MATHEMATICS TEACHERS’ CONCEPTIONS ABOUT LINEAR AND QUADRATIC EQUATIONS

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**ABSTRACT**

** ***The purpose of this study was to determine secondary school mathematics teachers’ conceptions about linear and quadratic equations. Six research questions and four null hypotheses guided this study. The study was conducted in Ika education zone of Delta-state. The sample was 124 secondary school mathematics teachers. Instrument for Data collection was a questionnaire titled linear and quadratic equation questionnaire (LQEQ). Descriptive analysis involving means and standard deviations were used to answer the research questions while Analysis of variance (ANOVA) and t-test were used to test the null hypotheses at 0.05 level of significance. The results showed that mathematics teachers have various conceptions about linear and quadratic equations and that mathematics teachers’ conception can influence their action in the classroom. The study also revealed that gender and teaching experience have a great influence in the teaching and learning of linear and quadratic equations. The result showed that there is significant difference between male and female mathematics teachers in their conception about linear and quadratic equations. Again, that there is significant difference between experienced and inexperienced mathematics teachers in their conceptions about linear and quadratic equations*

**CHAPTER ONE**

**INTRODUCTION**

**Background to the Study**

Mathematics is one of the core subjects that is taught in Nigerian secondary schools. It is the science that deals with the logic of shape, quantity and arrangement. It also deals with logical reasoning and quantitative calculation (Attah and Guwam, 2014). Mathematics holds the potency of making individuals relate its knowledge to everyday problems being encountered by individuals in the society. This is why mathematics is an important subject in the education of a child. With the knowledge of mathematics, the child can easily actualize some of the educational objectives and will be able to function well in the society. This is why the schools mathematics curriculum formulated most of these stated objectives,which can be actualized through the mathematics teachers.

The teacher being at the centre of the entire education at all levels is the most potent instrument for ensuring the fullest possible development of the student. The mathematics teachers play a vital role in the teaching and learning of the students and also help to expose the learners to numerous possibilities, which can be explored to deal with individual and group problems. This is why they are the initiator of creativity (Ugwuda, 2014). The mathematics teachers actualize the above goals through the school mathematics curriculum.

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The mathematics curriculum is a set of planned learning experiences which are taught in schools, directed and evaluated by the mathematics teachers to attain specified goals with recommended materials (Attah and Guwam, 2014). The mathematics curriculum is an official document stating educational aims, goals and objectives for a particular subject in the school system. Thus, the curriculum specifies what should be taught, why it should be taught, how it should be taught and to whom it should be taught (Ughamadu, 2006). This document is further broken down to syllabus, scheme of work, unit of lesson and lesson plan. The mathematics teachers prepare the lesson plan for teaching a particular topic from the scheme of work in which equationsis included. For effective teaching and learning the mathematics teachers use the scheme of work. This scheme of work is usually a guide in planning what is to be done per week over a term, and for the three terms in an academic year in school. The various topics are stipulated for student learning in the scheme of work, including equations, which are taught in second term in SS II class.

In mathematics, equations are statements that show that two algebraic expressions are equal in value. For example, 6x – 4 = 2x – 1 with an unknown x value. This equation is only true when x has a particularknown numerical value. In mathematics, letters of the alphabet are used, to stand for numbers e.g. 8 + p. Any letter can be used. For example 8 + a would be just as good as 8 + p. Capital letters are not used, only small letters are used. When using a letter instead of a number, the letter can stand for any number in general. When letters and numbers are used together in this way the mathematics is called algebra.

The word algebra comes from a book written by Mohammed Musa al Khnowarizmi- around AD836. The title of the book was Al-jabrwa’IMuqabalah. The statement y + 3 = 14 is called an algebraic expression, which means an unknown number y plus 3 equal or make 14. The number that the letter stands for should make the sentence true. To solve an equation means to find the real number value of the unknown that makes the equation true. The two most common equations taught at secondary school are linear equations and quadratic equations, (Macra et al, 2007).

