EDITORIAL

Learning, teaching, and using measurement:introduction to the issue

John P. Smith III • Marja van den Heuvel-Panhuizen •

Anne R. Teppo

Accepted: 13 September 2011 / Published online: 24 September 2011

� FIZ Karlsruhe 2011

This issue presents a collection of empirical research

reports that have examined different aspects of the learn-

ing, teaching, and use of measurement. The work reported

addresses measurement as an important domain of school

mathematics, including vocational education, and mea-

surement in use in various occupations and workplaces.

The collection is diverse in many ways, as characterized

below. Though the focus of many articles is the measure-

ment of space (length, area, or volume), attention is also

given in some to non-spatial quantities such as time,

weight, and money. The appearance of this issue in ZDM

reflects the concern felt in many countries that measure-

ment is an important elementary mathematical and scien-

tific competence, but one that—as evidence considered

below suggests—appears to be poorly learned. Weak

learning of measurement—particularly of the conceptual

principles that underlie measurement procedures—under-

mines students’ ability to learn and understand more

advanced mathematical and scientific content and hence

their access to important kinds of skilled work—both

professional and not. The research reported in this issue

will not solve that problem. Instead, the issue targets a

more modest goal: That more researchers across the globe

will reconsider the importance of measurement (in school

and out), its place in elementary mathematics, and the need

to pursue research that will produce partial answers the

basic question, ‘‘why are we doing so poorly teaching and

learning measurement?’’ We hope these partial answers, as

they assemble, will help curriculum developers design

more potent materials, teachers teach the measurement

content more effectively, and assessment professionals

develop more revealing assessments of learning.

In this introduction, we seek to orient the reader to the

collected articles in two ways. First, we briefly review

some of the issues that make measurement ‘‘basic and

fundamental’’ content in mathematics and science, in order

to orient and frame the inquiries reported in the articles.

We also identify some of the principal themes pursued and

central results reported in the articles. While this overview

is approximate, leaving out important messages particular

to individual articles, it is offered to the reader as a partial

‘‘roadmap’’ to the issue—and as motivation to explore

further.

1 Why is measurement important content

in elementary mathematics?

Measurement, the coordination of continuous quantity and

number, has a long-standing and important place in

mathematics. Spatial measurement, the coordination of

space and number, dates from human kind’s initial efforts

to understand and master the physical world (Lehrer,

2003). Though many countries expect elementary mathe-

matics curriculum and teaching to support students’ mea-

surement learning, typically beginning with length,

measurement is even more fundamental to the content and

practice of science (Michaels, Shouse, & Schweingruber,

J. P. Smith III (&)

Michigan State University, 509C Erickson Hall,

East Lansing, MI 48824, USA

e-mail: jsmith@msu.edu

M. van den Heuvel-Panhuizen

Utrecht University, Freudenthal Institute,

PO Box 85.170, 3508 AD Utrecht, The Netherlands

e-mail: m.vandenheuvel@fi.uu.nl

A. R. Teppo

PO Box 570, Livingston, MT 59047, USA

e-mail: arteppo@gmail.com

123

ZDM Mathematics Education (2011) 43:617–620

DOI 10.1007/s11858-011-0369-7

2008). In both mathematics and science, the measurement

of tangible and directly experienced quantities (e.g., mass/

weight, length, volume/capacity) leads quickly to the

measurement and study of composite quantities in both

mathematics (e.g., speed as a rate) and science (e.g., den-

sity, force). For these reasons, measurement is a founda-

tional competence for all fields and careers that build upon

mathematical and scientific knowledge. Also, measurement

may be unique in its inherent meaningfulness among

mathematical topics. Where the hallmark of mathematics is

the abstraction of structure from patterns observed in the

physical and social world (Steen, 1990), measurement—

particularly spatial measurement—remains strongly con-

nected to the measurer’s physical world. For that reason,

measurement is among the most sensible, contextually

situated, and practical domains of mathematics for

students.

