WELCOME TO KENTERPRO:
EFFECTS OF DYSLEXIA ON THE READING ABILITIES OF PRIMARY PUPILS IN ENGLISH LANGUAGE OF TARAUNI LOCAL GOVERNMENT AREA IN KANO STATE

CHAPTER ONE

INTRODUCTION

1. 1 BACKGROUND OF THE STUDY

Mathematics from Wikipedia, the free encyclopedia Britannica (2017) defines mathematics as the science of structures, order and relationship that involves elementary practice of counting, measuring, and describing the shapes of objects. It also deals with logical reasoning and quantitative measurement in everything we do. Oxford advance dictionary (2015), defines Mathematics as the science of numbers and shapes with arithmetic, algebra, geometry and trigonometry as branches. Mathematics dictionary (2019) defines it as a science of numbers, shapes, and quantitative reasoning.

Mathematics is the life wire of every technological advancement which is an indispensable tool for the development of science and technology. It is for this reason  that Ekeweme and Meremikwu (2013), stated that no nation can hope to achieve any measure of scientific  and technological advancement without  proper foundation in school  mathematics.

It is generally observed that one of the major function of the school system is to produce a pool of skiled manpower which a nation needs to grow. To this effect, countries all over the universe depends on their educational system for the development of the future workforce. ( Ekeh ,2019). It is in line with this notion that Nigeria on its national policy on education stated that education is an instrument per excellence for effective and adequate national development. (Federal Government of Nigeria (FGN,2013)).

In the same vein Adeneye and Oludola (2018) stated that Education is an instruments per excellence for effective social, psychological, economical, political, intellectual and other developmental transformations. Therefore, no nation can afford to neglect education.

Mathematics is the heart of every educational system in any successful national development. Mathematics  knowledge  in formal  education curricular has been the focus. It is in this view that Okpala, Okoye and Anene (2018) posited  that the role played by mathematics in  day-to-day activities of a man is suggestive of the fact that mathematics  is needed by all which places the subject at an advantage position for all human development.

Mathematics is to a nation what protein is to a young human organism. As a vital tool for the understanding and application of science and technology, the subject plays the necessary role of a precursor and harbinger to the much needed technological and natural development of the developing nations of the world.

In the contemporary Nigeria, subsequent to the nation’s endorsement of international protocols for Education for All (EFA); the Millennium Development Goals (MDGs) and the adoption of a National Economic Empowerment and Development Strategies (NEEDS), a greater emphasis is now being placed on industrial and technological development (NERDC, 2007). Consequently, students are being encouraged to take up sciences and technology related disciplines which mathematics forms the basis.

 It is an essential school subject which has its relevance in all fields of human endeavors. It is a key element in our  day-to-day activities where every human being practice  one form or the other, which includes the farmer in his farming business, the driver on the steering, carpenter in his work shop, the woman in the kitchen, the administrator in an organization just to mention a few.

It is an indispensable fact that any student who wants to be valuable in life must pass mathematics because is is a subject that needly networked in every other subject through integration of topics in curriculum planning and development. Knowledge obtained from mathematics in lower level serve as a base foundation and pre-requisites in the higher levels of learning.

It is an all-embracing  body of knowledge that open up the mind to logical reasoning, analytical thinking, and ignites or challenges one to develop attribute  for creative thinking, deep focusing, and  clarity of thoughts(Sidue,2006). This justifies why Federal Government  of Nigeria  through the  National Policy of Education (FGN,2013) recommended  mathematics  as one of the core subject to be studied by all at the primary and secondary schools level. In order to ensure admissions in to tertiary institutions of learning a credit pass in mathematics is a pre-requite. This makes the subject compulsory.

In spite of the importance  and popularity  of mathematics to all aspects of human endeavors , researchs  show that there is an increasingly  poor performance in the subject among senior secondary school students .(Ogunkola,2010,Awolola,2010, Abakpa and Iji2011,  Ongishi 2012, Iwendi ,2014).

According to Asuru (2016), the West African Examination Council (WAEC) is one of the major examining bodies in Nigeria. The others are the National Teacher’s Institute (NTI), National Examination Council (NECO), National Business and Technical Examination Board (NABTEB) and Joint Admissions and Matriculation Board (JAMB).