The Babylonians did not use algebraic symbols in the modern sense. Instead, they would state problems entirely in terms of words. The solution would be verbal instructions explaining how to solve the given problem, but there was no sense of generality or a formula. Many of the problems concerned partitioning of land and so on involved areas and quadratic. The ancient Egyptians could only solve quadratic equations involving x^{2} terms and constant terms, but not “mixed” equations involving both x^{2} and x terms. Diophantus is sometimes called the “Father of algebra”. Little is known about Diophantus. Diophantus major work was Arithmetical, which mostly deals with number theory and to move forward the symbolic form of equations from the purely rhetorical style of the Babylonians to something more similar to modern notation (Bradley and Michael, 2006).

The Indian mathematician and astronomer Brahmagupta was the first to solve quadratic equations that involved negative numbers. The Indian mathematician stated the rules for multiplying or dividing positive and negative numbers as: “The product or ratio of two debts is a fortune; the product or ratio of a debt and a fortune is a debt.” Muhammad Ibn Musa at Khwarizmi literally gave algebra its name when it was published in Baghdad; Kitab al-jabr we al-muqabalah (The condensed Book on Restoration and Balancing). In this book, the first systematic solution of quadratic equations was given and this book remained the quintessential reference on the theory of equations for centuries. The quadratic equation was finally published in its general form in Europe in the 12^{th} century by Abraham Bar Hiyya ha-nasi in the book Treatise on measurement and calculation, (Bob, 2012).

A linear equation is an algebraic equation of degree one. It is also equation in which the power of the variable is one. On a baseline level, a linear equation refers to a particular equation that is graphed on a straight line. Additionally, a linear equation possesses on the line one variable that is commonly referred to as x and x will always be of a degree that is one at most (Sterling and Mary, 2010). A common example of a linear equation would be n + 3 = 5. The value of n would equal 2 in this particular example and it can be figured out by merely using a little algebra on the equation to find out the value of n. To solve this n + 3 = 5, n = 5 – 3, n = 2. Therefore, the value of n is two, but not all linear equations are that easy, they can also come in complex form. Linear equation can also take fractional forms in some cases, for instance;

Another form of linear equation is simultaneous linear equations, which consist of two or more equations; with linear variables one of whose highest power represents one and represents the same items. For example, the equations x + y = 4 and 2x + 2y = 6, form a system of equation. In order to solve the system one must find values for the variables that both statements are true. In essence to solve simultaneous equations means finding the values of x and y that will make both equations true. Two linear functions with the same variables form a system of equations. Simultaneous equations can be solved using a variety of methods. One method is to graph the linear equations as two straight lines and examine them to see if the linear intersect at exactly one point or if the lines are on top of one another or if the two lines are parallel. The other methods are substitution method and elimination method (Arigbabu and Salau, 2013).

A quadratic equation is any mathematical statement with the highest power of the variable two. A quadratic equation is a situation where one or more of the unknown is squared. The word quadratic is derived from the Latin word for squared. There are different forms of solving quadratic equations such as factorization method, completing the square method, graphical method and using the formula method. The formula method is often the most convenient way. The general quadratic equation is of the form: ax^{2}+ bx + c = 0. Here x represents an unknown, while a, b, and c are constants with ‘a’ not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation, by inserting the formulae into the letter. Each of the solutions given by the quadratic formula is called a root of the quadratic equation

Examples of quadratic equation are:

Equations can be expressed in word form, for either linear or quadratic equation. This is why it is named word problems leading to equation. To solve word problems leading to linear equations, first read the problem carefully, for clear and proper understanding of the problem. Then take the following processes.

- Represent what you want to find with one type of letter
- If there is another unknown in the problem represent it using another type or specific letter.
- Consider the relationship between the different parts of the problem.
- Use the relationship to form an equation and solve it.
- Check the solution.

Examples of word problems leading to linear equations are:

- When two is multiplied by a number and four added to it, the result is eight. Find the number.
- Nine is added to a certain number. The result is four times the number. Find the number.