But as a growing body of research has indicated (Baturo

& Nason, 1996; Chappell & Thompson, 1999; Clements &

Bright, 2003; Irwin, Vistro-Yu, & Ell, 2004; Zacharos,

2006), measurement is poorly learned in school classrooms

in many countries. There is also evidence that elementary

teachers who are responsible for guiding students’ mea-

surement learning struggle with shallow understanding

themselves (Menon, 1998; Simon & Blume, 1994). The

interpretations offered by researchers examining these

results share a common theme: Students perform relatively

well on well-practiced tasks and very poorly on tasks that

are conceptually simple but frame measurement in non-

standard ways. Students around the world, it seems, have

been successful learning the standard procedures of mea-

surement (the typical focus of classroom instruction)

without learning the conceptual principles that stand behind

and justify those procedures (Irwin, Vistro-Yu, & Ell, 2004;

Stephan & Clements, 2003). Factors related to curricular

content and teachers’ knowledge and instructional practices

have been suggested as root causes (Menon, 1998).

In at least some countries, this pattern of poor learning is

also the result of less classroom attention to the measure-

ment of continuous quantities than to developing students’

understanding of base-10 number and arithmetic operations.

Weaker attention to measurement is more than a difference

in instructional time and textbook pages. Number is built on

the foundation of counting discrete quantities—collections

of objects that are physically separated. By contrast, mea-

surement addresses questions of ‘‘how much?’’ and ‘‘how

much more?’’ for initially continuous quantities—typically

length, time, and volume/capacity in the early grades. In

measurement, a continuous quantity becomes discrete when

we choose a suitable unit and iterate that unit to determine

the number of copies required to exhaust the quantity. Its

measure is the count of those units.

The predominant attention to discrete quantities,

counting, and number, place value, and the arithmetic of

whole numbers over the measurement of continuous

quantities appears to contribute to students’ challenges in

learning measurement. Not only do they suffer from lim-

ited time on measurement tasks, but they also apply a

discrete quantity orientation to measurement tools and

representations of space, e.g., counting marks on rulers to

determine the lengths of objects and counting dots on grid

paper to determine the area of shapes (Blume, Galindo &

Walcott, 2007; Kamii & Kysh, 2006; Nunes & Bryant,

1996). Moreover, they are also ill-prepared to draw on

continuous quantities, especially spatial quantities, when

more advanced mathematical topics, such as multiplication

and division of rational numbers, call for them. Similarly,

the coordination of discrete and continuous quantity nec-

essary to achieve a deep understanding of calculus is dif-

ficult to navigate when students’ knowledge of continuous

quantities and measurement is weak.

When students move into skilled work or technical

careers, measurement remains ubiquitous, but often

removed from view and direct physical activity. Techno-

logical interfaces and systems produce measures of quan-

tities that are central to work enterprises, but in ways that

introduce many layers of intermediate process and com-

putation between the workers and the measured quantity.

The embedding of measurement processes in complex

technological systems raises the importance of measure-

ment knowledge in workers who use these systems and at

the same time creates substantial challenges for workers’

interpretation of the measurements these systems produce.

2 What aspects of teaching, learning, and using

measurement are explored in the issue?

The ten articles that comprise this issue collectively rep-

resent a diverse set of theoretical and methodological

approaches, phenomena under study (e.g., curriculum,

assessment, student thinking, teaching, workplace prac-

tices), national contexts, and central themes and results.

There are large-scale empirical analyses of student

achievement, intensive qualitative analyses of the thinking

of small numbers of students, and experiments that com-

pare student outcomes from different instructional experi-

ences. There are analyses of the written curriculum and

classroom-based studies that focus on the ‘‘enacted’’ cur-

riculum—how the teaching of measurement actually

unfolds. And there are two studies that replace classrooms

with workplaces to examine how measurement practices

are carried out in support of productive work. The articles

are ordered roughly by the age of the learners reported in

618 J. P. Smith III et al.

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the articles, from the beginning to the end of schooling and

then into the workforce.

But among these different analyses, some commonali-

ties are evident. These are not the only meaningful con-

nections that can be drawn between two or more articles,

but they are worthy of note.

