Table of Contents
One of the most important subjects in our human life is mathematics because nothing is possible in the world without the knowledge of mathematics (Acharya, 2017, p. 8). Integers are a part of the mathematics curriculum which is considered to be a very important part of middle school as it symbolizes a move from concrete thinking to abstract thinking (Lamb & Thanheiser, 2006 p. 163). Also, Harris (2010), who entered the Doctor of Philosophy (Ph.D.) program in Mathematics Education at the University of Texas at Austin, stated that its number and operations concept is foundational in mathematics that can be applied in other fields like statistics, business, and science. It also contains a positive and negative sign (-,+) therefore, a minor error that affects the sign in solving integers can have a serious impact on the answer as it may result in an inaccurate interpretation of the solution (p. 1).
Mathematics can be brought in action in different areas like in technology, economic production and transaction, management and decision-making, information and communications procession, automatization and security procedures, and in people’s daily routines (Skovsmose, 2016, p. 9). Charles and Lester (1984), a professor at San Jose University, California and an author from Indiana University, added that this is also essential in almost every profession, such as business, science, weather prediction, medicine, engineering, architecture, and economics (as cited in Mkomange, Ilembo, & Ajagbe, 2012, p. 477). It has significant use in all human activities including school subjects like Biology, Physics, and Chemistry (Unodiaku, n.d., p. 1). Another thing is, good competency in mathematics is an important methodology in economics and social sciences. It is important in technology as mathematics can be considered as a language of nature and technology. Technology provides a way how to teach the students in solving different kinds of tasks (Pohjolainen, Nykanen, Venho, & Kangas, 2018, p. 1221; Mkomange et al., 2012, p. 483).
The activities in mathematics like numerical weather predictions, analyses of financial risks, and internet searches are crucial to economic growth, national competitiveness, and national security (National Research Council, 2013). Poorghorban, Jabbari, & Chamandar (2018), faculty member in College of Education, researcher at Ministry of Science and Technology, and child psychologist at Pooyesh Counseling Center, pointed out that the knowledge of math helps people to gain a precise understanding of information patterns and reasoning (p. 121). National Governors Association Center for Best Practices [NGA Center] (2010), a research and development firm that directly serves the nation’s governors (NGA, 2015), added that it also enables them to solve the problems existing in the society, workplace, and in everyday life wherein they can analyze and draw conclusions mathematically (as cited in Miller, 2013, p. 5).
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According to Suherman (2001), staff directory in Syarif Hidayatullah, Jakarta (Pustipanda, n.d.), one of the roles of mathematics at schools was to prepare students to be able to face changes in life circumstances in a changing world and to prepare students to use mathematics functionally in everyday life and in coping with other sciences (as cited in Maarif, 2016, p. 116). Moreover, the purpose of mathematics in basic education was to facilitate participation in productive life activities, providing a way of making sense of the world, serving as a means of communication, and operating as a gateway to national progress (Science Education Institute, Department of Science and Technology, & Philippine Council of Mathematics Teacher Educators [SEI-DOST & MATHTED], 2011, p. 1). Therefore, mathematics is very important in human lives so the students must value the knowledge they obtain in studying mathematics together with its purpose in society.
Factors Affecting Students’ Difficulties in Solving Integers
There are different kinds of factors that can affect the students’ performance in solving integers. To easily identify these factors, the researchers categorize each factor into home/economic environment factors, school-related factors, teacher-related factors, and student-related factors.
Home Environment Factors
Home was the first school of a child and their mother serves as their first teacher. All education of a child depends on their home because a good home environment enhances the students’ achievement in mathematics (Acharya, 2017, p. 13). Those students who came from a family with low SES status will gain low initial scores in their achievement in mathematics (Ma & Bradley, n.d., p. 15). Adams (1996), a theoretical physicist, explained that SES is one of the maximum researched and debated factors among educational professionals that contributed to the academic performance of students. It has a bad effect on the educational performance of the students since the primary needs of students stay unfulfilled hence, they do not academically perform better (as cited in Farooq, Chaudhry, Shafiq & Berhanu, 2011, p. 2).
Besides, the mathematics performance of the student also correlates with the parent educational attainment. But the educational attainment of the mother best predicts students’ mathematics performance. Therefore, if parents have high educational attainment the students can perform better in mathematics (Cordova & Tan, 2018, p. 109). Family affects students’ mathematics achievement, this is proven by Henderson and Mapp (2002), founder of the National Coalition for Parent Involvement in Education and president of the Institute for Responsive Education in Boston. They stated that the evidence on a study about the families of the children has a major influence on their achievement in school and life are convincing, positive, and consistent (p. 7).
According to Abdullah (2011), associate professor from 2004-2007 at Quatar University and Lumuli (2009) who worked at St Mary’s Kibabii Boys High School, Kenya, one of the school-related factors that affect students’ performance is the learning facilities in school at all levels including the equipment and human resources. Hence, providing adequate learning facilities enhances the quality and relevance of imparted skills of learners (as cited in S. Singh, P. Singh, & Malik, 2016, p. 176; Nambuya, 2013, p. 9).
