Pedoman Untuk Penulis
P-ISSN 2527-5615
E-ISSN 2527-5607
Kalamatika: Jurnal Pendidikan Matematika
Volume 6, No. 2, November 2021 pages 127-142
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127
THE VALIDITY AND RELIABILITY OF INSTRUMENTS FOR
MEASURING ELEMENTARY SCHOOL STUDENTS' EARLY
MATHEMATICAL ABILITY
Aan Yuliyanto1, Turmudi2, Ernawulan Syaodih3, Dadan Rusdiana Saputra4, Arie
Dharmawan5, and Cahya Karisma Pertiwi6
1Universitas Pendidikan Indonesia, Bandung, Indonesia.
aanyuliyanto@upi.edu
2Universitas Pendidikan Indonesia, Bandung, Indonesia.
turmudi@upi.edu
3Universitas Pendidikan Indonesia, Bandung, Indonesia.
ernawulansy@upi.edu
4Benda 1 Elementary School, Indramayu, Indonesia
saputradadan690@gmail.com
5Klender 06 Elementary School, Jakarta Timur, Indonesia
ariedharmawan007@gmail.com
6Universitas Pendidikan Indonesia Purwakarta Campus, Purwakarta, Indonesia
cahyakarisma@upi.edu
ABSTRACT
Teachers need to understand students' early mathematical abilities before continuing learning on the next topic
to retain the knowledge. This study aims to produce appropriate and reliable instruments for quality research
related to early mathematical abilities. This research implemented R&D. The subjects were 113 sixth-grade
students of the elementary school in Karawang. The instrument used was a test to measure early mathematical
ability. Validity and reliability tests indicated that the five initial mathematical ability test items were
considered valid, with r count > r table and p-value <0.05. The Cronbach's Alpha value was 0.875 (above 0.8 or
high reliability). Thus, the five items of the early mathematical ability instrument on the volume of cubes and
rectangular prisms can be used for further research to measure the same variables accurately. The results are not
significantly different for the same subject even though the time and place are different.
ARTICLE INFORMATION
Keywords Article History
Early Mathematical Ability
Instruments
Cubes and Rectangular Prisms
Submitted Jan 22, 2021
Revised Sep 1, 2021
Accepted Oct 26, 2021
Corresponding Author
Aan Yuliyanto
Primary Education Study Program, School of Postgraduate Studies, Universitas Pendidikan Indonesia
Jl. Dr. Setiabudhi N. 229 Bandung 40154
Email: aanyuliyanto@upi.edu
How to Cite
Yuliyanto, A, Turmudi, T., Syaodih, E., Saputra, D.R., Dharmawan, A., & Pertiwi, C.K. (2021). The Validity
and Reability of Instruments for Measuring Elementary School Students’ Early Mathematical Ability.
Kalamatika: Jurnal Pendidikan Matematika, 6(2), 127-142.
https://doi.org/10.22236/KALAMATIKA.vol6no2.2021pp127-142
http://creativecommons.org/licenses/by-sa/4.0/
http://creativecommons.org/licenses/by-sa/4.0/
mailto:aanyuliyanto@upi.edu
mailto:turmudi@upi.edu
mailto:ernawulansy@upi.edu
mailto:saputradadan690@gmail.com
mailto:ariedharmawan007@gmail.com
mailto:cahyakarisma@upi.edu
mailto:aanyuliyanto@upi.edu
128 KALAMATIKA, Volume 6, No.2, November 2021, pages 127-142
INTRODUCTION
Mathematics is taught to students to help them organize logical reasoning, display
personalities, and use mathematics and mathematical reasoning in real-life situations
(Soedjadi, 2004). Mathematics in schools has its own goals, often meaningless
manipulation of numbers, while mathematics serves as a means to achieve other goals,
gives meaning to calculations related to everyday life, as a useful context for
concentration, and can support persistence (Galla, Esposito, & Fiore, 2020; Rampal, 2003).
This goal will be achieved if students' understanding of the previous topic is good. Prior
understanding is very important for students and is a target for the learning process
(Shabrina & Sumiati, 2020).
However, mathematics teachers in elementary schools sometimes tend to convey
material directly without paying attention to students' understanding of the previous
material. This is evident in studies that found several obstacles faced by teachers. For
example, when the teacher teaches thematic material in the fourth grade, the acquisition of
students' knowledge competencies is not optimal; this can be proven by the competence of
previous knowledge below the Minimum Criteria of Mastery Learning (Luh, Santiasih,
Ganing, & Sujana, 2016). Previous understanding will affect how a person is involved in
the new understanding (Ardiyanti, 2016). Previous understanding can build and develop
mathematical concepts learned from understanding (Fatqurhohman, 2016). Prior
understanding can be used to be aware of situations and determine strategies through
thinking (Salayan & Ariswoyo, 2020).