Assessing achievement and characterizing measurement

knowledge Given the pattern of poor achievement and

learning around the globe, it is not surprising that

researchers have looked more carefully at what is learned

(and not) and when it is learned. The Marja van den

Heuvel-Panhuizen and Iliada Elia article, for example,

reports on the particular competencies of length measure-

ment mastered by kindergartners and on the efficacy of

reading them picture books highlighting length measure-

ment to develop these competences. They also investigated

the components of these children’s length measurement

performance. Research to assess measurement achievement

requires the analysis of different categorizations of mea-

surement knowledge and items to assess that knowledge,

either prior to or after assessment. Such analysis is also

shown in the articles authored by Jasmin Hanninghofer and

colleagues and by KoSze Lee and John Smith. Hanni-

nghofer and colleagues’ analysis of German primary school

students’ responses to measurement items led them to

replace a standards-based items classification with a more

effective distinction between instrumental knowledge and

measurement sense. Lee and Smith applied the tripartite

distinction between conceptual, procedural, and conven-

tional knowledge to the textual treatment of length mea-

surement in two countries, revealing a strong emphasis on

measurement procedures in both.

Measurement in relation to other mathematical domains

In most elementary mathematics textbooks, measurement

is a specific topic area, taught beside but in less depth than

base-10 number and arithmetic. This view follows and may

strengthen the separation between discrete and continuous

quantities outlined above. An alternative approach sees

measurement of continuous quantities as a suitable, even

desirable entry point into other mathematical topics areas,

including number and operations. Richard Lehrer and

colleagues show how length measurement can serve as an

entry point for meaningful work in statistics that focuses on

the key constructs of variation and measures of variation.

The work reported by Jeffrey Barrett and colleagues

illustrates how the comparison of spatial quantities can

support the development of notions of ratio and eventually

algebraic reasoning. Arthur Bakker and colleagues’ anal-

ysis of measurement competencies in a variety of skilled

occupations also reveals an intimate relation between

measurement and knowledge and reasoning in other

mathematical domains.

Learning trajectories for measurement As the collected

international body of research on students’ achievements

and struggles in learning measurement continues to grow,

in number and depth, some researchers have begun to

develop multi-year learning pathways (or trajectories) that

characterize the nature of students’ progress from initial to

more sophisticated understandings (Sarama & Clements,

2009; Van den Heuvel-Panhuizen & Buys, 2008). Julie

Sarama and colleagues’ article and the Barrett et al. article

report such learning trajectories, provide empirical support

for them, and show how these trajectories have been

revised to accommodate empirical testing. Sarama et al.

discuss length measurement in the early years; Barrett et al.

investigate length and area measurement across the ele-

mentary years. While many studies (including some in this

issue) target measurement of a single quantity, the Barrett

et al. article provides evidence for an educational approach

that attends to common conceptual issues that appear

across quantities.

Estimation in measurement Curriculum and instruction in

measurement typically focus on the use of physical and

computational tools to produce relatively exact measures.

But many everyday situations where measurement is nee-

ded and/or useful either require or naturally call for esti-

mation processes that produce rough and inexact

measures—suitable to the measurer’s needs. Two articles

explore the nature of estimation processes in length mea-

surement. Kuo-Liang Chang and colleagues’ article shows

how the presentation of length estimation tasks in US

textbooks structure some parts of the estimation process for

students, while leaving others open to teachers’ and stu-

dents’ interpretation and decision. The partially open

character of estimation tasks creates both opportunities and

potential pitfalls for learning. Zahra Gooya and colleagues’

study of Iranian secondary students’ methods for estimat-

ing lengths in familiar physical contexts shows the indi-

vidual and inventive character of their estimates that

combine the visual application of personal units with fea-

tures of the physical situation.

Measurement in the workplace Measurement, especially

the measurement of space, is a common component of

work in many skilled, non-professional occupations

(Millroy, 1992; Masingila, 1994; Smith, 2002). Educators

cite use in the world as a principal motivation for learning

to measure and understanding measurement. But as both

articles in this issue that address measurement at work

show clearly, the measurement skills that are taught and

valued in school are quite different from those used and

valued in many workplaces. One common thread reported

in the articles from Philip Kent and colleagues and Arthur

Bakker and colleagues is that measurement at work is

mediated by digital technologies that separate the measurer

Learning, teaching, and using measurement 619

123

and the measured material much more so than in schools.

The Bakker et al. article shows the demand for measure-

ment can be uneven across occupational categories and

how complex it can be when it appears. The case studies

reported by Kent and colleagues clearly show that mea-

surement is valued as one, but only one component in

workplace decisions that aim to optimize production pro-

cesses. That is, measurement is an activity that appears in

much more complex reasoning and problem solving con-

texts than is usually the case in school.

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