According  to (Domyil,2015), and Akanmu & Fjemdagba (2019) stated that despite the huge importance attached to the study of mathematics in Nigeria and the efforts made by Government  through national and international  competition in mathematics, allocation of  bursary awards for mathematics and science students, organization of seminar’s through  Mathematical Association of Nigeria (MAN), establishment and implementation of policies on subject and recommendations’ made by other researchers in order to make the teaching of the subject more easy by the student’s . It has not yielded the desired result.

The  West African Examination council (WAEC) chief Examiner’s report (2019)   shows how poor the performance are in the may/june WAEC exams. The summary of the report shows that; For the years 2015,2016, 2017 and 2018 have (17.57%,26.91%,49.54%,25.55,) respectively. The number of students that pass with credit in the subject with five credit including English Language consistently remain below 50%.

Research shows that  plateau state have been placed on position 16th in terms of students performances in mathematics. WAEC chief Examiner’s (2020). Federal Ministry of Education, Nigeria Digest of Education Statistics (2017); National Bureau of Statistics (2019). The result above shows how students performances’ have been persistently low in May/June WAEC Examinations in plateau state which Jos-South local Government area forms part.

The former Minister of Education, Rufa’i, who stated in Abuja at the flagging-off of the 2012 Nigeria Mathematical Year held at the Shehu Musa Yar’Adua centre, Abuja with the theme: Mathematics, Key to National Transformation, that the trend of poor performance in mathematics must be checked if the country has to move forward. The minister expressed sadness that Nigeria has remained largely a consumer nation because of lack of investment in sciences and technology in the past. As said there could be no meaningful progress in the country without promoting the study of mathematics (Rufa’i, 2012).

Rufa’i also said that for vision 20:2020 of the federal government to be a reality, efforts must be vigorously pursued toward science education, research and development which she described as the bedrock of national development.  The minister added that the very existence of any area of human endeavour was based on mathematics and mathematical sciences saying even ICT which is a product of mathematics has taken over on the global stage.

            Further expressed the need to motivate and encourage upcoming mathematicians, saying “For the nation, Nigeria to keep abreast with global trends of technological advancement, and for our pupils and students to achieve international recognition; our secondary school students and upcoming mathematical scientists should be motivated to participate in the mathematics improvement programme. She further added that for Nigeria to achieve her quest for sustainable economic growth and development there was the need for collective promotion of excellence in science and technology because “investment in science is vital for developing nations and a country that neglects science education for its citizens does so at its own peril” (Rufa’i, 2012).

These prompted the researcher to carry out research on how to improve student’s  performance and interest in learning mathematics which longitude and latitude forms part. the reasons behind  this low performances may be due to several factors such as poor teaching methods, psychological  factors, poor learning environment, evaluation process, bad attitudes of students and teachers, lack of interest and motivation during learning processes, teachers qualifications  and Governmental policies’ (Asikhia,2010 in Akonmoute an Olabisi,2018).

Longitude and Latitude is an interesting topic being taught at SS3 according to reviewed mathematics curriculum. Longitude are angular distances along the equator measured in degrees. it is of 0 degree to 180 degrees east or west of the Greenwich meridian while latitude are the angular distances of points north or south of the equator measured in degree.

Just as mathematics is essential in all human endeavors, longitude and latitude are also used in different areas of human endeavor’s in spite of the importance of mathematics to the society which longitude and latitude form part but yet both students and teachers always find some challenges in teaching and learning these concepts.

This difficulties might have emanated from misconceptions of the two concepts thereby using them interchangeably, poor knowledge   about lengths of arch, weak background on some theorems on circle geometry, confused formula on distance along great circles and small circles, difficulties in location of points around the equator and small circles, difficulties in calculating time zones and difficulty in translating words problems on longitude and latitude to a mathematical statement. The major problem relating to these difficulties is lack of appropriate instructional materials or mathematical models and lack of interest in teaching and learning of the topics.

Moreover, it is advocated that teaching of mathematics concepts should be concretized to enhance the students’ interest in teaching and learning. No course in science and mathematics can be considered as complete without including some practical work. Improvisation and utilization of instructional materials have potentials in teaching of abstract concepts. Gambari & Ghana (2015) emphasized that the use of mathematical models stimulates learning and assist the teachers to properly convey the topic content to the learner in order to achieve better understanding and performance. Afolabi (2018) maintained that achievement of objectives depends largely on the use of improvised mathematical models in mathematics teaching and learning.

