ABSTRACT
This study was designed to explore the misconceptions in secondary school geometry among students. Four research questions and two hypotheses were formulated to guide the study. The study adopted ex-post facto design and was restricted to Nsukka education zone of Enugu state. Six hundred SS 1 students selected through stratified random sampling were involved in the study. The instrument for data collection was a Geometry Diagnostic Test (G.D.T) which made use of both essay-type and multiple-choice questions. The scores obtained from the multiple-choice questions were subjected to reliability test using Kudder- Richardson (K-R 20) while the scores from the essay-type questions were subjected to reliability test using Scorer reliability. The formulated research questions for the study were answered using frequency and percentages, while the formulated hypotheses were tested by the use of Mann-Whitney U test. The result of the study revealed the misconceptions the students encountered in the study of SS1 geometry. The result of the study also revealed that gender has no significant influence on students’ misconceptions in geometry while school location has significant influence on students’ misconceptions in geometry with students from rural schools having the greater misconceptions in geometry than the urban students. Based on these findings, the researcher recommended among others that mathematics teachers in the rural areas should encourage the students to study mathematics paying more attention to geometry and devote more time and effort to teaching and learning of SS1 geometry as this will help the students in the rural schools measure up with their urban counterparts in the study of mathematics
particularly in geometry.
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CHAPTER ONE
INTRODUCTION
Background of the Study
The world has changed and is still changing day by day. Nigeria therefore needs the type of education which is geared towards helping her citizens adapt to the changing world. This is in agreement with one of the goals of revised edition of the National Mathematics Curriculum for Basic Education in Nigeria which is to cultivate the understanding and application of mathematics skills and concepts necessary to thrive in the ever changing technological world (Federal Ministry of Education (FME), 2007).
Mathematics skills are mathematical operations and processes involved in the solutions of a problem or study of some scientific field (Mathematics,n.d. Retrieved December 25th,2015). Cultivating the understanding of mathematics skills and concepts is very crucial as it would be helpful in learning of mathematics and solving mathematics problems. Four basic mathematics skills students should learn are the following: problem solving skill, applied mathematics skill, estimation and approximation skill and computational skill (Basic mathematics skills, n.d.Retieved December 25th ,2015 ). In other to cultivate the understanding of the skills, the students should:
• Understand mathematical symbols and can visualize patterns, mathematics concepts, and the parts of a problem in his/her head.
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• Understand mathematics vocabulary words and is able to build mathematics knowledge through the use of mathematics language.
• Understand how concepts are related as in the relationship between addition and subtraction or between ratio and proportion.
• See how mathematics concepts (such as proportion or measurement) apply to everyday life (Understanding mathematics concepts, n.d.Retieved December 25th, 2015).
Cultivating the understanding of these skills will help the students to
connect ideas, develop logical and abstract thinking, and to question, analyze and understand the world around them.
The world is moving scientifically and technologically in handling her everyday activities and resolving problems. This is applicable in all areas of life including all fields of learning which mathematics is one of them. According to Agwagah (2008), Mathematics is the study of quantity, structure, space and change. Mathematics is a subject that is applied in all aspects of life. Galadima in Usman (2005) states that without the application of mathematics in our daily lives, we might not be able to develop any scientific and technological capacity, no matter the efforts. This could be attributed to the fact that knowledge of mathematics helps in a clear and logical reasoning. Azuka (2013) records that, among all the academic subjects studied in the school, mathematics has definitely contributed more to objective of general education of man than other
subjects. In Nigeria, the federal government made mathematics a compulsory subject and one of the core subjects at the primary and secondary level of education (Federal Republic of Nigeria, 2004).
