CALCULATOR FOR CRITICAL THINKING IN MATHEMATICSLESSON: EXPERIENCE WITH 4TH GRADE STUDENTS
Sumardyono
SEAMEO QITEP IN Mathematics
Russasmita Sri Padmi
SEAMEO QITEP IN Mathematics
There is perception about calculator within school mathematics setting is that it has no
purpose aside from computing tool and can have negative impact on the students’mathematics ability, especially in lower educational stage. In Indonesia, using calculator in
the classroom is frowned upon and it is prohibited in examination. In this study, we attempted
to answer to this issue by investigating the impact of calculators on the students’ criticalthinking. Through design research, we developed mathematics learning material involving
calculator for 4th grade students. The implementation of the learning materials is conductedin nine schools in Yogyakarta. The data is collected in the form of video recording, which isthen analyzed according to critical thinking framework. This paper reports the result from thefirst cycle of this study. The result is expected to contribute to the literature of mathematics
education and help rectify the perception on the use of calculator in primary schoolmathematics.
Keywords: technology, calculator, critical thinking
Introduction
Ever since calculator reached global calculation in 1970-s, its use in school environmenthas been met with endless debate. Overwhelming majority of education practitioners,
parents, and stakeholders fear that continuous use of calculator will hinder the students’
understanding of basic mathematical procedures and their skills acquisition, especially inlower grades (Banks, 2011). There is also concern that the use of calculator will be detrimental
on their ability to perform mental computation which result in dependency (Lee, 2006).
Calculator is part of technology, yet the all-encompassing term of ICT, which is commonlysuggested as a way to make mathematics lessons more innovative and enjoyable, often
excludes calculator (Savelsbergh et al., 2016).
The widespread apprehension toward calculators manifest itself in the form ofgovernment policies. Many western countries, such as the United States, endorsed
calculators use not only in daily lesson, but also during exams (NCTM, 2000). East andSoutheast Asian countries are typically stricter when it comes to using calculator in school. In

Indonesia, calculator is banned in national examinations (BNSP, 2020), even though there isno specific policies on its use in everyday classroom activities.
However, the bad reputation suffered by calculator regarding its use in primary school
is not supported by research findings. The use of calculator in tandem with pen-and-paper
exercise has been found to positively affect their problem solving skills, since calculator lift
the burden of computation away and let the students focus on their problem solving
strategies (Stacey & Groves, 1994; Wheatley, 1980). The students who use calculators is also
reported to possess better attitude toward mathematics compared to those who do not use
calculators (Ellington, 2003; Hembree & Dessart, 1986). Its availability and affordability is also
viewed in positive light (Kissane & Kemp, 2012).
Considering the favorable findings regarding calculator, it seems plausible that theconcern toward its educational use stands on weak ground. Technology in the classroom,
including calculators, does not have to make traditional pen-and-paper exercise obsolete; infact, with the right strategy, it can improve and reinforce it. It is up to the teachers to not see
it as something to avoid and instead embrace it as a challenge.
In order to encourage teachers to explore calculator’s use in the classroom, further
research centered on best practice is needed. One of the topics, which has not been
mentioned in calculator research, is critical thinking.
Critical thinking is a very important ability, so important that it is hailed as one of theskills needed to persist in 21st century. It is a complex subject encompassing many
characteristic, such as rationality, skepticism, or unbiased analysis. Many definitions exist;
one of them is Facione (2011), which defines that critical thinking consist of six core cognitive
skills, namely interpretation, analysis, inference, explanation, evaluation, and self-regulation.
This study aims to see whether using calculator in primary school can stimulate thestudents’ critical thinking. The research question we attempt to answer through this study is
how does the use of calculator support 4th grade students critical thinking in mathematics
classroom?
Methods
This study is part of a four-cycle design research aimed to design learning materials
incorporating calculators for primary school mathematics. Design research is an approach
whose purpose is to develop theories about the process of learning and designing the meansthat support that learning (Gravemeijer & Cobb, 2006), which we consider suitable for theaim of this study.
Design research consist of three phases namely design, teaching experiment, andretrospective analysis (Eerde, 2013), as depicted in Figure 1.

Figure 1 Phases of Design Research
Based on the current knowledge (K), which comprises literature review, curriculum
documents, and school textbook, the researcher design (D) learning activities. This is the
Design phase. In the Teaching Experiment phase, the activities are put into practice inteaching experiment (E). In the Retrospective Analysis phase (R), the researcher reflect on the
result of teaching experiment, which contribute to new knowledge. The new knowledge thenstarts a new cycle. This study consists of four cycles; this article presents the initial analysis
from the first cycle.
Designing classroom activities
To refute the stereotype that calculator merely serves as computing tools, Ruthven(2003) and Kissane (2017) proposed four ways calculator can be used in the classroom thatemphasis its educational purpose, namely:
1. computation implementation or calculation, which is using calculator to perform
computation otherwise impossible to perform manually by hand,
2. result checking or affirmation, indicating the use of calculator to confirm prediction orassess estimation,
3. trial improving or exploration, implying calculator as a means for students to explore
several possibilities and derive conclusions, and4. structure modelling or representation, which is using calculator to demonstrate numericalstructure or concept.
Therefore, when designing the activities, we made sure to focus on these four strategies
and minimize the use of calculator simply as computing device. The example of the activities,
which are in the topic of fraction representation (namely representing regular fraction asmixed fraction and vice versa), are depicted in Figure 2. There are two parts in this worksheet;the second part more or less follow the same structure and poses similar questions, but the
problems uses different numbers.
The activities are designed for 4th grade students. According to Indonesian national
curriculum, the students in 4th grade already possesses knowledge on division as arithmetic