Offorma (2014) stressed that teachers should be able to produce simple and inexpensive materials such as charts, posters, maps, pictures, drawings and models for effective teaching and learning. Improvisation is the act of using alternative resources to facilitate instructions whenever there is lack or shortage of some specific first hand teaching aids (Eniayeju, 2015).

Deductively, it is defined as the act of providing teaching materials from our locality when there is shortage or lack of standard ones. It could also be described as sourcing, selection, deployment of relevant instructional elements of teaching and learning for a meaningful realization of specified educational goals and objectives.

Mathematical models are those resource materials used by mathematics teachers in the classroom. They are resources which both the teachers and students use for the purpose of ensuring effective teaching and learning Obodo (2014). Azuka (2012) identified the use of mathematical models as a means of improving the teaching and learning in mathematics. Shih, Kno and Liu (2012) developed and evaluated mathematical models and learning system and found that the model enhanced mathematical achievement. Joshua (2017) reported that using geometrical globe model for teaching Mathematics concept such as longitude and latitude at senior secondary schools enhanced students’ performance.

For teaching to be effective functional mathematical models are required in modern scientist and technologest, among others. Idris, (2018). Education is the focal point to a country genuine growth and development for every Nigerian child in whatever moral, mental, emotional, psychological and condition of health.

The teachers, who are to implement the (U.B.E) curriculum, are also expected to use a wide range of quality instructional materials for effective and efficient teaching and learning classroom activities. Mathematical models are essential tools in learning every subject in the school curriculum. They allow the students to interact with words, symbols and ideas in ways that develop their abilities in reading, listening, solving, viewing, thinking, speaking, writing, using media and technology. According to Faize and Dahan (2011) mathematical models are print and non-print items that are designed to impact information to students in the educational process. Instructional materials include items such as prints, textbooks, magazines, newspapers, slides, pictures, workbooks, electronic media, among others.

These models play a very important role in the teaching-learning processes the availabilities of textbook, appropriate chalkboard, Mathematics kits, Science kit, teaching guide, science guide, audio-visual aids, globe models, overhead projector, among others are the important instructional models (Yusuf, 2015), However many facilities are missing in approximately almost all secondary schools in the state. According to Raw (2010) the first instructional model is the textbook. Various definitions to textbook emphasize the role of textbook as tool for learning.

This mathematical models are instructional materials designed either by industry or classroom teacher to make the concepts comprehensively meaningful. Examples of this models are a models of wire or the clay models to explain the location of point on either longitude or latitude, wire models of ball , sketch maps models made from metal sheet, balloon and paper Cutouts.

But the researcher will use the improvised models made from metal sheet rather than the industrialzsed or commercially made ones because the improvised made are completely controlled by the teacher in the classroom setting. While the industrialized ones may not be designed for the purpose of teaching angles and distance. Therefore this can be absolutely achieved if the students and their teachers must have interest towards teaching and learning the concept.

 Interest is a powerful motivational process that energizes learning, guides academic and career trajectories, and is essential to academic success. Interest is both a psychological state of attention and effect toward a particular object or topic, and an enduring predisposition to reengage over time. Integrating these two definitions, the four-phase model of interest development guides interventions that promote interest and capitalize on existing interests.

Four interest-enhancing interventions seem useful: attention-getting settings, contexts evoking prior individual interest, problem-based learning, and enhancing utility value. Promoting interest can contribute to a more engaged, motivated, learning experience for students.value information (as when teachers simply tell students that material is useful) in promoting interest and performance (Canning &Harackiewicz, 2015).

A utility-value intervention can help spark situational interest in a topic, and it may help students connect that topic to their own interests, which can build on individual interest.The efficacy of the intervention for promoting interest and performance was first demonstrated in ninth-grade science classes, with the strongest benefits for less confident students (Hulleman&Harackiewicz, 2019); the intervention improved performance for these at-risk students by nearly two thirds of a letter grade, and enhanced their interest in science. Moreover, interest predicted students’ science-related career plans, suggesting that this simple intervention promotes important academic outcomes.

 This duality not only highlights the richness of the interest concept but also contributes to the complexity of defining interest precisely. Interest combines affective qualities, such as feelings enjoyment and excitement, with cognitive qualities, such as focused attention and perceived value, all fostered by features of the situation (Hidi & Renninger, 2016).

