Pedoman Untuk Penulis
P-ISSN 2527-5615
E-ISSN 2527-5607
Kalamatika: Jurnal Pendidikan Matematika
Volume 6, No. 1, April 2021, pages 45-56
45
STUDENT EVALUATION MATHEMATICAL EXPLANATION IN
DIFFERENTIAL CALCULUS CLASS
Gabariela Purnama Ningsi1, Fransiskus Nendi2, Lana Sugiarti3, Ferdinandus Ardian Ali4
1Indonesian Catholic University of Saint Paulus Ruteng, Jalan Ahmad Yani 10 NTT, Indonesia.
ningsilatib5@gmail.com
2Indonesian Catholic University of Saint Paulus Ruteng, Jalan Ahmad Yani 10 NTT, Indonesia
fransiskusnendi@gmail.com
3Indonesian Catholic University of Saint Paulus Ruteng, Jalan Ahmad Yani 10 NTT, Indonesia
lanasugiarti09@mail.com
4Indonesian Catholic University of Saint Paulus Ruteng, Jalan Ahmad Yani 10 NTT, Indonesia
Ardi0807068703@mail.com
ABSTRACT
This study a im ed to determine tha t the fa ilure of students to eva lua te ma thema tica l expla na tions ba sed on
ma thema tics is influenced by socioma thema tica l norms, tea ching a uthority, a nd cla ssroom ma thema tics
pra ctice. The resea rch method used is the ca se study method. The resea rch da ta were obta ined from inside a nd
outside the resea rch cla ss. The da ta in the resea rch cla ss were in the form of field notes, video recordings of the
cla ss, video recordings of student group work, a nd student work. Da ta outside the resea rch cla ss is the result of
interviews with three interview subjects. By studying the three eva lua tion methods students u sed in eva lua ting
expla na tions, it wa s found tha t ea ch student a pplied a different eva lua tion method a t different times. T h e t h re e
eva lua tion methods contributed to some of the difficulties students experience in eva lua ting their ma thematic a l
descriptions. The results indica te tha t the fa ilure of students in eva lua ting expla na tions is not solely due to
errors in choosing the method, a pproa ch, or lea rning model used but ca n be ca used by socioma thema tica l
norms, a uthority, a nd cla ssroom ma thematics pra ctices a pplied in the cla ssroom .
ARTICLE INFORMATION
Keywords Article History
Cla ssroom ma thematics pra ctices
Eva lua tion ma thematical expla na tion
Socioma tema tic norms
Tea ching a uthority
Submitted Aug 30, 2020
Revised Apr 16, 2021
Accepted Apr 16, 2021
Corresponding Author
Ga ba riela Purna ma Ningsi
Indonesia Ca tholic University of Sa int Pa ulus Ruteng
Ja la n Ahma d Ya ni 10 Ma ngga ra i NTT Tenda , Wa tu, Ruteng, Indonesia
Ema il: ningsila tib5@gma il.com
How to Cite
Ningsi, G.P., Nendi, F., Sugia rti, L., Ali. F.A. (2021). Student Eva lua tion Ma thema tica l Expla na tion in
Differentia l Ca lculus Cla ss. Kalamatika: Jurnal Pendidikan Matematika , 6(1), 45-56.
https://doi.org/10.22236/KALAMATIKA.vol6no1.2021pp45 -56
46 KALAMATIKA, Volume 6, No. 1, April 2021, pages 45-56
INTRODUCTION
One of the essential parts of reforming mathematics education in the mathematics
education study program is to teach students to reason and communicate well mathematically.
It is intended that these students become mathematics educators who can serve their students
in carrying out mathematics learning activities properly and can assess or measure the level of
learning success of these students with specified procedures (Sagala, 2005). Mathematical
reasoning and communication can be developed by mathematics learning activities that do not
emphasize the teaching system of referring to procedures and teacher-centered and viewing
the performance of correct procedures as evidence of the effectiveness of mathematics
learning (Fatimah, 2020). Mathematics learning not only aims to make someone master the
procedure but also develop various mathematical abilities applicable in everyday life.
Mathematical reasoning and communication skills are part of the five standards that describe
mathematical competencies described in the Principles and Standards for School Mathematics
in 2000. The five standards referred to are problem-solving, reasoning and proof,
communication, connection, and representation (NCTM, 2000). According to Qohar &
Sumarmo (2013), mathematical communication is a way for students to express and interpret
mathematical ideas verbally or in writing, either in the form of pictures, tables, diagrams,
formulas, or demonstrations. According to Basir (2015), mathematical reasoning can be used
as a foundation in understanding and doing mathematics as well as an integral part of
problem-solving. Reasoning is different from thinking; mathematical reasoning is the most
crucial part of thinking that involves forming generalizations and describing valid conclusions
about ideas and how they are related (Turmudi, 2008).