operation and fraction which is introduced in previous grades. However, the concept offraction introduced in the 3rd grades is part-whole relationship, therefore prior to the teachingexperiment, the teacher introduced the students to the concept of fraction as division.
To bridge the students’ prior understanding on regular fraction to the learning objectives
which is mixed fraction, we start the worksheet with division problems (question 1). The
context introduced is about building solar panels from solar cells, in which the students haveto guess by themselves the mathematics operation needed to solve the problem. This
problem concludes in division with remainder, and when represented as fraction, results inlarger numerator than denominator. By the end of question 1, the students are conjectured
to already identify the result and the remainder of the division.

Figure 2 Example of calculator-enhanced activities in the topic of fraction representation

In Question 2, the students are asked to solve the problem with calculator. However,prior to pressing the button, they have to guess first. We predict that the students will guesswith the result and the remainder they counted in Question 1, hence the conjecture is thatthey will be surprised upon discovering that the calculator displays fraction. The teacher canuse connect this moment to their prior understanding on fraction as division. In Question 3,
the students are asked to press q then n button, resulting in the regular fraction changing tomixed fraction.
In this part of the problem, the students use calculator for its structure modelling or
representation purpose. By demonstrating how regular fraction changes into mixed fraction,and relating it to the result of division, the students are expected to make a connection
between the result of the division and the components of mixed fraction. Through several ofthese examples, the students then derive a conclusion on what a mixed fraction is and how
to represent regular fraction as mixed fraction.
The problem used in this study is contextual and open-ended problems which encourage
critical thinking because it requires students to ask questions, analyse situation, and decide
the right knowledge to apply to solve problems (Bruning, 2005). Throughout the worksheet,
we incorporate many fill-in exercises to make it more interactive for the students. The
students are also encouraged to use calculators with written exercise, because similar to theirteacher, the students are also very cautious about using calculators in the classroom.
Research setting
Prior to the research, all the participating teachers were enrolled in four-day workshop
on utilizing calculators for primary mathematics teaching and learning. The purpose of thisworkshop is, aside from supporting the teachers with technical knowledge about calculator,
is also to introduce how it can be used in educational setting and to socialize the worksheet
and lesson plan. The teachers also installed calculator emulator in their personal computer,to make it easier in displaying calculator interface to the entire class.
This study was conducted in a primary school in Yogyakarta. Twenty 4th grade students
participated in the first cycle. Most come from middle to lower socio-economic class and arefamiliar with calculators, yet its use is prohibited in school. The teacher who normally teachthe class, taught the lesson. During the lesson, the students were split into groups of four.
Data Collection and Analysis
We collected the data in the form of video recording of the classroom discussion. Threecameras, one static and two moving around the classroom, are used to capture the classroom
interaction. Not all interactions were recorded, however, since the decision on what and how
to capture is somewhat influenced by the research question. Therefore, the cameras moved
around the classroom capturing any group that visibly had productive discussion, whichmeans the video recordings are not continuous and chronological documentation of thewhole lesson. Instead, it comprises snippets of discussion and group work that potentiallygives meaningful information to answer the research question.
Watch, discuss

Generate hypothesis

Link to video

Analyze, interpret

Develop Code

Apply Code

Figure 3 Cycle of coding and analysis of video data (Jacobs et al., 1999)
To analyze the data, we followed cyclical process of coding and analysis developed by
Jacobs, Kawanaka, and Stigler (1999) as depicted in Figure 3. First, the team of researchers
watch and discuss the video to acquire the overall result of the teaching experiment andformulate hypothesis. Next, the researcher develop code based on the critical thinking
framework proposed by Facione (2011). Codes are tags or labels used to assign meaning todata collected during the study, to ensure the objectivity and the reliability of the qualitative
data analysis (DeCuir-Gunby, Marshall, & McCulloch, 2011; Jacobs et al., 1999). Codes are notmeant to stay fixed; they will be constantly revised during the data analysis. While no priorresearch have used Facione’s framework to develop video data analysis codes, we argue thatthe framework is general enough to apply for any population and age group, including 4 thgrade students.
The initial version of the code developed for this study is as follows.
Table 1 Codes for critical thinking
Code Meaning
Description
I1

Interpretation

Comprehending and expressing the students’ own understandingof the problem

I2

Inference

To identify and secure elements needed to draw reasonableconclusion; to form conjectures and hypothesisA