In learning mathematics, students need a solid foundation to understand the
previous material to reach the next level. For example, to understand the topic of the
volume of cubes and rectangular prisms, students must understand previous topics or
prerequisites, such as arithmetic operations, area, and perimeter of shapes, measurement of
units of length and volume, story problems, and other prerequisite topics. The previous
understanding is referred to as Early Mathematical Ability. Early Mathematics Ability is
the student's ability to understand the prerequisite material (Yuliyanto & Turmudi, 2020).
Students who know previous materials could follow and do the next learning (Jamaan,
Musnir, & Syahrial, 2020; MacDonald & Carmichael, 2018). Early mathematical abilities
can affect stable characteristics, affecting mathematics achievement over time (Watts et al.,
2017). Students' early mathematical abilities also affect students' mathematical dispositions
(Noviana, Hadi, & Handayani, 2020). Thus, early mathematical ability is a prerequisite
ability to support mathematics learning in subsequent topics.
Yuliyanto, Turmudi, Syaodih, Saputra, Dharmawan, & Pertiwi 129
Previous research related to Early Mathematical Ability stated that the Realistic
Mathematics Education Approach not only improves mathematical literacy in eighth-grade
junior high school students with high early mathematical abilities, but also students with
low early mathematical abilities on statistics topics (Sutisna, Budi, & Noornia, 2018).
Regarding the interaction of early mathematical abilities with learning, it was found that
there was no interaction between learning and Early Mathematical Ability in improving the
reasoning abilities of eighth-grade junior high school students (Ayal, Kusuma, Subandar,
& Dahlan, 2016). Through qualitative research, the development of students' early math
abilities could be handled well by not only focusing on early math concepts at home but
also by focusing on developing learning behaviors, such as engagement, resilience,
curiosity, and challenge seeking (Kritzer, 2012). The parental reports about the amount of
children's activity at home predicted the children's performance on standardized early math
ability tests (Blevins-Knabe & Musun-Miller, 1996). The problem-based learning approach
had a significant effect on the critical thinking skills of high school students in terms of
school level and early mathematics abilities (Widyatiningtyas, Kusumah, Sumarmo, &
Sabandar, 2015).
Previous studies above have revealed that early mathematical abilities have been
examined quantitatively and qualitatively for junior high school and high school students.
It also has been examined based on the parents and related statistics. However, instruments
for measuring students' early mathematical abilities, especially in elementary schools, are
limited. Therefore, this research revealed an instrument to measure the early mathematical
ability of fifth-grade elementary school students on the topic of cubes and rectangular
prism using the R&D research method.
A tool is needed to measure students' early mathematical abilities appropriately to
conduct research or learning mathematics by reviewing students' early mathematical
abilities before proceeding to the next topic. Thus, it can be used to determine the
students’ previous understanding of the topic being taught. That is because to produce
high-quality research; reliable instruments are needed (Putri, Wahyudy, Yuliyanto, &
Nuraeni, 2020). An assessment instrument is a tool used to make an assessment or
evaluation, an instrument can be a test or non-test, and observation can be carried out in
two ways, namely systematic and non-systematic observation. (Rafianti, Anriani, &
Iskandar, 2018). The Good instruments meet certain rules, provide accurate data according
to their function, and only measure samples of certain behaviors. The characteristics of a
good evaluation instrument are valid, reliable, relevant, representative, practical,
130 KALAMATIKA, Volume 6, No.2, November 2021, pages 127-142
discriminatory, specific, and proportional (Siswantari & Maretha, 2020). Furthermore,
empirical validation is carried out through validity and reliability tests to ensure that they
can measure the measured variables and produce similar results even though they are used
repeatedly (Putri et al., 2020). The validation of the research instrument aims to measure
whether the instrument made is per the measurement assessment by the validator
(Setyansah, 2020).