Learning also involves the interaction of students with the environment. Teaching and learning resources include classrooms, laboratories, libraries, playing fields, textbooks, and others. Indeed, material resources move an extended way in creating a conducive environment that promotes effective teaching and mastering. In relation to this is the Draft Report on Cost and Financing of Education in the Republic of Kenya [Rok] (1995), were they distinguish the ratio between textbook and school facilities as some yardsticks used to gauge the quality of education in secondary school (as cited in Nambuya, 2013, p. 9).
National Council of Teachers of Mathematics [NCTM] (2000), the world’s largest organization (NCTM, 2019), and Grouwns and Koehler (1988), explained that teachers’ beliefs and attitudes about mathematics and teaching are a significant factor. For instance, some teachers believe that the students learn by repetition and explicit examples and others believe that they learn by investigation or discovery. The attitude of the teacher in the classroom is often declared that could influence their actions, which become critical to the learning of the students. Thus, the understanding of the students in mathematics, their ability to use that understanding to solve problems, their confidence in mathematics and their disposition towards mathematics are shaped by the teaching they experience in school (as cited in Menge, 2019, p. 1069).
Understaffing, inadequate teaching/learning materials, and the attitudes of the teacher also contributes to the poor performance of the students (Mbugua, Kibet, Muthaa, & Nkonke, 2012, p. 87). Turner, Meyer, Kang, Patrick, Midgley, Gheen, & Anderman (2002), professors, research scientist, faculty member, and an associate dean in different universities and colleges, added that teacher’s personality and teaching approach also affects the students’ mathematics performance cause sometimes it depends on how the teacher teach and how they interact with their students (as cited in Segumpan and Tan, 2018, p. 6).
In terms of integers, the incorrect methods used in teaching integers, learners’ mistakes when working with integers or the language of learning and teaching (LoLT) used may lead to the difficulty experienced by the students (Soga, 2017, p. 1). Another point is, the teachers who teach without teaching the students from the conceptual perspective and only focuses on the operations of students may become one of the barriers experienced by their students (Karadeniz, Kaya, & Bozkus, 2017, p. 81).
Ma and Kishor (1997), senior lecturer at Monash University and professor in Motilal Nehru National Institute of Technology , Allahabad; Cote and Levine (2000), professors emeritus at the University of Western Ontario; Effandi and Normah (2009), senior lecturers at the National University of Malaysia and Malacca Matriculation College, Malaysia, declared that attitude is one of the most potent factors that relate to achievements because a student with positive attitude in mathematics allowed an individual to be more successful at solving mathematical problems compared to those students who had a negative views about mathematics. This negative attitudes must also overcome by the students so that they will not suffer from poor problem-solving skills in the future (as cited in Mohd, Mahmood, & Ismail, 2011, p. 49; Magno, 2011, p. 67; Silao, 2018, p. 490).
Additionally, patience and willingness, lack of student hard work and motivation, mental ability, math anxiety, study habits, and reading comprehension influence or affects the mathematical achievement of the students especially on their academic performance (Mohd et al., 2011,p. 49; Acharya, 2017, p. 11).
Borasi (1990), author and serve as a dean from 2001-2018 at the University of Rochester and Faridah (2004), associate professor at University of Malaysia, Sarawak, also added that the students’ conceptions, attitudes, and expectancy regarding mathematics and mathematics teaching is considered to be very important factors in the achievement of the students in school. Most of the students immediately attempt working the problem without planning first the strategy they can do. Thus, only moderate numbers of students can solve the mathematical questions (as cited in Mohd et al., 2011, pp. 49-50).
Self-confidence can also be considered as one of the factors that affect students’ achievement as it has a high association with the ability of the students (Hendriana, Johanto, & Sumarmo, 2018, p. 297). Hendriana, Sumarmo, & Slamet (2014), lecturers at IKIP Siliwangi Bandung, Indonesia and studies Educational Technology, defined self-confidence as the act of believing in their capability to finish the given task (as cited in Hendriana, Johanto, & Sumarmo, 2018, p. 292). Acharya (2017) found out on his study that the students are afraid to answer mathematical equations because they are worried on what their companion will think if they cannot answer the equation and somebody might laugh at them which is the cause why the students lack self-confidence (p. 10).
Therefore, the students must avoid all these factors to achieve a better performance in mathematics especially in solving integers which is essential in life because all factors stated above may lead to such difficulties as formalizing rules for integer arithmetic, creating negative numbers as mathematical objects, conceptualizing numbers less than zero (Stephan & Akyuz, 2012, p. 429). Vlassis (2001), associate professor at the University of Luxembourg; Melemezoğlu (2005), a researcher from Selçuk University, Konya; Avcu and Durmaz (2011), a teacher at Eskisehir Osmangazi University and faculty member at Süleyman Demirel University, Turkey, indicated, determined, and revealed that the misconceptions of the students about integers caused difficulties in algebra. They may also have misconceptions about the subjects of operations, producing a model with integers, word problems, doing operations with integers, and in ordering integers (as cited in the Ural, 2016, p. 171).
