Mathematics educators frequently recommend that students use strategies for measurement estimation, such as the reference point or benchmark strategy; however, little is known about the effects of using this strategy on estimation accuracy or representations of standard measurement units. One reason for the paucity of research in this area is that students rarely make use of this strategy spontaneously. In order to boost students' strategy use so that we could investigate the relationships among strategy use, accuracy of students' representations of standard measurement units, and estimation accuracy, 22 third-grade students received instruction on use of the reference point strategy and another 22 third-grade students received instruction on the guessand-check procedure. Analyses reveal that children's strategy use predicts the accuracy of their representations of standard linear measurement units and their estimates. Relative to students who did not use a reference point, students who used a reference point had more accurate representations of standard units and estimates of length.

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### Elana Joram, Anthony J. Gabriele, Myrna Bertheau, Rochel Gelman, and Kaveri Subrahmanyam

### Janka Szilágyi, Douglas H. Clements, and Julie Sarama

This study investigated the development of length measurement ideas in students from prekindergarten through 2nd grade. The main purpose was to evaluate and elaborate the developmental progression, or levels of thinking, of a hypothesized learning trajectory for length measurement to ensure that the sequence of levels of thinking is consistent with observed behaviors of most young children. The findings generally validate the developmental progression, including the tasks and the mental actions on objects that define each level, with several elaborations of the levels of thinking and minor modifications of the levels themselves.

### Lynne N. Outhred and Michael C. Mitchelmore

The focus of this article is the strategies young children use to solve rectangular covering tasks before they have been taught area measurement. One hundred fifteen children from Grades 1 to 4 were observed while they solved various array-based tasks, and their drawings were collected and analyzed. Children's solution strategies were classified into 5 developmental levels; we suggest that children sequentially learn 4 principles underlying rectangular covering. In the analysis we emphasize the importance of understanding the relation between the size of the unit and the dimensions of the rectangle in learning about rectangular covering, clarify the role of multiplication, and identify the significance of a relational understanding of length measurement. Implications for the learning of area measurement are addressed.

### M. Katherine Gavin, Tutita M. Casa, Jill L. Adelson, and Janine M. Firmender

The primary goal of Project M^{2} was to develop and field–test challenging geometry and measurement units for all K—2 students. This article reports on the achievement results for students in Grade 2 at 12 urban and suburban sites in 4 states using the Iowa Tests of Basic Skills (ITBS) mathematics concepts subtest and an open–response assessment. Hierarchical linear modeling indicated no significant differences between the experimental (n = 193) and comparison group (n = 192) on the ITBS (84% of items focused on number); thus, mathematics concepts were not negatively impacted by this 12–week study of geometry and measurement. Statistically significant differences (p < .001) with a large effect size (d = 0.89) favored the experimental group on the open–response assessment. Thus, the experimental group exhibited a deeper understanding of geometry and measurement concepts as measured by the open–response assessment while still performing as well on a traditional measure covering all mathematics content.

### Jeffery E. Barrett, Douglas H. Clements, David Klanderman, Sarah Jean Pennisi, and Mokaeane V. Polaki

This article examines students' development of levels of understanding for measurement by describing the coordination of geometric reasoning with measurement and numerical strategies. In analyzing the reasoning and argumentation of 38 Grade 2 through Grade 10 students on linear measure tasks, we found support for the application and elaboration of our previously established categorization of children's length measurement levels: (1) guessing of length values by nai've visual observation, (2) making inconsistent, uncoordinated reference to markers as units, and (3) using consistent and coordinated identification of units. We elaborated two of these categories. Observations supported sublevel distinctions between inconsistent identification (2a) and consistent yet only partially coordinated identification of units (2b). Evidence also supported a distinction between static (3a) and dynamic (3b) ways of coordinating length; we distinguish *integrated abstraction* (3b) from *nonintegrated abstraction* (3a) by examining whether students coordinate number and space schemes across multiple cases, or merely associate cases without coordinating schemes.

### James R. Smart and John L. Marks

Measurement is a major topic in elementary school mathematics, appearing in every grade from one through eight. The importance of measurement is indicated not only by its wide application in everyday life, but also because its basic ideas are studied in mathematics classes through the graduate level of the university. Despite the wide applicability of measmcment and its continuous appearance in the curriculum, there is a great gulf between how the practical man measures, reports his measurements, and judges their preciseness, and the mathematician's convention for dealing with measurements.

### Cheryl A. Lubinski and Diane Thiessen

This article focuses on how the children's book *How Big Is a Foot?* (MyHer 1990) was used to prompt measurement experiences that reflect ideas embedded in the *Curriculum and Evaluation Standards for School Mathematics* (NCTM 1989).

### William R. Harrison

In “Building a Metric City or Town” Alec (1980) states, “The mathematical skills prerequisite to learning to measure the length of an object are minimal. The major skill a student needs to have is to be able to read a numeral.” In other words, measuring the length of an object consists of lining up one end of the ruler with an end of the object and reading off the numeral that corresponds to the other end. If a child can perform this task, we conclude that he or she knows how to measure. No wonder most students have such a superficial understanding of measurement; we do not teach basic measurement concepts.

### Truman Botts

No one needs to tell teachers that children have active imaginations. Yet the theme of this first lecture—that mathematics is a thing of the imagination—may seem surprising at first. This is because we are so used to thinking of mathematics as impersonal, coldly logical, cut and dried, hardly a very human activity and certainly not an imaginative one. Nevertheless, I hope we shall see in this lecture that mathematics is in a very essential way imaginative, and that the study of measurement shows this particularly clearly.

### Vera Sarina

, who determined the correct value of absolute zero (–273.15˚C, –459.67˚F) “Electrical Units of Measurement,” delivered May 3, 1883 ( Popular Lectures and Addresses, vol. 1 [1889]) Two basic human activities—counting and measuring—gave birth to