Thus, a good instrument must have validity, namely the accuracy of an instrument
in measuring the variable to be measured, and reliability, namely the instrument's stability
in measuring the measured variable repeatedly, at different times and places with the same
subject. Validity can identify and be useful in determining the management of education
compared to tests based on theoretical frameworks and other perspectives (Scheuer,
Herrmann, & Bund, 2019). A valid instrument can estimate the effect of an unbiased
treatment. However, at the same time, it is not possible to ensure that all the assumptions
necessary for the validity of the instrument have been met. (Rassen, Brookhart, Glynn,
Mittleman, & Schneeweiss, 2009). Thus, the validity measurement must be executed
through logical validity with expert considerations and empirical validity by testing on a
predetermined sample that is not part of the sample to be studied, at least one level above
it. Validity can be quantified using Pearson’s product-moment correlation coefficient (r)
(McNamara, Hudson, & Taylor, 2010).
Meanwhile, reliability is measured by determining the Cronbach's Alpha value
because this study will use an essay-formed test instrument. Cronbach's Alpha is suitable
for instruments in the form of essays or questionnaires (Yusup, 2018). Reliability can be
understood as the consistency of test measurements when the measuring procedure is
repeated (Baumgartner, Oh, Chung, & Hales, 2002; Scheuer et al., 2019). Reliable
instruments will achieve the same conclusions when applied to the same subjects at
different times (Fan, 2018). Thus, this research will produce an appropriate and reliable
instrument to measure the early mathematical abilities of fifth-grade elementary school
students on the topic of volume cubes and rectangular prisms through validity and
reliability trial.
METHOD
This research employed the R&D method. R&D is a systematic process to develop,
improve, and assess education programs and materials (Gall, Gall, & Borg, 2010; Jackson,
2009). The design used was ADDIE (Analysis, Design, Development, Implementation, and
Yuliyanto, Turmudi, Syaodih, Saputra, Dharmawan, & Pertiwi 131
Evaluation). The purpose of this study was to produce an accurate and reliable instrument
for measuring the early mathematical abilities of fifth-grade elementary school students on
the volume of cubes and rectangular prisms. This study was conducted because some
teachers still pay less attention to the extent of students' early mathematical abilities and
tend to directly continue the topics that must be taught to their students. At the same time,
the early mathematical ability of students is necessary to be understood by teachers before
continuing with more complex materials requiring fundamental concepts. Studies
explained that early mathematical abilities can be in the information of concepts,
principles, procedures, and facts that a person already has (Nismawati, Nindiasari, &
Mutaqin, 2019). Sixth-grade elementary school students in West Java were the population
in this research. The samples were 113 elementary school students in Karawang Regency,
West Java, selected by purposive sampling. The results of the development of this test
instrument used to measure the early mathematical abilities of fifth-grade elementary
school students. The topic developed was related to the volume of cubes and rectangular
prisms, so students must understand prerequisite topics, including numerical count
operations, calculating square roots, determining the area of a two-dimensional shape,
measuring long units, and finally, problem-solving.
The scoring guidelines in this study were modified from Facione (1994), as
presented in Table.
Table 1. Guidelines for Scoring Students' Early Mathematics Ability
Score Criteria
4 The solution is explained in full, almost all instructions are followed, the presentation is logical according to mathematical
concepts, and there are no drawing/calculation errors.
3 The solution is explained correctly, there are few errors in calculation/drawing, and the presentation is logical.
2 Solutions are explained incorrectly, answers appear to be trial and error, and the presentation is less logical
1 The interpretation is incorrect, the answers seem trial and error, and the presentation is not logical
0 No response
The development of instruments was carried out based on logical validity and
empirical validity. In logical validity, by considering three math experts in elementary
schools based on the accuracy according to the content studied, the accuracy of wording,
and psychological constructs, a readability test was carried out on students. Furthermore,
empirical validity is taken by testing it on students who are not research samples,
specifically at least one level above it, i.e., students in six-grades elementary schools; this
is because these students have been deemed to have mastered the topic be tested. Data
analysis using SPSS 25. The validity decision is identified based on the value of the
correlation coefficient (rxy) and reliability based on the value of Cronbach's Alpha. If the
sign on the validity test < 0.05 and tcount is positive and > rtable, the instrument is considered
132 KALAMATIKA, Volume 6, No.2, November 2021, pages 127-142
valid (Mahendra, 2015). Meanwhile, if the value of Cronbach's Alpha> 0.70 then the
instrument is accepted, while Cronbach's Alpha> 0.8, the reliability is very good (Wells,
Russell, Haraoi, Bissonnette, & Ware, 2011). The interpretation of validity and reliability
is based on the criteria developed by Guilford (1956) in Table 2.
