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Non-standard and standard units and tools for early linear measurement

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Abstract

Taking into consideration that research results on kindergarten children's capabilities in linear measurement are not always in agreement, we assumed that the auxiliary means used for early linear measurement may play a crucial role. To investigate kindergarten children's actions when using non-standard and standard units and tools for linear measurement, we conducted an inquiry-based classroom experiment in which the designed task created an environment for children to investigate linear measurement in teams. The results showed that through children's measurement actions specific parameters arose that varied according to the different characteristics of units and tools.
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
NON-STANDARD AND STANDARD UNITS AND TOOLS FOR EARLY
LINEAR MEASUREMENT
Chrysanthi Skoumpourdi
University of the Aegean
Taking into consideration that research results on kindergarten children’s
capabilities in linear measurement are not always in agreement, we assumed that the
auxiliary means used for early linear measurement may play a crucial role. To
investigate kindergarten children’s actions when using non-standard and standard
units and tools for linear measurement, we conducted an inquiry-based classroom
experiment in which the designed task created an environment for children to
investigate linear measurement in teams. The results showed that through children’s
measurement actions specific parameters arose that varied according to the different
characteristics of units and tools.
INTRODUCTION-THEORETICAL FRAMEWORK
Length is one of the main magnitudes in the content area of measurement in early
childhood mathematics curriculums (Smith, Tan-Sisman, Figueras, Lee, Dietiker &
Lehrer, 2008; Smith, van den Heuvel-Panhuizen, & Teppo, 2011). Its importance is
highlighted by the fact that it is the simplest form of measurement (quantification of
continuous quantities) and thus is considered an accessible and understandable
magnitude even by young children (Tan-Sisman & Aksu, 2012). It is also
fundamental for perceiving other magnitudes, such as perimeter, area and volume; for
connecting mathematical content areas for example number and geometry; as well as
for linking mathematics to the real world that children live in.
The perception, comparison, and measurement of length as a magnitude, as a length
or width of two-dimensional shapes, as a height of a three-dimensional shape, as a
distance or as a movement between two points, is a slow and evolving process and
develops through several stages (Sarama, Clements, Barrett, Van Dine & McDonel,
2011). According to Clements & Sarama (2007) there are eight main concepts that are
fundamental for children’s understanding of length measurement, 1. Understanding of
the attribute of length. 2. Conservation of length. 3. Transitivity. 4. Equal partitioning
of the object to be measured. 5. Iteration of the unit; the placing of the unit end to end
alongside the object and the counting of these iterations. 6. Accumulation of distance;
the number words of the counted iterations signify the space covered by the units up
to that point. 7. Origin; any point on a ratio scale can be used as the origin, 8. Relation
between number and measurement. The sequence that these concepts are developed in
is not commonly accepted yet, since it is influenced by age, experience and
instruction. Different pedagogical approaches
.
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
do not seem to influence children’s performance on linear measurement
(Kotsopoulos, Makosz, Zambrzycka & McCarthy, 2015) whereas the complexity of
measurement tasks does (van den Heuvel-Panhuizen & Elia, 2011).
Curriculums suggest starting to teach length with the qualitative perception of the
concept using relevant words such as big-small, long-short, as well as the ability to
make direct comparisons, such as length-based ordering of objects. After that they
suggest continuing with estimations, with indirect comparisons and with the ability to
quantify length, giving it a numerical value. Indirect comparisons can be made both
by placing multiple units or by iterating a unit. Initially, non-standard units are used
and then standard units. The final stage of teaching is the cultivation of the ability to
use measurement tools, such as rulers (Ministry of Education 2010; NCTM, 2006;
ΠΣN, 2011). This sequence of instruction which is also proposed by many researchers
(Barrett, Cullen, Sarama, Clements, Klanderman, Miller, et al. 2011), is based on
Piaget’s theory of measurement. However, there is also research that suggests
beginning the instruction with standard units and rulers, for an initial understanding of
measurement, and a later introduction of non-standard units. This suggestion comes
from the fact that young children show a preference for rulers and are able to use them
before they fully understand the unit represented on rulers (Clements, 1999; Mac-
Donald & Lowrie, 2011; van den Heuvel-Panhuizen & Elia, 2011).