In recognition of the peculiar place of science, mathematics in particular, in the overall development of the country, the present revised edition of the national mathematics curriculum for basic education programme by the federal ministry of education focuses on giving the children the opportunity to: acquire mathematics literacy necessary to function in an information age , cultivate the understanding and application of mathematics skills and concepts necessary to thrive in the ever changing technological world, develop the essential element of problem solving, communication, reasoning and connection within their study of mathematics and lastly, understand the major ideas of mathematics, bearing in mind that the world has changed and is still changing since the first National Mathematics Curriculum was developed in 1977 (Federal Ministry of Education, 2007,page viii). Much effort has been made by both federal and state governments in promoting the study of mathematics in our schools. The federal ministry of education develops curricula and syllabuses at the national level in conjunction with other agencies. The ministry also conducts the common entrance mathematics and quantitative aptitude examinations for use in selecting candidates into federal government colleges, the junior school certificate mathematics examination for federal
government colleges and special schools like university secondary schools. The state ministry of education develops questions for various examinations of the ministry (including mathematics examinations) and coordinates the marking\vetting of the examinations. The ministry, through the examination centres, conducts mathematics examination in the First School Leaving Certificates (FSLC) examination, junior secondary school certificate examination, state common entrance examination into junior secondary school, entrance examinations into nursing schools, and special school entrance examination (Obodo, 1997). These efforts by the government towards the learning of mathematics could go a long way in solving the problems associated with teaching and learning of mathematics which misconception of concept is a major one.
Despite these efforts by both federal and state government towards the learning of mathematics and the world wide recognition of the importance of mathematics to national development, the teaching and learning of the subject have been in a dismal state in most of our schools. Students dread the subject and perform poorly in it. For instance, the West African Examination Council results of students in Nigeria show that students perform poorly. In the year 2008, 2009, 2010 and 2011, the percentage pass with credit and above in Nigeria were 23.0%, 31.0%, 24.94% and 38.98% respectively (Kurumeh & Imoko; Moseli, Onwuka & Iweka, and Iyi, in Azuka, 2013).
The poor performance of students in mathematics in external examinations as recorded by WAEC (2011) in her chief examiner’s report which states that students have been performing poorly in mathematics while others avoid questions in geometry and also in internal examinations could be as a result of problems faced by the teaching and learning of mathematics. These problems include; poor instructional materials, poor attitudes to mathematics, nature of mathematics, the teacher factor, lack of mathematics laboratory and misconception of concepts. In other to handle these problems, the teacher should adopt some strategies to generate and sustain students’ interest in mathematics which can be done by the use of: motivation (both extrinsic and intrinsic), use of relevant set induction technique, statement of clear instructional objectives, effective use of instructional materials, use of mathematics games, students participation in mathematics activities, use of mathematics laboratory, relating mathematics to other disciples, etc).
One major problem in the study of mathematics among others is the misconception of concepts. According to Hornby (2006), misconception is a belief or an idea that is not based on correct information, or that is not understood by people. The noun misconception comes from the prefix mis- meaning “bad, wrong” and the word conception-meaning “act of conceiving.” A misconception usually results from incorrect thinking or a flawed understanding. Therefore, a misconception is a conclusion that is
wrong because it is based on faulty thinking or facts that are wrong (Vocabulary. Com, 2014). Misconceptions can also be referred to as a preconceived notion or a conceptual misunderstanding (New York Science Teacher, 2014). There are cases in which something a person knows and believes does not match what is known to be scientifically correct. New York Science Teacher (2014) states that a lot of people who hold misconceptions do not even know that their ideas are false or incorrect. When they are told they are wrong, they often have a hard time giving up their misconceptions, especially if they have had a misconception for a very long time. What is specially worrisome about misconceptions is that people continue to build knowledge on their current understandings. Possessing misconceptions could have serious impacts on an individual’s learning.
Students posses misconceptions in learning, especially in the study of mathematics, geometry in particular. Redmond (2009) defines geometry as a branch of mathematics that deals with shapes and sizes, and may be thought of as the science of space. Most students in our secondary schools cope with other topics in mathematics but not geometry (WAEC, 2013).The reason according to WAEC (2009) is that students find it difficult to differentiate geometric shapes and are unable to recall theorems in geometry, complaining that there are a lot of diagrams, proofs, logic reasoning in geometry. Misconceptions in geometry by students could also be attributed to the inability of the teachers to present the concept well to the students.
The students on the other hand, find it difficult to see the significance and relatedness of geometry to everyday life activities and its usefulness outside the school which could also lead to misconceptions in geometry.
There are many misconceptions that are encountered by students in the study of geometry. Vygotsky (2012) identified some common misconceptions in geometry which include the following:
Identifying the Base and Height of a triangle.
Conservation misconception.
Angles- Larger space means larger angle.