According to Imoko and Agwagah, (2016) and (2017) defined interest as the attraction which forces or compelled a child to respond to a particular  stimulus . George, (2008) sees it as an aspect of affective domain which has to do with one readiness to like or dislike something. It is a subjective feelings of concentration or persistent tendency to pay attention to an activity or content.

Research has shown that every interested and motivated student is also good at cognitive, affective and psychomotive level of learning. (Bloom,2015) .Therefore interest affects learning of the concept because research has shown in several cases that highly motivated and interested students always perform best because they are not studying to just pass but to make use of the content  learned in real life situation .

1.2       STATEMENT OF THE PROBLEM.

The teaching and Learning of longitude and latitude is not easy due to the difficulties associated with this content area. which include misconception of the two concepts there by using them interchangeably, poor knowledge   about lengths of arch, poor background on some theorems on circle geometry, confused formula on distance on a great circle and small circle, difficulties in location of points around the equator and small circles, difficulties in calculating time zones, location and date ,calculation of shortest distance, speed and difficulty in translating words problems on longitude and latitude. A large number of question in this area of mathematics are reflected yearly in WAEC or NECO or NABTEB or JAMB examination.

Evedences show that geometry contribute over 60% of the questions with latitude and longitude forming a significant aspect. A large number of students in Jos-South LGA of plateau state failed to understand these areas of difficulties. There by leading to poor performance in WAEC and NECO examinations over the years. The poor performance has been a source of worry to parents, teachers, students and Government. The poor performance may be as a result of lack of teaching mathematical models, gender issues and low interest level in learning the topic. This three factors has been a major contributing factors exhibited in the study of mathematics as a subject which longitude and latitude form part as a content area under SS3  mathematics curriculum.

(Afolabi and Adeleke,2010) identifies inadequate provision of mathematical models in learning of longitude and latitude to serve as a major causes of students lack of interest leading to poor performance in mathematics. Although researchers have recommended some innovative teaching  methods’  that will enhance students performance for years. Yet WAEC chief examiner’s annual report (2019) shows that student’s performance in mathematics has not improve especially in geometry including longitude and latitude. This necessitated further Investigation on ways of improving the situation especially through the use of improvised mathematical models to determine if students interest and performance could be enhance.

1.3       AIM AND OBJECTIVES

      The general purpose of this study was to investigate the effects of mathematical  models on  students interest and performance in longitude and latitude among SS3 students in jos-south LGA of plateau state.  The specific objectives of the study include:

1. To compare the mean interest scores of students taught longitude and latitude using mathematical models and those taught using the traditional method.
2. To determine the influence of mathematical models on students interest in longitude and latitude base on gender.
3. To compare the mean performance scores of students taught longitude and latitude using mathematical models and those taught using traditional method.
4. To determine the influence of mathematical models on students performance in longitude and latitude base on gender.

1.4     RESEARCH QUESTIONS

The following research question was  posted to guide the study.

1. What is the mean interest scores of students taught longitude and latitude using mathematical models and those taught using the traditional method?
2. To what extent could mathematical models influence students’ interest in learning longitude and latitude base on gender?
3. What is the mean performance scores of students taught longitude and latitude using mathematical models and those taught using traditional method?
4. To what extent could mathematical models influence students mean performance scores in longitude and latitude  base on gender ?

1.5    RESEARCH HYPOTHESIS

The following research hypothesis was formulated and tested at 0.05 significant level.

1. There is no significant difference in the mean interest scores of students taught longitude and latitude using mathematical model and those taught using traditional method.
2. There is no significant difference in the mean interest score of male and female students taught longitude and latitude using mathematical models.
3. There is no significant difference in the mean performance scores of students taught longitude and Latitude using mathematical models and those taught using traditional method.
4. There is no significant difference in the mean performance scores of male and female students taught longitude and latitude  using mathematical models.

1.6    SIGNIFICANCE OF THE STUDY

            The significance of this study is both theoretical and practical in dimensions.

Theoretical, the findings are important to Robert Gagnes learning theory condition because the students will actively participate in handling the models which the theory should support. This study will help the theory to capture a step-by-step procedure in handling hence the use of improve models will enable student to think critically, generate skills, predict, co-ordinate and control attempts to learn and solve problem to enhance performance and interest in learning latitude and longitude, this comes as a result of an environmental experience that Robert Gagnes emphasized on.