Research results on kindergarten children’s capabilities in linear measurement are not
always in agreement. Most of the research suggests that young children have an
intuitive understanding of length (Clements & Sarama, 2007) and are able to make
direct comparisons and classifications of objects according to their length (Barrett,
Jones, Thornton & Dickson, 2003; Clarke, Cheeseman, McDonoug & Clarke, 2007).
They perform length estimations by activating the cognitive processes of holistic
visual recognition, classification and unification (Van den Heuvel-Panhuizen & Elia,
2011). They can measure the objects’ length by following the necessary procedures
such as placing the units from one end to the next, without gaps and overlays,
measuring the number of units and communicating the result of the measurement
(Sarama, et al., 2011).
However, there is also research, suggesting that young children use units in a non-
systematic way and are not able to determine the length of an object (Barrett, et al.,
2003; Castle & Needham, 2007; Clarke, et al., 2007). Nevertheless, even if they
measure length using an appropriate method, not all of them give the right numerical
value. This is affected by the measuring material and the object to be measured, which
could lead them to meaningless measurement results (Skoumpourdi, 2015).
Children’s main strategies in measuring length are the linear, the perimetrical and the
spatial placement
.
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
strategy. Additionally, research results indicate that young children, although they
show a preference for the use of rulers (Kotsopoulos, et al., 2015), they find
difficulties in using them methodically, despite their repeated use during teaching
experiments (Sarama, et al., 2011).
The most reported students’ errors during length measurement are (Tan-Sisman &
Aksu, 2012): units overlapping, mixing length units with other measurement units,
confusing the concept of perimeter with area, incorrect alignment with a ruler, starting
from 1 rather than 0, counting hash marks or numbers on a ruler/scale instead of
intervals and focusing on end point while measuring with a ruler. Problems arise also
when children have to iterate units-blocks to measure a length when blocks are fewer
than the necessary (Kotsopoulos, et al., 2015).
From the above mentioned we made the assumption that the role of the auxiliary
means used in a linear measurement may be crucial. The type of the magnitude to be
measured, as well as the units and tools that are used for the measurement influence
children’s ability to measure accurately. Thus, the purpose of this paper is to
investigate kindergarten children’s actions when using non-standard and standard
units and tools for linear measurement. The research questions posed were the
following:
1. How do kindergarten children use anglegs
1
and Cuisenaire rods
2
as non-standard
units for linear measurement?
2. How do kindergarten children use a ribbon as a non-standard tool for linear
measurement?
3. How do kindergarten children use snap cubes
3
as standard units for linear
measurement?
4. How do kindergarten children use a ruler as a standardized measurement tool for
linear measurement?
5. What characteristics of the units and the tools used seemed to influence early linear
measurement?
METHOD
To investigate kindergarten children’s actions when using non-standard and standard
units and tools for linear measurement, an inquiry-based classroom experiment took
1
Anglegs (One set contains 48 snap-together plastic pieces, in 6 different lengths/colors)
2
Cuisenaire rods (One set contains 74 rods: 4 each of the orange, blue, brown, black, dark green and yellow, 6 purple,
10 light green, 12 red and 22 white)
3
Snap cubes (One set contains 100 snap-together plastic cubes, in 10 different colors)
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
place. A pre-service kindergarten teacher, through a designed linear measurement
task, created an environment for the children to explore, experiment with and
investigate linear measurement in teams.
The task was implemented in a public kindergarten
4
, with 18 students (6 girls and 12
boys) divided in five teams of three or four persons. In the designed linear
measurement task, children had to measure the length of the four sides of a field, for
ordering a fence to protect the planted carrots. Common non-standard and standard
units and tools for linear measurement, different for each team, were used to
investigate children’s actions. Anglegs and Cuisenaire rods were used as non-standard
measurement units, because of their multiple sizes and colors, but also because
anglegs could be snapped together, whereas Cuisenaire rods could not. A roll of 3
meters ribbon was used as a non-standard measurement tool that, due to its
continuousness, covers a length easily. Snap cubes were used as standard
measurement units because of their consistent size and their multiple colors. Also, a
ruler was used as a standardized measurement tool. All the units and tools were
familiar to the children with no specific knowledge of their used required, except for
the ruler.
