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Epistemological Obstacles of Negative Numbers
Abstract
A student asks, "What is the difference between a negative sign and a minus sign?" I cringe. How do I address what I think the student is thinking? Why is it that as soon as a negative number is involved in a mathematical procedure, it suddenly becomes a significantly more difficult question. In this article, I explore the epistemological obstacles around negative numbers.
... Those three things are tightly related with abstract concept. This abstract concept of negative number is known as result in epistemological obstacle in development of mathematical knowledge (Larsen, 2012). ...
... Furthermore, based on Table 4 also found that more than 50% of respondents unable to represent negative number in all contexts. The abstract nature of negative number become main epistemological obstacles which occur in students (Almeida & Bruno, 2014;Altiparmak & Ozdogan, 2010;Blair , 2012;Larsen, 2012). This can occur because thus far student is introduced to number concept as a real thing. ...
... Negative numbers are an abstract concept for which students need phenomenological guidance in order to avoid epistemological obstacles (Larsen, 2012). Heeffer (2011) states that it is important for mathematics teachers to understand the history of the epistemological obstacle of negative numbers in the history of development of mathematical concepts to explore their students' difficulties when learning concepts of negative numbers. ...
This study revealed how students’ understanding of negative numbers and identified their difficulties related with the concept of integer and its counting operation as part of identifying epistemological obstacles about negative numbers. Even though teachers have explained counting operation procedure of integer, but there was concept misunderstanding among students. The concept difference between what was comprehended by respondents in their learning process and knowledge science has resulted in wrong perceptions about the negative numbers. In this article, the authors explained how these misunderstanding in concepts occurred among students and how these ideas were expressed by the students in solving the problem which was related with counting operation of negative integer. This study was part of Didactical Design Research using qualitative approach in negative number learning by involving 96 students of 7th grade as participants in three different schools. This study showed some difficulties by the student to understanding negative numbers in terms of prerequisite knowledge, understanding concepts, procedures, principles and problem solving.
... Berkenaan dengan pembelajaran bilangan negatif, Blair et.al (2012) berpendapat bahwa tidak seperti bilangan positif, bilangan negatif tidak memiliki rujukan persepsi yang jelas, dan karena itu, para siswa harus berusaha lebih keras belajar mengenaibilangan negatif Penelitian yang telah dilakukan yang berkaitan dengan bilangan negatif menunjukkan bahwa bilangan negatif telah menimbulkan persoalan tersediri bagi banyak siswa tingkat menengah (Almeida dan Bruno, 2014 (Larsen, 2012). Brousseau (2002) menegaskan bahwa peran guru adalah untuk mendorong ideide matematika dalam konteks melalui proses penyelidikan. ...
Penelitian ini bertujuan untuk mendesain Hypothetical Learning Trajectory (HLT) pada pembelajaran bilangan negatif sebagai hasil dari tahap pertama Didactical Design Research yaitu Analisis Prospektif. HLT ini merupakan tindak lanjut dari hasil identifikasi Learning Obstacle yang yang dilakukan peneliti dalam pembelajaran bilangan negatif yang terintegrasi dalam materi Bilangan Bulat di kelas 7 sekolah menengah pertama. Observasi mendalam terhadap proses belajar mengajar di kelas yang diamati peneliti memperlihatkan kesulitan guru dalam menanamkan konsep bilangan negatif dan operasi bilangan yang melibatkan bilangan negatif serta beberapa kesalahan konstruksi konsep yang dialami oleh siswa. Istilah HLT merujuk pada rencana pembelajaran berdasarkan antisipasi belajar siswa yang mungkin dicapai dalam proses pembelajaran yang didasari pada tujuan pembelajaran matematika yang diharapkan pada siswa, pengetahuan, dan perkiraan tingkat pemahaman siswa, serta pilihan aktivitas matematika secara berurut. HLT ini disusun berdasarkan analisis terhadap Learning Obstacle, tahap berpikir siswa, dan analisis terhadap kurikulum dengan tetap berpijak pada konsep materi yang harus dipahami siswa.This research aims to design Hypothetical Learning Trajectory (HLT) on learning of negative numbers as a result of the first stage on Didactical Design Research specifically in Prospective Analysis. This HLT is a follow up of the results of the identification of the Learning Obstacle conducted by researcher in learning negative numbers that integrated in the material Integer in the 7th grade secondary school. From the observations conducted by researcher on the teaching and learning process in the classroom showed the difficulties of teachers in embedding the concept of negative numbers and arithmetic operations involving negative numbers and several construction errors of concepts that experienced by the students. Term of HLT refers to a lesson plan based on the anticipation of student learning possibly achievable in the learning process which is based on mathematics learning goals expected on students, knowledge, estimates of the level of students understanding, and the selection of mathematical activity sequentially. This HLT is compiled based on an analysis of the Learning Obstacle, level of students thinking, and analysis of the curriculum that standing on the concept that must be understood by the students.
... In turn, negative integers (negative numbers) and operations with signed numbers can be learned more meaningfully when taught within an algebrafied curriculum (Peled & Carraher, 2006) In Indonesia, negative integers are introduced to fourth-grade elementary students after they have learned whole number and its arithmetic operation in the previous year (Ministry of Education and Culture, 2013). Studies related to negative numbers have shown that the main problem many students experience in learning integer operation is understanding negative number due to its abstract nature (Altiparmak & Ozdogan, 2010;Blair, Rosenberg-Lee, Tsang, Schwartz, & Menon, 2012;Bishop et al., 2014;Larsen, 2012;Heeffer, 2011). Besides, students' learning experience at primary school also influences their comprehension of this concept. ...
This study is a series of Didactical Design Research in prospective analysis stage to comprehensively identify some obstacles in learning negative integer in order to capture how the learning process used by teachers and how the impact on students in understanding the concept of negative numbers. This research is a case study administered to seventh-grade students in one of the classes in a junior secondary school in Indonesia to investigate how the teacher managed the learning process of integers and identify students' thinking construction. Involving a mathematics teacher and 37 students, this qualitative study collected data from field observations, interviews and document review. The data were analyzed by the perspectives of learning obstacles based on Theory of Didactical Situation. Findings show how teacher describes the negative integers and the arithmetic concepts to the students and its contribution in causing misunderstanding on the students. Within the framework of epistemological obstacles, the result of this study is used as a basis for designing a learning program which can overcome problems related to teaching and learning of negative integers.
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