Differential calculus is one of the subjects studied by the students of the mathematics
education study program. This course is a course that can be a suitable means for students to
develop mathematical abilities. Matter in differential calculus involves the derivative of a
function. In the differentiation process, the activities involve analyzing the rate of change in
quantity and making predictions about its behavior. To be able to analyze correctly, students
need adequate mathematical skills. Mathematics reasoning skills and mathematical
communication are two mathematical abilities that are indispensable in understanding the
material of differential calculus. With mathematical reasoning, students can formulate proof
and check the truth of an argument against a mathematical problem being solved and draw
Ningsi, dkk 47
conclusions properly and correctly (Rizqi & Surya, 2017). Besides, mathematical
communication skills provide opportunities for students to express rational reasons for a
statement, model mathematical problems into mathematical models, and illustrate
mathematical ideas into a relevant description (Masykur, Syazali & Utami, 2018).
There are many ways to help and encourage students to communicate and reason
mathematically. One way is to give students the opport unity and responsibility to evaluate
mathematical explanations based on mathematics itself (for example, whether the explanation
is valid or makes sense based on mathematics?). Students must be able to question their
explanations as well as the explanations of others based on mathematics. Unfortunately,
students rarely evaluate mathematical explanations but often turn to knowledgeable lecturers
or classmates to solve a problem. Students rarely explain a problem solution. This also
happened to the first semester students of the Mathematics Education Study Program of
UNIKA, Santu Paulus Ruteng. In differential calculus courses, many students have not been
able to evaluate the mathematical explanations they make. Many students are still hesitant to
explain their work and also question their classmates' explanations.
There are several reasons why students in the early semester were unable to evaluate
mathematical explanations: the mathematics teaching and learning activities in secondary
schools are teacher-centered, in which the teacher still holds the authority as the sole evaluator
in learning activities; students reject the role of evaluating mathematics on their own because
they do not like it; students refrain from evaluating because they cannot evaluate the
explanation. Even this is still a habit for these students when they become students in their
early semesters.
Evaluating mathematical explanations in differential calculus courses can be a very
complex activity, given the complexity of the components that make up the explanation. To
evaluate mathematical explanations, we must understand the author's mathematical thinking
and determine whether the mathematics presented in the explanation is convincing or justified.
Therefore, it takes more than developing an understanding of the material differential calculus,
but also being fluent in providing explanations following the norms prevailing in the class, the
authority, and the practice of the mathematics classroom. The norms in question are the
sociomathematical norms that apply in the class. Sociomathematical norms are normative
behavior of students in mathematics class, which is a way to participate in all mathematics
48 KALAMATIKA, Volume 6, No. 1, April 2021, pages 45-56
activities in the class community (Kadir, Jafar, Jazuli, & Ikman, 2018). Sociomatematic norms
are social norms related to the nuances of mathematics because sociomathematical norms
specialize in learning mathematics rather than other learning. If students talk mathematics,
they must be learning about mathematics (Piccolo, Harbaugh & Carter, 2008). Sociomatematic
norms are linked explicitly to mathematical argumentation, which is how students carry out
interaction and negotiation to understand mathematical concepts such as understanding what
kind of arguments can be accepted mathematically (Sulfikawati, Suharto, & Kurniati, 2016).
The authority referred to in this study is the authority of teachers/lecturers in dominating all
learning processes, causing low student activity, which results in low student/student ability to
evaluate mathematical explanations based on mathematics, which leads to low mathematics
learning outcomes (Sumaryati et al., 2013) ). Classroom mathematics practice is a learning
practice applied in mathematics classrooms that includes all mathematics learning processes
from beginning to end of learning (Pramudya et al. 2020).
This study aimed to better understand how authority, classroom mathematics practice,
and sociomathematical norms can prevent students from evaluating mathematical explanations
based on mathematics.
METHOD
This research is limited to the student's evaluation of mathematical explanations in the
differential calculus class, so the researcher uses the case study method. This method was
chosen so that researchers get several things that contain the various difficulties of students in
evaluating the mathematical explanations contained in the differential calculus class.
