EARLY CHILDHOOD CORNER
A n d r e w M . Ty m i n s k i , M o n i c a We i l b a c h e r, N i c o l e L e n b u r g ,
and Cindy Brown
Ladybug Lengths:
Beginning Measurement
M
easurement is one of five Content Standards promoted by the National Council
of Teachers of Mathematics as crucial
facets of students’ mathematical knowledge
(NCTM 2000). Skills and reasoning that develop
in association with measurement are applicable in
everyday life as well as in many career choices.
A significant topic within elementary school curricula, measurement is addressed through the use
of investigative activities. Unfortunately, students
traditionally struggle with measurement tasks,
which points to limited understandings of concepts involved in such tasks (Kamii 2006; Thompson and Preston 2004). Therefore, examining the
manner in which measurement is approached
within the early elementary classroom is a germane pursuit.
Figure 1
A lack of conceptual understanding of measurement can result in
answers such as 10 1/2 when students measure the object below
(Thompson and Preston 2004).
Students are typically exposed to tasks that focus
only on developing skills for measuring length,
area, volume or capacity, and weight. Opportunities
to develop conceptual understandings associated
with measuring each of these attributes (Van de
Walle 2007) are often missing from the tasks. For
example, when children measure length by counting
along a ruler, they may not be aware of what they
are actually counting. Children may count numbers
on the ruler but may not necessarily understand that
they are actually counting the spaces between the
numbers (see fig. 1).
Third graders consistently have difficulty with
measurement tasks such as the one in Figure 1
(Lindquist and Kouba 1989), suggesting that learning measurement is more complex than learning
how to use a ruler. Primary grade teachers can play
an integral role in planning and implementing lessons that support students’ growing understanding
of measurement. Developing a concept of measurement and strategies for approaching measurement
tasks is important, especially to the development
of children in grades pre-K–2. Young children need
opportunities to engage in activities that allow them
to construct their own measurement concepts, such
as what it means when we say something is five
inches long. At an early age, measurement activities
should involve use of concrete materials and follow
a learning trajectory that—
Andrew M. Tyminski, [email protected], is an assistant professor of mathematics education at Purdue University, West Lafayette, Indiana. His interests include the professional
development of mathematics teachers and preservice and in-service mathematics teachers’
pedagogical decisions in the moment of teaching. Monica Weilbacher, mweilbac@purdue
.edu, and Nicole Lenburg, [email protected], are elementary education majors at Purdue
University. They both enjoy working with children, especially in the area of mathematics.
Cindy Brown, [email protected], teaches kindergarten at Burnett Creek Elementary School
in West Lafayette, Indiana.
• allows for explorations using iterations of nonstandard units of measurement;
• provides opportunities for children to develop
their own measurement instruments; and
• transitions toward the use of tools involving
standard units of measure.
Edited by Signe E. Kastberg, [email protected], an assistant professor of mathematics
education at Indiana University-Purdue University Indianapolis. “Early Childhood Corner”
addresses the early childhood teacher’s need to support young children’s emerging mathematics understandings and skills in a context that conforms with current knowledge about the way
that children in prekindergarten and kindergarten learn mathematics. Readers are encouraged
to send submissions to this department by accessing tcm.msubmit.net. Manuscripts should
be double-spaced and must not exceed eight double-spaced typed pages.
This article describes a series of activities used
to support children’s development of the concept of
measuring length. Monica Weilbacher and Nicole
Lenburg developed the activities as part of their
elementary mathematics field experience. Input
and feedback from their cooperating kindergarten
34
Teaching Children Mathematics / August 2008
Copyright © 2008 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.
Photograph by Monica A. Weilbacher; all rights reserved
teacher, Cindy Brown, and their university professor, Andrew M. Tyminski, were used to extend the
activity series and provide opportunities for kindergarten students to begin developing a sense of
length measurement.
The Lesson
The first activities were designed to give children
the opportunity to use iterations of nonstandard units
to determine the length of various objects. Because
this was one of the first exposures to the concept of
measuring length for Brown’s kindergartners, Weilbacher and Lenburg decided to read a story to help
introduce the concept. Using literature within mathematics lessons can help students make text-to-self
connections, which are especially important in the
primary grades. A story also serves as an appropriate context to introduce and promote a discussion
of length. The teachers chose Ladybug on the Move
(Fowler 1993) because of the engaging ladybug
character and its interesting travel paths. In the story,
the ladybug travels from various locations, such as
from the leaf to the door and from the watering can
to the stones. The teachers recognized the book as
an excellent starting point for investigating length, in
this case, the length of the ladybug’s path.