Shape properties.
Orientation and Rotation of shapes.
Perpendicular lines
Lines of symmetry.
‘… a rectangle is a long shape… ’ and ‘… a square is not a rectangle…’
These misconceptions students posses in the study of geometry as identified by Vygotsky (2012) which hinder the students from learning geometry will be elucidated in chapter two of this study.
Misconception of concepts leads to a very poor performance in mathematics especially in geometry among students and could be influenced by school location and gender factors. School location (which is connected with town or city where the school is cited) could be urban or rural and could have influence on students’ performance in geometry. An urban area
is a location characterized by high human population density and vast human built features in comparison to the area surrounding it (Wikipedia,
2015). A rural area is a geographical area that is located outside cities and towns (Wikipedia, 2015). For this study, schools located along coal-tarred roads are considered as urban schools, otherwise considered as rural schools.
Could school location (urban and rural locations) influence the misconceptions students posses in the study of mathematics? Ugwu (1992) states that environment plays a major role in achievement of students in tests and examination. According to Ugwu, differences may exist between the performance of rural and urban schools but such differences could not be concluded as regularly occurring pattern. Some factors that may be responsible for the disparity in the achievement of urban and rural students include: quality of teachers, learning environment, areas covered by the students as at the time of the administration of the test and students’ experiences. Misconception as a major problem students have in the study of mathematics could also not be influenced by school locations depending on the area. This is supported by Ogoke (2003) who conducted a study on the misconceptions in algebra of junior secondary school mathematics and discovered that the misconceptions students have in understanding of algebra concepts is not influenced by the school location. Ogoke (2003) found that most local families are richer than their urban counterparts. They
are now aware of the importance of education and therefore invest their resources to ensure that their children are educated. Since the issue of school location as a factor influencing students’ misconceptions has not been concluded, it is therefore necessary to carry out a further research to find out if misconceptions in the study of geometry among the students are influenced by school location or not.
Another important factor that could influence misconceptions in geometry is the gender factor. Gender has been defined as cultural differences between women and men based on the biological division between male and female (Connell, 2003:9). According to Okeke (2001), gender refers to the social or cultural construct, characteristics, behaviours and role which society ascribes to males and females. Gender could be a factor that influences the performance of students in science subjects which mathematics is inclusive. This is supported by Okwo and Otubar (2007) who show that science achievement depend on gender. Specifically, in the study of geometry, gender is a great factor that could affect the performance of students. In Odo (2000), the result of the analysis carried out indicated that the performance of students was generally low and that gender is a significant predictor of students. Furthermore, students’ misconceptions in the study of mathematics could also influenced by gender. In a study conducted by Ogoke (2003), the result showed that female students have more misconceptions than male students. Must gender affect the
performance of students in the study of mathematics? According to Douglas
(2009) in the study of intelligence: sex differences states that: Many studies have examined whether gender
differences exist in mathematically ability, but the results have been inconsistent. In 1990, American
researchers statistically combined the results of more than 100 studies on gender differences in mathematics using a technique known as meta- analysis. They found no significant differences in the average scores of males and females on mathematics tests. Research also indicates that the average girls’ grades in mathematics courses equal or exceed those of the average boys. Other studies have found that boys and girls perform equally well in mathematics achievement tests during elementary school, but girls begin to fall behind boys in later years (para 4)
This inconclusive fact about the influence school location and gender on the misconceptions of secondary school geometry among students has prompted the need to conduct an investigation to find out if misconceptions in secondary school geometry among students are independent of school location and gender. How are misconceptions in geometry influenced by school locations and gender differences?
Statement of Problem
Geometry is one of the themes in the current Senior Secondary Education Curriculum for mathematics. It is one of the basic concepts and most widely applied aspects of mathematics. A lot of efforts have been made by individuals and government towards the study of mathematics in
general, which geometry is a branch. In spite of the importance of mathematics, individuals and government efforts towards the study of mathematics and the utilitarian value of geometry, students’ achievement in geometry shows a progressive decline in quality and number of passes. From WAEC (2011) in her chief examiner’s report records that students have been performing poorly in this area while some avoid answering questions from geometry. The poor performance in geometry could be based on the students’ misconceptions in geometry. It is now based on this that the researcher wants to answer the following question; what is the extent of misconceptions in secondary school geometry among students and what factors influence the misconceptions among the students in Nsukka Education Zone ?