According to Polit & Beck (2004), case study research is a form of qualitative research
based on human understanding and behavior based on human opinion, and subjects in research
can be individuals, groups, agencies, or communities. The research steps used were adapted
from the steps proposed by Rahardjo (2017). First, selecting themes, topics and cases, in this
step the researchers selected cases in the field of interest and from the results of observations
in the differential calculus class. Second, reading the literature, in this stage the researchers
read all sources of information related to the predetermined research topic. Third, the
formulation of the focus and research problems, this stage aimed to make the researchers
concentrate on one point: the center of attention. Fourth, data collection was carried out in the
research classroom consisting of 21 students. The data consisted of field notes, video
Ningsi, dkk 49
recordings taken in class, video recordings of student group work, student work, and data
outside the classroom, namely data from interviews from three students who were taken after
the researcher classified the 21 students in the class into three small groups based on their
level of confidence and ability in their performance in the classroom and in mathematics as a
whole. Fifth, data improvement; in this stage, the researchers read the entire data by referring
to the formulation of the problem posed. The data were considered perfect if the problem
statement was believed to be answered with the available data. Sixth, data processing; in this
stage, the researchers checked the correctness of the data, compiled the data, carried out
coding, classified data, and corrected unclear interview answers. Seventh, data analysis, the
researchers read the entire transcript to obtain general information from each transcript. The
general information was compiled to take specific information; from this specific information,
general patterns are known. The data was then grouped based on the sequence of events,
categories, and typology.
RESULT AND DISCUSSION
In this section, the interviews of the three research subjects are explained. The form of
the evaluation method used by the three students are also described .
Subject 1
Background. Subject 1 is a student who has a positive view of mathematics. When asked
about his experiences in learning mathematics, he said:
"I liked math from an early age because my parents always helped me learn math
(arithmetic) from an early age."
Because of his fondness for mathematics, his motivation to learn mathematics is very high. At
the beginning of the lecture, Subject 1 admitted that he had difficulty keeping up with the
rhythm of the lecture. This is due to the unfamiliar learning situation and the different teaching
methods of the lecturers from those experienced in high school. Initially, he did not like how
lecturers taught that they did not directly provide material and explained the correct way to
solve some problems. He said:
"If the lecturer does not directly explain the material and how to solve the questions, it
will take much time, and we will not understand the material."
However, at the end of the calculus lecture, Subject 1 ad mitted that learning from experience
was more meaningful than just providing material and memorizing it.
50 KALAMATIKA, Volume 6, No. 1, April 2021, pages 45-56
Evaluation Method. The evaluation method used by subject 1 has three distinctive features.
First, he used his lecturer to evaluate his explanation by listening to the explanation, using the
feedback given by the lecturer in commenting on his work, and then explaining it in classroom
mathematics practice when the lecturer approved it. Evidence of this subject using lecturers'
expressions to evaluate their explanations can be found through interviews where he said:
"I hold the principle that whatever my lecturer says is absolute truth. When the
lecturer asked me the truth of my work over and over again, I would think that my job
was wrong, even though this is not necessarily the case ”.
Second, he used classroom mathematics practice to evaluate his work and explanation by
showing his work in a mathematics classroom in front of all class members. The proof of this
subject using classroom mathematics practice in evaluating the correctness of its explanation
is identified in the interview where he said:
"… I will ask questions and see the expressions of my friends and lecturers in
understanding my explanation. When their expressions are good/happy, then I am sure
my work is correct, but when they show confused expressions, I will worry that my
work is wrong, and I start to see every concept I use…. ".
Third, Subject 1 also used the authority given by his lecturer in evaluating his explanation.
When the lecturer allowed him to present his work in front of the class members, he believed
that the explanation was correct; this can be known through the results of an interview where
he said:
"I believe my job is right when the lecturer gives the opportunity to account for the
results in front of all class members."
Based on several evaluation methods used by Subject 1 above, it can be said that Subject 1 can
evaluate his explanation/work when there is an expert/expert in the class, namely the lecturer
himself. When there is no expert guiding, Subject 1 will have difficulty evaluating his work.
This also shows that classroom mathematics practice is very influential for subject 1 in
evaluating mathematical explanations.
Subject 2
Background. Subject 2 is one of the students who started to like mathematics when taking
mathematics lessons in junior high school. When asked about his experiences and views on
mathematics, he said:
Ningsi, dkk 51
"At first, I did not like mathematics because I did not understand the material being
studied, and the teachers who taught mathematics tended to be tough and coercive in
teaching math material, so I did not like mathematics. I liked mathematics when I
entered junior high school in grade VIII because the teacher taught math material in a
fun way".
Evaluation Method. In answering some of the questions in the interview, Subject 2 said:
"Initially, I preferred the lecturer to tell us the concepts used in solving problems and
immediately give an assessment of my work. I do not like wordy things. Usually, I
judge whether my work is right or wrong through comments from my lecturers and
classmates, whom I think have good abilities. Sometimes even though my work is right,
I do not feel confident when I see the expression from the lecturer, even though the
lecturer is only testing my understanding. Even though my lecturer's expression
showed a "negative" reaction (which I saw), he did not immediately blame my work
but asked, "why do you use this concept, why not this one? Why should this be done?