Weilbacher and Lenburg began by reading the
story aloud to the students. Using a white board at
the front of the room, they replicated the insect’s
travels with a ladybug model and pictures of various landmarks from the story. Straight lines drawn
between pairs of landmarks represented each portion of the ladybug’s path. To begin the investigation, they drew two different ladybug paths (lines)
on the board. One line was approximately six inches
long; the second was closer to twenty inches long.
Students were to choose the path on which the ladybug would travel the farthest. Kamii suggests that
beginning with this type of question helps children
engage and provides a reason for them to answer the
more specific questions that will follow (2006).
The children were quick to choose the longer
line as their answer. When asked how they knew
the ladybug traveled farther on the longer line,
they replied that it “looked bigger.” The students’
ability to use perceptual judgment in this case was
expected, as the teachers had purposefully drawn
lines of clearly different lengths. However, wanting to motivate the children to gather information
beyond observation, they asked the students how
they could “be sure” that one line actually was longer than the other. Because this was an introductory
Teaching Children Mathematics / August 2008
lesson on length and measurement, students were
understandably confused by the question. Unsure of
how to proceed, Weilbacher and Lenburg decided
to introduce the idea of iterating a series of units
as a way to measure and compare two line lengths.
They began to line up cutouts of the ladybug along
the shorter path, at which point a student shouted,
“You, like, measure!” The students began making
connections between the term measure and the use
of units.
Before using the ladybugs to measure the lines,
students estimated how many ladybugs, lined up
end to end, would match the length of the two lines.
Estimation should play a role in children’s development of measurement concepts; it allows students
to develop a familiarity with the unit and to focus
on both the measuring process and the attribute
being measured (Van de Walle 2007):
• What does it mean to estimate how long this line
is?
• How many of these units will I need to match the
given length?
• What number seems reasonable?
The shorter line was a little longer than one ladybug unit. When students made their estimates, they
gave reasonable answers, such as three or four ladybugs. For the longer line, most students predicted
inappropriately large numbers, such as seventy-five
or one hundred ladybugs. A few students made
more accurate predictions with numbers in the
twenties. Students’ answers gave some information
on their current estimation abilities as well as their
developing number sense. Students who predicted
35
that a large number of ladybugs would be necessary
to match the length of the line likely viewed the
line as “very” long. For most kindergarten students,
one hundred is not just a big number, but the big
number. All responses were recorded on the board,
and the process of using ladybugs to measure the
lines began. As children helped place ladybugs on
the line, they voiced concerns with the placements:
Should the ladybugs face the same way? Should the
ladybugs be right next to each other? The teachers’
goal was to encourage the children to develop a
consistent method of lining up units, in this case,
ladybugs. Students themselves would have an
opportunity to practice measuring with nonstandard
units in the next stage of the lesson.
Using lima beans as another nonstandard unit of
measure and their own activity booklets containing pages with ladybug paths of various lengths,
students were to estimate and record the number
of units needed to match line lengths. Then they
used lima beans to actually measure the lines. For
the most part, the students worked alone, but some
discussion among peers took place. The teachers
walked around and observed interesting problemsolving methods. Some students had no difficulty
completing the task, but others ran into trouble
using the lima beans as a unit of measure. Modeling the procedure for lining up the objects on the
line (see fig. 2) seemed to eliminate the problem
of students overlapping beans. Weilbacher and
Lenburg had anticipated teachable moments when
students would furnish slightly different answers
for the same line, but unanticipated problems also
arose. Several students arranged their beans so that
they almost resembled a wave: some beans placed
Figure 2
Photograph by Monica A. Weilbacher; all rights reserved
The teachers modeled the procedure for lining up the lima beans to
measure the lines.
36
slightly above the line, some slightly below, and
some right on top of the line. Students also oriented
their beans differently on the lines. Some placed
the lima beans side by side lengthwise, and others lined them up widthwise. The differences in
lengths measured in beans provided an opportunity
to talk about orientation and consistency in how
the children used the beans (units) to measure the
lines. The primary goal of the task was to provide
students with an opportunity to develop the concept that measurement is a number that represents
“a comparison between the attribute of the object
being measured and the same attribute of a given
unit of measure” (Van de Walle 2007, p. 375).
The activity also encouraged students to ask,
“What if you measure short of the line or over the
line?” As the teaching team had planned the task,
they had predicted that some students would not
completely cover the line, leaving a portion of the
line less than one bean length uncovered and resulting in an answer of one less bean. Other students
added an extra bean to their line to make sure that
no part of the line was showing, although the beans
then extended past the line. One student explained
his reasoning: “Well, there still was line showing,
so I knew I could fit more and measure it better.”
