not understand units of length explain some of the poor results reported by NAEP. and gave the answer 6 cm. Teaching Children Mathematics / October 2006 155 Figure 1 need to bring from another room to cover the bul- letin board. In this situation, children have to think An inverted T used in the indirect of a tool to use to make an indirect comparison. comparison task Students are likely to think about a piece of string, a A paper strip, a stick, a book, a yardstick, or a 12-inch ruler. Debate about the quantity of paper to bring, without wasting any, will motivate students to think logically about the length and width of the paper they need to bring. When students are asked if a doorway is wide enough for a certain table to go through or if a certain space is big enough to put a large carton in, B they also have to make an indirect comparison. The problems in figure 2 also require indirect compari- sons, and students can be asked to make drawings that are twice the size of a photocopied model to Two aspects of instruction explain why so many take home and amuse their families.