Table 2. Interpretation of Test Validity and Reliability of Instruments
r11 Interpretation of Reliability rxy Interpretation of Validity
0.80 to 1.00 Very High 0.90 to 1.00 Very high
0,60 to 0,80 High 0,70 to 0,90 High
0,40 to 0,60 Intermediate 0,40 to 0,70 Moderate
0,20 to 0,40 Low 0,20 to 0,40 Low
< 0.20 Very Low 0.00 to 0.20 Very Low
< 0,00 Not Valid
RESULTS AND DISCUSSION
Early Mathematical Ability
It has been explained that the early mathematical ability is the prerequisite ability
that students have for understanding the next topic. In this research, early mathematical
abilities were developed to understand the prerequisite abilities of students to study the
topic of volume cubes and rectangular prisms. The blueprint for the initial mathematical
ability instrument produced is shown in Table 3.
Table 3. Blueprint early mathematical ability instrument
Items Indicator Questions
Level of
Difficulty
1 Doing arithmetic calculations in
numerical count operations
8 - 6 x 8 : 10 + 2 = .... Easy
2 Calculating the square root √225 x (3√1000 - 3√125) = .... Moderate
3 Determining the area of a two-
dimensional
Pay attention to the shape below. Calculate the area of the
three shapes!
Difficult
4 Calculating the measurement of
length units
5,000 cm + 15 km x 5 dm - 8 dam = ...... m Difficult
5 Solve problem-solving problems Mr. Adi wants to make a terrace on the left and right of his
villa with a length of 7 meters and a width of 5 meters for the
left side terrace and the right-side terrace, the size of each
side is 4 meters. Then how wide are the two terraces?
Difficult
Yuliyanto, Turmudi, Syaodih, Saputra, Dharmawan, & Pertiwi 133
Students are required to understand several topics at the previous meeting
developed in this research to understand the volume of cubes and rectangular prisms, such
as in item 1 about doing arithmetic calculations in numerical count operations with
problems 8 - 6 x 8: 10 + 2 = ... This problem will encourage students' abilities when
performing volume calculation operations for cubes and rectangular prisms. Furthermore,
in item 2 about calculating the square root with the problem √225 x (3√1000 - 3√125) = ...
This problem will help students when learning the operations to calculate squares and
cubic on the volume of cubes and rectangular prisms. Furthermore, in item 3 about
determining the area of a two-dimensional like the problem in Figure 1.
Pay attention to the shape above. Calculate the area of the three shapes! In this
problem, students are trained to discover the area of combined shapes; the questions will
be useful for understanding the calculation of the volume of cubes and rectangular prisms
combined. Furthermore, item 4 is about calculating the measurement of length units such
as 15,000 cm + 15 km x 5 dm - 8 dam = ...... m. This problem will help students remember
how to modify the unit of length, which will help students how to modify the volume unit.
Furthermore, item 5 is about solving problem-solving problems such as the following
questions: Mr. Adi wants to make a terrace on the left and right of his villa with a length of
7 meters and a width of 5 meters for the left side terrace and the right-side terrace, the size
of each side is 4 meters. Then how wide are the two terraces? This question will train
students on story problems in determining the volume of cubes and rectangular prisms, for
example, in questions of determining the volume of bath water and dipping water. After
the five instruments were developed, then the instrument validation was conducted. The
validation of the research instrument aims to measure whether the instrument made is by
the measurement assessment by the validator (Haryanti & Saputro, 2016; Setyansah,
2020).
Figure 1. Problem Finding the Combined Area of the Construct.
134 KALAMATIKA, Volume 6, No.2, November 2021, pages 127-142
Validity Test Analysis
The five items that have been constructed through logical validity are then tested
for empirical validity. Following are the results of the validity trial presented in Table 4.
Table 4. Analysis of Early Mathematical Ability Validity Test
Item
Correlation
Value (r count)
r table (⍺=5%, k= n-
2=111)
Direction of
Correlation
p-
value
Criteria Conclusion
1 0.730
0.178
positive, rcount > rtable 0.000 High Valid
2 0.877 positive, rcount > rtable 0.000 Very High Valid
3 0.871 positive, rcount > rtable 0.000 Very High Valid
4 0.809 positive, rcount > rtable 0.000 Very High Valid
5 0.840 positive, rcount > rtable 0.000 Very High Valid
Based on the results of the item validity test in table 4, all items have a significant
value with p-value < 0.05 and all rcount > 0.178 = rtable and are positive so that all items are
valid with the criteria item 1 is high, and others are very high in terms of the correlation
coefficient according to (Guilford, 1956). The study said that the instrument is classified as
valid if it has a positive and significant coefficient < 0.05 (Muhsin, Slamet, & Wahyudin,
2017; Yusof, Bahari, & Adnan, 2014). It is also known that the lowest correlation value is
0.730. Meanwhile, the correlation coefficient <0.3 is low, 0.3-0.5 is moderate, while> 0.5
is high (Tsang, Royse, & Terkawi, 2017). Instruments with a minimum correlation of 0.5
are considered capable of uncovering important and relevant issues to be observed
(Masood, Masood, Saub, & Newton, 2014). Thus, all items are considered to measure the
early mathematical ability of the volume of cubes and rectangular with high accuracy.