To solve word problems leading to quadratic equations, first read the problem carefully, where necessary, do more than one reading for clear and proper understanding of the problem. Then take the following steps.

- Represent the unknown with one type of letters
- Examine the relationship between the different parts of the problem
- Use the relationship to form an equation
- Solve the equation using the appropriate method.
- Verify or check the result.

The above steps can guide the teachers and students to solve quadratic equation easily.

Examples of word problems leading to quadratic equation are:

- A woman’s age is four times the age of her son. 6 years ago the product of their ages was 36. Find their present ages.
- Two numbers differ by 6, the difference between their squarers is 120. Find the numbers.

However, quadratic equation as a topic, despite the steps stated above that can be used to solve it, has been associated with variety of conceptions by the mathematics teachers. These conceptions about equation, affect students’ learning efforts as it exert powerful influence on how to evaluate their students’ ability.Researches abound regarding the conceptions which mathematics teachers and students hold about mathematics such as mathematics is difficult, boring, abstract and so on(Kojigili, 2013).

Teachers’ conceptions about the purposes of teaching a subject matter or content have an impact on what to teach and how they to teach. It has been noted that mathematics teachers’ conceptions about the subject matter, teaching and learning influence teachers’ action in the classroom. One’s conception of what mathematics is, affects one’s conception of how it should be presented. One’s manner of presenting it is an indication of what one believes to be most essential in it (Golafshani, 2006). This explains that for teachers to be more effective in teaching mathematics depends on the conceptions about mathematics. Thompson (1992) believes that the teachers’ subject conception resides in their belief system by indicating that the key belief components of the mathematics teacher is the teachers’ conception of the nature of mathematics and the belief system concerning the nature of mathematics as a whole. The belief system is a metaphor for describing the manner in which one’s beliefs are organized in a cluster, generally around a particular idea or object. Beliefs systems are associated with three aspects:

- Beliefs within a beliefs system may be primary or derivative
- Beliefs within a beliefs system may be central,
- Beliefs are never held in isolation and might be thought of as existing in clusters. (Ernest, 1989).

Conception’sof the subject matter are part of teachers’ pedagogical content knowledge (Grossman, 1990). Conception is a general notion or mental structure encompassing beliefs, meanings, concepts, propositions, rules, mental images and preferences (Thompson, 1994).

Pehkonen, (2001) defines conception as conscious beliefs, regarded as higher order beliefs and based on such reasoning processes which are at least justified and accepted by the individual himself. The researcher can say that conceptions about linear and quadratic equations can be viewed under four parts which are: conceptions about the nature of linear and quadratic equations, about the teaching and learning of equations, about the self in context of the teaching and learning of equations and about the nature of knowing and the process of knowing of equations. The teachers’ pedagogical content conceptions show the range of conceptions that a teacher holds about equations when presenting the concept to students.

The teachers’ conceptions of mathematics are in line with the traditional absolutist view and non-traditional constructivist views of mathematics (Roulet, 1998). The teachers’ with absolutist conception of mathematics describe the mathematics as a vast collection of fixed and infallible concepts and skills and useful but unrelated collection of facts and rules. The constructivist view emphasizes the practice of mathematics and the reconstruction of mathematical knowledge. Teachers holding the constructivist view of mathematics take the subject as a language developed by human to describe their observation of the world. The teacher views mathematics as continually growing, changing and being revised as solution to new problems explored by the learners with the teachers as facilitators (Nahid, 2006). Other words, the way and manner through which the mathematics teachers teach is of paramount importance as they play significant role in shaping the conceptions of students about equation.