RESULTS AND DISCUSSION
At the beginning of the process, and before children’s separation into teams, the
scenario and the carrot field were presented to the students, who were asked both to
show what they should measure and to estimate the length of the fence. Children
seemed to understand what they should measure, and a child showed how to do it by
moving his hands and saying "this, all around". They seemed to be willing to
estimate, but offered answers at random without much consideration. Their
estimations were numbers with a measurement unit, such as “3 meters”, “2 meters”,
“10 meters”, “20 meters”, etc. Because of the variety of the estimations, the need for a
more accurate measurement came up. To the teacher’s question about how they
should measure in order to have an accurate result, all of them answered with a
ruler.
After that episode, children in teams had to measure the length of the fence and write
the result of their measurement on a piece of paper. The first team had to measure
with a ribbon, the second team with the anglegs, the third team with the cubes, the
fourth team with the Cuisenaire rods and the fifth team with a 50cm ruler.
.
4
This kindergarten (students from 3 years and 9 months to 6 years and 6 months old) was chosen a) because of the frequent
cooperation we have with the teacher who likes integrating innovations in her teaching, b) because the children in that classroom
were able to compare two objects directly and recognize their equality or inequality, c) because they were also able to place in
order objects according to their length and d) because they knew to count and write numbers up to 100.
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
Measuring with a ribbon
The first team, 2 boys and 2 girls, had to measure with the use of a ribbon. One of the
boys, who had the ribbon in his hands, asked the teacher some clarifying questions
about how they should measure, while one of the girls started to count the carrots.
Counting objects is a common activity in their class. The teacher reminded to the girl
that they had to measure the carrot field to order the fence and not count the carrots
and at the same time she gave the initiative to the boy to decide with his team what to
measure and how to measure it. The boy showed with his hands where to measure
saying here all around”, and then, with the help of the other two team members he
started measuring with the ribbon. They placed the ribbon around the carrots, holding
the ribbon with their fingers firmly on the corners and saying, approximately this
much” (photo 1). To the teacher’s question about what the result of their measurement
was and what they were going to write on the paper, their answers varied from 1 to 15
meters. Finally, the team members came up with the result 4 meters”. They did not
determine the ribbon’s actual length nor did they attempt to cut the ribbon to match
the perimeter of the carrot field.
Measuring with the anglegs
The second team, 3 boys and 1 girl, had to measure the carrot field with the use of a
set of anglegs. The measurement started with a boy who placed 2 red pieces. Then he
picked purple pieces, which were shorter, intending to place them beside the red ones
but the girl preceded and placed a blue one beside the red ones. At the same time the
other two boys placed multiple sized pieces along two other sides of the field. The
former placed first 1 yellow, then 1 blue, then 1 yellow and finally 1 purple piece. The
latter placed 1 blue, 1 blue and 1 yellow piece. The last piece to be placed was a
matter of concern, because in the meantime the first boy had already completed the
fourth side with 5 purple pieces but there was a small gap left. After several attempts,
two of the boys filled the gap by placing 2 orange pieces (photo 2). To the teacher’s
question about what the result of their measurement was and what they were going to
write on the paper, their answer was 17, the number of pieces they had placed around
the field, without regard to the different sizes.
Photo 1
Measure
with
ribbon
Photo 2
Measure
with
anglegs
Photo 3
Measure
with
cubes
Photo 4
Measure
with
Cuisenaire
rods
Photo 5
Measure
with
ruler
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
Measuring with cubes
The third team, 2 boys and 2 girls, had to measure the field with the use of cubes.
Three of the children (1 boy and 2 girls) connected some cubes and placed them along
one side of the field. The length of these cubes, however, was longer than the side of
the field and a girl removed the excess number and continued placing cubes on
another side. The two girls seemed to be concerned with the accuracy of the
placement and perhaps for this reason they were careful to place the cubes exactly
along the perimeter. Or perhaps they just wanted to use as many cubes from the box
as they could (Photo 3). The second boy did not place any cubes, but he tried to count
all the arranged cubes. To the teacher’s question about what the result of their
measurement was and what they were going to write on the paper, their answer was
56, a number not corresponding to the actual number of cubes (90).