Purpose of the Study
The purpose of the study is to investigate the misconceptions in secondary school geometry among students. The study is designed to achieve the following objectives:
(i) Find out the misconceptions in secondary school geometry among students.
(ii) Find out whether the misconceptions are peculiar to a particular gender; and
(iii) Find out whether the misconceptions are peculiar to a particular school location.
Significance of the Study
The theoretical significance of this study hinged on Piaget’s cognitive development theory of 1964.The basic tenet of Piaget’s theory is that when a learner is in the state of disequilibrium, the student seeks for balance, what Piaget called equilibration. The findings of this study will strengthen the theory by revealing whether or not students possess misconceptions in geometry and such misconceptions if any when remedied will make the students to regain cognitive equilibration as prostrated by Piaget.
Practically, the findings of the study will be beneficial to the students, teachers, parents, book authors, curriculum planners and the society. The study will help the students identify the areas of geometry the students possess misconceptions. Possibly with the help of the teacher, the misconceptions are remedied; the students will have better academics performance in geometry both in internal and external examinations. With the help of this study, when the misconceptions in geometry among the students are remedied, phobia and apathy for mathematics among the students will be a thing of the past especially as it concerns geometry.
From the findings of this study, the teachers will be exposed to the misconceptions in geometry students possess and the content areas of geometry the students posses the misconceptions. When the identified misconceptions are tackled and remedied, it will save teachers’ time and energy when teaching as the students’ level of understanding will be enhanced.
When the misconceptions in geometry among the students are remedied and the academic performance of the students increases, it will encourage the parents and the investments the parents make on their children will not be wasteful.
The findings of this study will expose the book authors on the areas of geometry the students posses misconceptions. This exposure will help the book authors to simplify the areas in geometry where the students have misconceptions in the course of writing the students’ texts. When this leads to remediation of the identified misconceptions among the students and the students’ level of understanding increases, it will bring about more number of students buying the texts and consequently leads to increase in the sales of the texts by the book authors, thereby making more money.
The findings of this study will enable the curriculum planners to be aware of the content areas of geometry where students have misconceptions, and then expunge the topics in geometry that are beyond the mental ability of the students in future review of the curriculum.
If by the findings of this study, the misconceptions students encounter in geometry are remedied and the students’ academic performance improved especially in external examinations, say West African Secondary School examinations, honour will be accredited to the society where the students belong to. Secondly, when the students’ misconceptions in geometry are remedied and their academic performance improved, the students will be more useful in the society by contributing meaningfully to the development of the society.
Scope of the Study
The study was carried out in Nsukka education zone of Enugu state. The study was limited to SS1 students. The reasons for the choice of SS 1 students were based on the fact that they must have been exposed to geometry in their junior secondary school education level and have begun to study geometry in the senior secondary school education level. This is the class where misconceptions in senior secondary school geometry are first encountered and need to be identified and handled before the students move to the next class. The area of mathematics that was used for this study is geometry as contained in senior secondary school education curriculum for mathematics for senior secondary one (SS 1) provided by the Federal Ministry of Education (FME) through Nigeria Education Research and Development Council. The topics include: construction, proof of some basic theorems, trigonometric ratios and mensuration.
Research Questions
The following research questions were used to guide the study:
(i) What are the misconceptions in secondary school geometry among students?
(ii) In what content area(s) of geometry do misconceptions exist among secondary school students?
(iii) What influence has gender on misconceptions in secondary school geometry among students?
(iv) What influence has school location on misconceptions in secondary school geometry among students?
Hypotheses
The following null hypotheses further guided the study and were tested at 0.05 level of significance:
(i) There is no significant difference between the percentage of misconceptions of male and female students in secondary school geometry. (ii) There is no significant difference between the percentage of misconceptions of urban and rural students in secondary school geometry.
This material content is developed to serve as a GUIDE for students to conduct academic research
MISCONCEPTIONS IN SECONDARY SCHOOL GEOMETRY AMONG STUDENTS IN NSUKKA EDUCATION ZONE OF ENUGU STATE NIGERIA
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