Researches have shown that the teachers’ conceptions about a concept can affect the students’ performance, especially in the external examination, such as WAEC, NECO etc (Kojigili, 2013). The chief examiners in mathematics have expressed their concern on the low performance of candidates in the subject through their reports (West African Examination Council, WAEC, 2012 – 2013). In the 2012 May/June Senior School Certificate Examination (SSCE), the Chief Examiners report that the mathematics paper was generally within the experience of the candidates, and that the paper compared favourably within those set in previous years. However, according to the report, the candidates’ performance was generally disappointing especially in the areas of solving equations and other topics. According to the report, the wrong answers to the equations indicate that the candidates did not understand the questions and could not differentiate between linear equation and quadratic equation. As a result of this ugly trend, the teaching and learning of mathematics equations should be revisited for constant failure to become a history especially in the teaching and learning of equations. This can be possible when the mathematics teachers, especiallyinIka education zone have good conceptions about the teaching and learning of equations.

The teachers’ conceptions influence teachers’ plans about what to teach, and how to teach, and these conceptions could be influenced by the following factors: teaching experience and teacher’s gender. Teachers’ teaching experience can be defined as the length or period the teachers have been involved in the teaching and learning of mathematics in school. The experienced teachers are those with more than five years teaching experience while the inexperienced teachers are those with less than five years teaching experience. Research has shown that teachers are significantly more effective when they have at least two years’ experience and if they entered the profession with adequate preparation, (Berry, Daughtrey and Wieder, 2009). Erickson, (1993) reported that novice and experienced teachers hold different mathematical conceptions, which is an outcome of personal experience. The more experienced mathematics teachers are, the more efficient and effective they become in the teaching and learning of mathematics thereby, influence the conception they have about mathematical concepts.

Another factor that influences conception is gender. One of the major problems of female students’ poor performance in mathematics in Nigerian schools could be linked to the belief they hold about mathematics as being a male domain. It is accepted that male students perform better in mathematics examination than female students. This implies that, the teaching of mathematics will be done effectively and better by male teachers since mathematics is male domain. However, female teachers seem to be more affected negatively by mathematics conceptions than male counterparts (Kojigili, 2013). Lalonde and Runk, (2004) revealed that even in a mathematically talented group of students, girls demonstrated poor confidence, which affects the beliefs they have about mathematics.

Scanty research had been done about mathematics teachers’ conceptions about linear and quadratic equations. Attorps, (2006) researched on mathematics teachers conceptions about equations. The study was carried out on equations generally, and on mathematics teachers in the higher institutions in Helsinki. All forms of equations were discussed in Attorps work. However, the present study is based on mathematics teachers in Ika education zone and on mathematics teachers’ conceptions about linear and quadratic equations only.

**Statement of the Problem**

In Nigeria, the students’ poor performance in mathematics can be attributed to many factors such as thestudents’ factor, teachers’ factor,environmental factor etc. The teacher factor has to do with the method the teacher uses for teaching and learning, the teacher’s conceptions about a particular concept and so on. The teachers’ conceptions play a vital role in human knowledge. The teachers’ conceptions about a concept can affect what they teach and how they teach. It has been shown that some mathematics teachers have wrong conceptions about equations. As a result of this, some teachers find it difficult to teach linear and quadratic equations effectively, which affect the students’ performance in the internal and external examination.

Scanty studies had been carried out about mathematics teachers’ conceptions about linear and quadratic equations. Then, the question, what conceptions do mathematics teachers in Ika secondary schools hold about linear and quadratic equations? What are the connections between the variables gender and teaching experience on teachers’ conceptions about linear and quadratic equations? These questions underline the problem of this study.

**Purpose of the Study**

The main purpose of the study is to investigate the mathematics teachers’ conceptions about linear and quadratic equations.

Specifically the study aims at determining:

- The conceptions mathematics teachers hold about linear equation.
- The conceptions mathematics teachers hold about quadratic equations.
- Gender has any influence on teachers’ conceptions about linear equations.
- Gender has any influence on teachers’ conceptions about quadratic equation.
- Experience has any influence on teachers’ conception about linear equations.
- Experience has any influence on teachers’ conception about quadratic equations.

**Significance of the Study**

This study has both theoretical and practical significance. The theoretical significance is anchored on cognitiveand constructivist theory of learning. The cognitive theory of learning was developed by a group of German psychologists known as Gresalt theorists. They view learning as condition of developing new insight or modifying earlier ones. Thus, learning by cognition is based purely on insight and not the trial and error learning. The cognitive theorist believe that learners actively process information and are able to make valued judgment based on the new information they receive in association with the previous knowledge.