Measuring with Cuisenaire rods
The fourth team, 2 boys and 1 girl, had to measure with the use of a set of Cuisenaire
rods. One boy started the placement with the orange rods, which were the longest
rods. He used them all (4) and he added 1 purple rod. Then he continued along the
next side with 1 black, 1 blue, 1 brown, 1 dark green, 1 yellow and 2 white rods. At
the same time the other two sides had already been covered by the rest of the team
members: the boy had covered one side with 2 blue, 1 black and 2 yellow rods, while
the girl had placed, 1 purple, 1 blue, 1 black, 2 dark green, 1 yellow, 1 light green and
1 purple rod along the other side (Photo 4). To the teacher’s question about what the
result of their measurement was and what they were going to write on the paper, their
answer was 25, the number of rods they had placed around the field, without
distinguishing the different sizes.
Measuring with ruler
The fifth team, 3 boys, had to measure with the ruler. The first boy placed the ruler
along one side of the field in such a way that the side of the field matched the middle
section of the ruler (the ruler was 50cm and the side 40cm). The second boy pushed
the ruler so that the its edge matches the side of the field’s edge. Then, the first boy
started counting imaginary units with his index finger ignoring the units on the ruler
(Photo 5). The second student interrupted him and told him "No need to count, its 40",
indicating the ruler's units. However, the first boy continued to use the same strategy
on the next side starting from 40 and ending at 54. To measure the 3rd side he placed
the ruler as he did in the beginning, counting his imaginary units along the length, 10
in total and continued along the next side in the same way, announcing “20” as the
measurement’s result. Essentially, the student acted on his own. The other children of
the group, apart from the original correction and the placement of the ruler, did not
interfere.
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
After the measurements with the units and tools the class had to decide on the final
length of the fence, so the carpenter would know how much material would need. But
the answers varied since each group reported a different number as the result of their
measurement without any justification. To the teacher's question about which of these
auxiliary means they considered to be the best for a measurement they replied that the
best for measuring was the ruler because it had numbers on it.
Non-standard units
Standard
units
Standard
tool
Anglegs
Cuisenaire
rods
Cubes
Ruler
Strategy
Linear
placement
Linear
placement
Linear
placement
Linear
placement
Team
work
Place pieces
along sides/
separately
Place rods
along sides/
separately
Place cubes
along sides/
separately
Placement
along sides
by a student
Answers
17
25
56
54+20
Measu-
rement
result
Pieces5:
2r, 2o, 3y,
4b, 6p
Rods6:
1lg, 1b, 2w,
3p, 3b, 3dg,
4b, 4y, 4o
90 cubes
~170cm
Time
3
minutes
2.5
minutes
9.5
minutes
1.5
minutes
Table 1: Parameters that vary according to the auxiliary means and its use
Children’s measurement actions set parameters that seemed to vary according to the
non-standard and standard units and tools specific characteristics and influenced the
measurement results (table 1). Regarding the anglegs and the Cuisenaire rods, as non-
standard units, it seemed that their multiple sizes did not seem to trouble the children
who used them but counted them as if they were of the same size. The time they spent
measuring with these materials was not long (2.5-3 minutes) and the numerical value
they gave when quantifying their result
.
5
r: red, o: orange, y: yellow, b: blue, p: purple
6
lg: light green, br: brown, w: white, p: purple, b: black, dg: dark green, b:blue, y: yellow, o: orange
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
was meaningless because it was equal to the non-standard units’ total number of
pieces. Τhe number of the pieces allowed all the children of the group to use them
independently, without cooperating with each other. In the case of the ribbon, a non-
standard tool, it was clear that although it made the measurement a quick and easy
process, it was not helpful for quantifying the measurement’s result. As for the cubes,
standard units, children used them in the same way as the non-standard units,
although the time they spent to complete the measurement differed significantly. The
uniformity of their size could have led to a correct quantification of the
measurement’s result, but that did not happen, because of their large number. The
ruler, a standard tool, also led to incorrect measurement results.
From the above results many questions arise: The method of measurement with the
cubes is considered to be a correct one but should the method with the anglegs and the
Cuisenaire rods are considered to be wrong? Since in both cases we do not know if
children have any understanding of (non) standard units. If we choose to give to the
children units of the same size, such as cubes and the children fit them correctly,
measure them accurately and quantify their measurement, should we consider that
children have understood linear measurement? Do they realize transitivity, equal
partitioning of the object to be measured, iteration of the unit, accumulation of
distance? And if the above does occur with the children of this age, should this be the
starting point for teaching measurement in the kindergarten? Should also emphasis be
given on the accuracy on the quality data of the measurement rather than solely to the
numerical result?
