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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)

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.

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

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.

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 tool

Non-standard units

Standard

units

Standard

tool

Ribbon

Anglegs

Cuisenaire

rods

Cubes

Ruler

Strategy

Perimetrical

along sides

Linear

placement

Linear

placement

Linear

placement

Linear

placement

Team

work

Placement

along sides/

holding on

corners

Place pieces

along sides/

separately

Place rods

along sides/

separately

Place cubes

along sides/

separately

Placement

along sides

by a student

Answers

4

meters

17

25

56

54+20

Measu-

rement

result

A length of

the ribbon

Pieces5:

2r, 2o, 3y,

4b, 6p

Rods6:

1lg, 1b, 2w,

3p, 3b, 3dg,

4b, 4y, 4o

90 cubes

~170cm

Time

1.5

minutes

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

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

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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