Reliability Test Analysis
The analysis of the validity test indicates that the five items of the instrument are
classified as valid and can measure correctly. To discover the instrument's stability in
measuring the same subject, but at different times and places, a reliability test was carried
out by determining the Cronbach's Alpha value. The summary of the instrument reliability
test is listed in Table 5.
Table 5. Results of Early Mathematical Ability Instrument Reliability Tests
Cronbach’s Alpha N of Items
0.875 5
Cronbach's Alpha value shows the instrument has very good stability. Research
says that an instrument with a Cronbach's Alpha value> 0.70 can be concluded that all
statements are reliable and can be used for further analysis (Astutik & Priantono, 2020;
Bolarinwa, 2015; Lima-Rodríguez, Lima-Serrano, & Domínguez-Sánchez, 2015; Tsang et
al., 2017). Because the instrument has excellent consistency, the instrument can be used to
measure variables that are measured repeatedly with similar subjects even though the place
Yuliyanto, Turmudi, Syaodih, Saputra, Dharmawan, & Pertiwi 135
and time are different.
Face Validity
Researchers in this study also investigated the advanced validity of the early
mathematical ability instruments based on two experts in the field of mathematics
education in elementary schools. The quality of the instrument is assessed based on three
aspects such as the suitability of the instrument with the indicators, the suitability of the
instrument with the material, and the readability of the instrument. Measurement of
instrument quality using a scale between 1-5. The following is the results of the instrument
quality assessment based on face validity in Table 6:
Table 6. Face Validity Results of Early Mathematics Ability Based on Expert Assessment
No Assessment Aspect Average Criteria
1 Suitability of the instrument with the indicators 3.5 Worth using/ testing with revision
2 The suitability of the instrument with the material 4.7 Worth using/ testing with revision
3 Instrument readability 2.8 Worth using/ testing with revision
Total Average 3.67 Worth using/ testing with revision
Based on Table 6, the five items of the instrument, according to the experts, are
considered suitable to be used to measure early mathematical abilities by requiring some
improvements. The research revealed that the eligibility criteria for a product were
observed based on a range of 1-5, namely 1.00-2.33 the product was considered unfit for
use/tested, 2.34-3.67 the product was considered Eligible to be used/tested with revisions,
and 3.68-5,00 products are deemed Eligible for use/ tested without revision (Arikunto,
2012). Reviewing the results of the instrument quality assessment according to the experts
shows that the five items are considered good enough and deserve to be tested on the
respondents to measure how well the students' early mathematical abilities are with a slight
improvement before being tested. An in-depth assessment of an instrument must be carried
out to find out how appropriate the measuring instrument is in measuring the aspects
measured when used in the field. Supporting this, before implementation, the product
developed was assessed for quality first by asking for an assessment from a team of experts
(Wijayanti, Saputro, & Nurhayati, 2015).
CONCLUSION
The development of early mathematical ability instruments related to the volume of
cubes and rectangular prisms have been consulted with experts and tested on 113 fifth-
grade elementary school students in Karawang. The validity and reliability test showed that
the five items have good accuracy and reliability in measuring early mathematical abilities
on the volume of cubes and rectangular prisms and can measure the same variables and
136 KALAMATIKA, Volume 6, No.2, November 2021, pages 127-142
subjects at different times and places in subsequent research. Early mathematical abilities
are considered the foundation for students in studying the topics to be studied, so teachers
are suggested to understand students' early mathematical abilities before continuing
learning. To understand students' early mathematical abilities, teachers can use an
instrument of a test with appropriate aspects and indicators as developed in this study.
ACKNOWLEDGMENTS
The researchers would like to thank the teachers and principals of Cikampek 2
Public Elementary School and the West Cikampek 3 Elementary School for allowing the
researchers to develop instruments for early mathematical abilities on the topic of cubes
and rectangular prisms for fifth-grade students.
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