CONCLUSIONS
The classroom experiment showed that the children effectively used non-standard and
standard units, but not tools, to perform a linear measurement yielding a result that
was logical for them. However, this result cannot be accepted as an actual result of
measurement.
Through children’s measurement actions specific parameters arose that varied
according to the different characteristics of the units and tools used as auxiliary means
for measurement. These were the strategy used, the type of the cooperation, the
children’s answers and the time spent for the measurement in relation to the
measurement result.
As it is often suggested, children have to be educated in the use of the ruler, as a
standardized tool. From this experiment it became clear that it is necessary to educate
children also in the use of any auxiliary means used for measuring, and mostly in the
announcement of an accurate measurement result with qualitative data related to the
unit/tool used. Thus, we can add to the dilemma, about how to start teaching linear
measurement from non-standard or from standard units and tools, the necessity to
educate children how to use units and tools for linear
Skoumpourdi, C. (2018). Non-standard and standard units and tools for early linear measurement. In B. Maj-
Tatsis, K. Tatsis, E. Swoboda (Eds.) CME Mathematics in the Real World (pp. 174-183), Wydawnictwo,
Uniwersytetu Rzeszowskiego, Poland.
measurement, as well as how to quantify their results giving the qualitative data that
come from the means they used.
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This paper addresses: firstly, kindergartners’ performance in length measurement, the components of their performance and its growth over time; secondly, the possibility to develop kindergartners’ performance in length measurement by reading to them from picture books. To answer the research questions, an experiment with a pretest–posttest experimental control group design was carried out involving nine experimental classes and nine control classes. The children in the experimental group participated in a 3-month picture book program that, among other things, spotlighted the measurement of length situated in meaningful contexts. Before and after the intervention, the children’s performance in length measurement was assessed in both groups. The responses of 308 kindergartners (4- to 6-year-olds) from two kindergarten years (K1 and K2) were analyzed. Analysis of the pretest data showed that the measurement tasks included in the test were not easy to solve. However, the children belonging to K2 did better than the younger children belonging to K1. Within children’s performance, three components could be identified: holistic visual recognition, ordering and unitizing. Finally, the effect of the intervention was investigated by comparing the performances of the experimental and control group in the pretest and the posttest. We found a weak but significant effect of reading picture books to children on their general measurement performance. However, this effect was only found for K1 children on the component of holistic visual recognition. KeywordsKindergartners–Picture books–Mathematics–Measurement tasks–Assessment
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This article presents data gathered from an investigation which focused on the experiences children have with measurement in the early years of schooling. The focus of this article is children’s understandings of length at this early stage. 32 children aged 4–6years at an Australian primary school were asked to draw a ruler and describe their drawing, once in February at the beginning of school, and again in November towards the end of their first year of school. The drawings and their accompanying descriptions are classified within a matrix which, informed by Bronfenbrenner’s ecological theory and literature regarding the development of length concepts, considers conceptual understanding and contextual richness. The responses revealed that children have a good understanding of length at the start of school, but that as their ability to contextualise develops so too does their conceptual understanding. This article suggests that participation in tasks such as these allows children to create their own understandings of length in meaningful ways. Additionally, the task and its matrix of analysis provide an assessment strategy for identifying children’s understandings about length and the contexts in which these understandings develop. KeywordsYoung children–Measurement–Representations–Context
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This study focused on the meaning of measurement to a group of 16 first grade students. A university professor and the teacher of the students partnered together using qualitative analysis of field notes, student interviews, and student work samples gathered from September through May of a school year. Findings indicate students’ knowledge of measurement including transitivity, unit iteration, conservation of number and length, and social knowledge of measurement terms and tools increased over the year. Researchers identified six themes of students’ measurement understanding including that children’s literature played a motivating role in student-initiated measurement activities. Recommendations call for first grade measurement activities focused on what it means to measure rather than on how to measure. Researchers caution that educators using mathematics curriculum and assessment should not assume that primary grade students understand conservation and unit iteration.