Researches have shown that, previous school experiences, teachers’ current practice and teacher education courses also influence teachers’ mathematical beliefs. The cognitive theory of learning will be of a great benefit to this study since; learning is a condition of developing new insight or modifying earlier ones. In this case, the mathematics teachers will modify the previous conceptions held about the teaching and learning of linear and quadratic equation.

Constructivistas its name implies, emphasizes on the building or the construction of knowledge that occurs in students’ minds when they learn. This construction of knowledge takes place within a context of social interaction and agreement. In the process of construction of this knowledge people develop relatively stable patterns of conceptions. The construction process is influenced by a variety of social experience. Then reconstructing the conceptions they had about mathematical concepts.Theoretically, the findings of the study will give information that will be of a great importance to the existing postulations of theories related to the teaching and learning of linear and quadratic equations, especially with the cognitive theory of learning. Therefore, processing information and the principles of constructivism thereby, adding to the philosophical explanation, psychological understanding and practical approaches to linear and quadratic equations.

The practical significance of this study is that, the study will be beneficial to mathematics teachers, students, researchers and curriculum planners. The findings of this study will help the mathematics teachers to realize that misconceptions about any concepts affect students’ performance. Then, teachers having wrong conceptions about linear and quadratic equation will see the need to change or modify that attitude or behaviour. The result will also help the teachers to improve on the teaching behaviour for effective and efficient teaching and learning.

The students will also benefit from the findings because when they are taught by mathematics teacher that have modified behaviour and good conception about linear and quadratic equations, there is every assurance that the learners will understand the concepts very well and could enhance the students’ performance in mathematics.

The result of this study will be of a great benefit to researchers that wish to carry out research on the students’ poor performance in mathematics and will act as foundation for further studies on mathematics teachers’ conceptions about linear and quadratic equations in other subject areas of education zone.

Finally, the result from this study will provide information to the curriculum planners on what teachers know, beliefs and think, and thus could influence the direction of future curriculum development. The curriculum planners using the findings of this study would get better information about teachers and bring out good strategies for proper evaluation of mathematics content.

**Scope of the Study**

The study was limited to secondary schools in Ika education zone of Delta State. The choice is based on the fact that the researcher is very familiar with the school location and this will give the researcher opportunity to effectively monitor and supervise the study. Again due to the level of poor performance of students in mathematics in the area, the researcher wishes to carry out this study in order to find out the factors that contribute to the low performance of students. The study was limited to teachers teaching mathematics at secondary school level. Teachers at this level are expected to have a minimum of NCE mathematics teaching qualification, which shows an indication of some exposure to mathematics pedagogy.

The content scope include: teachers’ conceptions about linear and quadratic equations, and the influence of gender and experience on conception about linear and quadratic equations.

**Research Questions **

The following research questions are posed to guide this study:

- What are the mean conceptions of mathematics teachers about linear equations?
- What are the mean conceptions of mathematics teachers about quadratic equations?
- What is the influence of gender on mathematics teachers’ conceptions about linear equations?
- What is the influence of gender on mathematics teachers’ conceptions about quadratic equation?
- What is the influence of experience on mathematics teachers’ conceptions about linear equations?
- What is the influence of experience on mathematics teachers’ conception about quadratic equations?

**Hypotheses**

Four null hypotheses guided this study and were tested at alpha level of 0.05.

**Ho _{1}: **There is no significant difference between the mean rating of conceptions of male and female mathematics teacher about linear equations.

**Ho _{2}: **There is no significant difference between the mean rating of conceptions of male and female mathematics teacher about quadratic equations.

**Ho _{3}: **There is no significant difference between the mean rating of conceptions of experienced and inexperienced mathematics teacher about linear equations.

**Ho _{4}: **There is no significant difference between the mean rating of conceptions of experienced and inexperienced mathematics teachers about quadratic equation.

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