This approach provided additional opportunities
to discuss estimation and precision, different bean
sizes, and different ways beans could be placed on
the line. Weilbacher and Lenburg concluded the
discussion by explaining that each student should
decide how to orient the beans and whether the
line should be completely covered with beans or
“overflowing” with beans. Most students accepted
this explanation, but a few were still very concerned
about getting the “right” answer.
Using an iteration of nonstandard units to measure an object’s length is the first stage in a learning
trajectory toward a conceptual understanding of
length measurement. Children should have multiple opportunities to use iteration of many units as
a means of determining length. These experiences
should also include the opportunity to measure the
same object with different-sized units. Tasks such
as those used in the ladybug activity help children
begin to develop an understanding of the relationship between the size of the unit and the number of
units needed to match a given object’s length.
The kindergartners reexamined their line lengths
with a new unit of measure—paper clips. Again, the
students were to make and record their estimates
and then measure their lines with paper clips. The
teachers asked, “Would it take more or fewer paper
Teaching Children Mathematics / August 2008
Photograph by Monica A. Weilbacher; all rights reserved
clips than it did lima beans to measure a certain
line?” Almost all the students seemed to understand
that fewer paper clips would cover the line because
the paper clips were larger than the lima beans.
Overall, students seemed excited and engaged
during the lesson. Weilbacher and Lenburg concluded that their use of nonstandard units helped
students begin developing a concept of what it
means to measure length. Further, the opportunities
to estimate and measure with two different-sized
units were beneficial in introducing important
mathematical relationships.
What Comes Next
After Weilbacher and Lenburg implemented the
activities described, it was up to Brown to continue
the learning trajectory initiated by this lesson.
Brown’s plans for the next few lessons on length
were designed to give her kindergartners more
practice with iterating units. The children were
engaged in measuring various objects using Unifix
cubes and learning links. In subsequent lessons,
Brown asked her students to answer questions,
such as, “Who has the longest shoe, the widest
hand, or the shortest hair?” These question types
are especially motivating for kindergarten students,
and Brown’s were eager to engage in measurement
activities in order to find the answers.
Brown chose Unifix cubes and learning links as
nonstandard measuring units because they can be
connected and used as “first rulers.” The development of students’ own measurement instruments
is the next logical stage in developing meaning for
measurement. Brown’s use of Unifix cubes helped
students make the cognitive connection between
measurements as counts of objects and the use of a
ruler. After giving students some experiences lining
up Unifix cubes along given objects, Brown asked
them to make a rod of ten cubes to measure classroom objects.
The last stage of her unit on measuring length
involved supporting her students’ transition from
the use of nonstandard to standard units of measure.
To accomplish this goal, Brown used inchworm
manipulatives. Students can use iterations of inchworm manipulatives to directly compare lengths of
objects, while engaging in comparisons of the standard unit. Manipulatives such as Unifix cubes can
also be connected to form rudimentary inch rulers.
Engaging tasks, however, are not always enough
to ensure that students are constructing the knowledge we hope they are. Talking with the students
Teaching Children Mathematics / August 2008
is important—during and after the measurement
tasks—to see what they have gleaned from the
activities. Following each measurement activity,
Brown led a community circle discussion in which
the students talked about their discoveries. She
found that her students became more precise in
their measuring as the instructional unit progressed,
and she was pleased with their developing understanding of units of measurement and rulers.
Conclusion
Measurement is an important life skill beyond
mathematics classrooms, and foundations for such
skills begin in the early primary grades. Teaching length measurement as solely a formula often
leaves students with misconceptions. On the other
hand, a series of well-planned experiences helps
children on their way to a meaningful understanding of length, unit, rulers, and measurement.
References
Fowler, Richard. Ladybug on the Move. San Diego, CA:
Harcourt Brace, 1993.
Kamii, Constance. “Measurement of Length: How Can
We Teach It Better?” Teaching Children Mathematics
13 (September 2006): 154–58.
Lindquist, Mary M., and Vicky L. Kouba. “Measurement.” In Results from the Fourth Mathematics Assessment of the National Assessment of Educational
Progress, edited by Mary M. Lindquist, pp. 35–43.
Reston, VA: National Council of Teachers of Mathematics, 1989.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics.
Reston, VA: NCTM, 2000.
Thompson, Tony D., and Ronald V. Preston. “Measurement in the Middle Grades: Insights from NAEP and
TIMSS.” Mathematics Teaching in the Middle School
9 (May 2004): 514–19.
Van de Walle, John. Elementary and Middle School Mathematics: Teaching Developmentally. 6th ed. Boston:
Pearson Education, 2007. s
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