". At the beginning of the lecture, I really did not like this method, but at the end of the
lecture, I realized the benefits of the method used by my lecturer. So I disagreed with
my friend, who said that the method used by my lecturer was not suitable. Besides all
that, I like to attend lectures because the class situation is very relaxed, not tense, and
the lecturer considers us to be discussion partners. However, it is not uncommon for
me to still be unable to evaluate my differential calculus work ".
Furthermore, when asked further about how he evaluated his work, he said that he still used
the expressions of lecturers, friends, and the answer key given to evaluate his work until now.
Even though, he liked how the lecturers helped them evaluate their work, he still did not have
the confidence to make a correct assessment of her work. From the results of this interview, it
can be seen that Subject 2 used classroom mathematics practice and the authority used by the
lecturer as a way to evaluate the truth of his explanation/work.
Subject 3
Background. Subject 3 is a student with a unique thought about choosing a mathematics
education study program. When asked about his experiences and views on mathematics, he
said:
"I did not like mathematics until now because I found it very difficult to understand
52 KALAMATIKA, Volume 6, No. 1, April 2021, pages 45-56
math material and how to teach teachers that were difficult to understand. However, I
aspire to become a math teacher. My goal arose because I wanted to help students
who have difficulties like me in learning mathematics. I also want to make them
understand that mathematics will not be difficult to understand if we persistently study
and study it repeatedly. Because these ideals make me always motivated to study
mathematics even though I have to work hard ".
Evaluation method. In evaluating his explanation, this subject used classroom mathematics
practice, as indicated by the results of the interview, where this subject said:
"I usually evaluate my work by asking the teacher/lecturer, classmates, and the math
books used. Even though my math skills are inadequate, I still try to solve the practice
questions following the concepts explained by the lecturer. When the learning method
used by the teacher is what I expected, it will really help me evaluate the explanation.
Sometimes, I like the method used by the teacher/lecturer in lectures, but it all depends
on the level of difficulty of the material we are studying. I do not like it when the
lecturer says, "ok, today you are divided into several groups and discuss some things."
This method will not make it easier for me to understand the material, especially since
the level of material in the lecture process is higher. When I do not understand the
material, it will be difficult for me to evaluate my work. I prefer the lecturers to
explain the material to be studied rather than discuss the material ourselves and
discover essential concepts ourselves ".
Besides, this subject also used sociomathematical norms in its evaluation. This was indicated
by the interview answer given by this subject, in which he said:
"Usually, to evaluate mathematical explanations, I first think about the reasons for
each of these steps being used in solving a problem. The explanation must also be
clear and valid. When the reason is invalid, the explanation is rather difficult to
follow; then, I believe that the explanation is wrong. In addition, an explanation must
be simple, and conclusions can be drawn. If not, then I would doubt the explanation
given either by friends or by the lecturers”.
Based on the interviews with the three subjects above, it can be concluded that the
three subjects experience difficulties or make mistakes in evaluating mathematical
explanations. Subject 1 experience difficulties when the lecturer does not evaluate his
Ningsi, dkk 53
explanation and does not express what he expected. In addition, this subject also make
mistakes in evaluating when the lecturer does not give him the authority or opportunity to
explain or present his work in front of class members. When the lecturer does not give the
opportunity, this subject think that his work is incorrect, even though this is not always the
case. Subject 2 experience difficulties in classroom mathematics practice when the lecturer
and his friends do not comment on their work. Besides, this subject incorrectly evaluate his
work when he misreads the expression of the lecturer when viewing or examining his work.
Subject 3 have difficulty evaluating the explanation when the learning method used is not
what he expected. In addition, when the explanation is complicated and difficult to understand,
this subject immediately say the explanation is wrong. This is fateful because a complicated
and not coherent explanation is not necessarily wrong.
CONCLUSION
Based on the data from interviews with three research subjects, it can be concluded
that the application of authority, sociomathematical norms, and mathematics practice in the
classroom has an impact on students' evaluation abilities and can contribute to the failure of
students to evaluate mathematical explanations based on mathematics. If the teacher applies
sociomathematical norms that contain excessive authority or teacher-centered learning in
classroom mathematics practice, it will negatively impact the students. The negative impact
can be in the form of low evaluation ability of students or in assessing or evaluating their math
work. Another impact can be in the form of student difficulties in solving mathematics
problems.
ACKNOWLEDGMENTS
Researchers would like to thank the Indonesian Catholic University Santu Paulus
Ruteng and the three research subjects who have provided the opportunities, facilities, and
infrastructure for researchers to carry out this research.
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