Analysis

To identify the intended and actual inferential relationship in the
problem, either among statements or information; making
connection

E1
E2
S

Evaluation
Explanation
Self-regulation

Judging and assessing information available in the problem.
Explaining the results of one’s reasoning
Related to metacognition, self-consciously monitoring,assessing, and evaluating one’s learning

The following step is applying the code to the video data. The researcher watches thevideo and assigning a code to unit of analysis, namely an instance of the students’ verbal or
non-verbal response to the instruction, that is identified throughout the lesson. This instancecan be in the form of the students’ answering the teacher’s question, students discussing with
their peers, or the written work of the students.
The result is compiled and analyzed to investigate to derive conclusion in regard to theresearch question. The conclusion is further compared to the video, which start the cycle allover again. Relevant section is chosen and transcribed to support the conclusion.
Results and Conclusion
While research involving video data usually transcript the video to be followed byanalyzing the transcript, due to time constraint, transcribing the whole video is deemed
impractical for this study. Therefore, the code is applied to the video itself, with timestamp
linking the code to the intended vignettes of the video.
The result suggest that the majority of instances related to critical thinking in the video
recording fall under Interpretation (I1) and Analysis (A). Minimal evidence on Explanation (E2)
was available, as the students had a hard time formulating their thoughts, either in writtenor spoken words. On the other hand, no evidence is identified on Evaluation (E1), Inference(I2), and Self-regulation (S). The evidence are not universally recorded among the students;
some students responded more actively to the teacher’s instruction and have moreproductive discussion than others.

Figure 4 The repeated addition the students did in an attempt to understand the problem.
Interpretation is identified the most frequently during question 1; both in first and second
part of worksheet. The students used different strategies in attempting to understand the
difficulties faced by Donny. One of the strategies are repeated addition (Figure 4), in which
the students add the divisor repeatedly. The student in above picture added the divisor twotimes before stopping and realize what the “difficulty” is. Other strategy that appears isgrouping the object, in this case the solar cell, either by drawing or hand (Figure 5).

Figure 5 The student group the objects by hand
While almost all students could arrive to conclusion that the difficulties faced by Donnyis because there are not enough cells to make whole panels, only a few students can interpretthat division is the mathematics operation needed to solve the problems.
On the other hand, Analysis is identified mostly in Question 3. Initially, the students arehaving difficulty identifying the previously unknown mixed fraction, with some students only
mention the fraction part. The following excerpt (Fragment 1) depicts whole-class discussion
on this question.
Fragment 1
1 Teacher
2 Students
3 Teacher

4 Students
5 Teacher
6 Student A
7 Student B
8 Teacher
9 Student C
10 Student D
11 Teacher
12 Student B
13 Teacher

14 Student B
15 Teacher: .. now the number on the screen is 22/9. What number is this?
: Fraction!
: Now we are going to learn using his calculator. Let’s push the q button –this one – and then n. If we push enter, what do you think we are going toget?
: [inaudible]: let’s try, shall we? One, two, three … what do we get?
: Four-over-nine?
: Four-over-nine!
: Really? Come on, how should we read it?
: There is a four … four over …
: Two, four-over-nine?
: How should we read it? Two …
: … four-over-nine!
: That’s correct, we should read it two four-over-nine. But why two? How
does it connect to your previous calculation? Two four-over-nine, how
come?
: The two comes from there [pointing to the board] the division: This one?

16 Student B
17 Teacher: and the four comes from the remainder. And the nine is the divisor: that is correct!

There is evidence that by connecting the calculator display to the result of divisioncalculated previously, some students are able to conclude that the whole number denotes
the result of division, while the numerator denotes the remainder.
Discussion
In this section, we will discuss how the design of activities contribute to the findings
regarding cognitive skills related to critical thinking found in the video. According to the result
discussed in previous section, it was evident that the cognitive skills related to critical thinking
that occurred with the use of calculator is Analysis. Other cognitive skills that implies criticalthinking emerged too, such as Interpretation, however it mostly coincided with noncalculator question.
There are some explanations that might shed light on this phenomenon. Criticalthinking is learnable, even with young children. Open-ended problems that does not provide
any clues about mathematics concept needed to solve it are often cited as beneficial forhigher order thinking skills or HOTS (Fong, 2000), however associating activities are alsosuggested strategies to develop critical thinking of young learners (Rahman, 2014). In
Question 2, where the students use calculator as means of representation, the calculator
itself support the students to make a connection between the result of division and therepresentation of mixed fraction. Therefore, the use of calculator in primary school can
support the students’ critical thinking, specifically analysis skills, through its Structure
Modelling or Representation purpose.
We suggest future research to venture into other purposes of calculators and othercognitive skills in critical thinking, as well as the connection between the two. The use of
calculator in primary school mathematics is still an interesting idea which is rarely explored,
hence it is still potential for future researcher. However, we also suggest future researchers
to explore other mathematics content.
Lastly, this study is not without limits. We acknowledge the small sample and the limited
timeframe as the main drawbacks of this study. Future researchers are encouraged to
conduct longer studies with larger sample.
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