Practically, the study will be of benefits to mathematics teacher, educational curriculum planners, resource person, students and government agencies.

The research findings will be of benefit to students because it will help the students to realize the importance of mathematics and its value and why it has been made as a prerequisite subject for entering into higher institutions of learning. To stimulate their interest in learning mathematics and longitude and latitude in particular. To improve their performance, require the use  of the instructional models to practically motivate the learners.

            The research findings will also be important to the mathematics teacher for strategic planning of effective teaching and learning of mathematics and longitude and latitude in particular.

            The research work will also be helpful to educational curriculum planners to review curriculum in mathematics in such a way that it will be helpful in learning the subject in order to achieve our national goals of the education.

The findings will be of help to resource persons in education to write journals that will encourage the use of improvise mathematical models. The study will also be of benefit to governmental agencies on the need to allocate more funds in the education sector for development of models for effective teaching and learning of Mathematics.

This study will help Parents, school counselors and public; the parent will help in assessing the performance of private and public secondary school students’ and to also assist students to choose their career choice. It will also help parent to allow their children to select their career choice themselves and also assist the school councilors to guide their student’s well on how to stimulate student interest in learning the concept of longitude and latitude.

Generally, the research findings will be of help Jos South LGA of plateau state ,ministry of education and other educational bodies in secondary school, primary and tertiary education to formulate and implement policies that will ensure the development of improvised Mathematical models for teaching and learning of mathematics so that the subject will be meaningfully taught

Further more  the application of mathematical models on teaching will help students to have a better understanding of  the connection between the real world and the object in the world, the study will help geography teachers to use different improvised models to explain better the concepts for their students  because the study have given them hints on how models can be made .

The study will also motivate stake holders in the society to be able to see the relevance of mathematics  and its application. This can be done through organizing talk show in order to stimulate and engage students interest in the study of the subject. The study will also help association like mathematical association of Nigerian students (MANS), mathematical teachers association of Nigeria (MTAN) and other governmental agencies to be able to organize seminars that will encourage teaching through improvisation rather than depending absolutely on the industries made materials.

1 .7      THEORETACAL FRAME WORK

Robert Gagnes theory of learning will be adopted in this study because the study is in agreement with this learning theory.

1.7.1    ROBERT GAGNE’S THEORY OF LEARNING

The psychologist Robert M. Gagne has done the research into the phases of learning sequence and the types of learning. His research is particularly relevant for teaching mathematics. Gagne used mathematics as a medium for testing and applying his theories about learning and has collaborated with the University of Maryland Mathematics Project in studies of mathematics learning and its curriculum development.

According to Gagne theory of learning four important points must be put in to consideration which include:

1. The role of the students:-students should engage actively in the learning process.
2. The role of the teacher­:-the teacher should follow a consistent step-by-step using the model to explain the mathematical concept effectively especially in longitude and latitude.
3. The learning environment:-the learning environment should be condusive for the learning activities.
4. The implications of for teaching and learning-the teacher should make sure his teaching method should be student centre.

1.7.2    Categories of learning:

Signal Learning, Stimulus-Response Learning, Chaining, Verbal Association, Discrimination Learning, Concept Learning, Rule Learning and Problem-Solving. Below are the descriptions of each type of learning which are explain in details in relation to longitude and latitude.

1. Signal Learning

Signal learning is involuntary learning resulting from either a single instance or a number of repetitions of a stimulus which will evoke an emotional response in an individual. In order for signal learning to occur, there must be a neutral signal stimulus and a second, unexpected stimulus that will evoke an emotional response in the learner which he or she will associate with the neutral stimulus which the model sent signal to the students. In the example of the person who learned to fear group signing in a first grade music class, the neutral signal stimulus was singing in a group and the unexpected stimuli were a shout and a slap.

As a mathematics teacher, we should attempt to generate unconditional stimuli which will evoke pleasant emotions in our students and hope that they will associate some of these pleasant sensations with the natural signal which is our mathematics classroom.

2. Stimulus-Response Learning

Stimulus-response learning is also learning to respond to a signal. It is voluntary and physical. Stimulus-response learning involves voluntary movements of the learner’s skeletal muscles in response to stimuli so that the learner can carry out an action when he or she wants to do.

Most examples of pure stimulus-response learning in people are found in young children. They are learning to say words, carry out various life-supporting functions, use simple tools, and display socially acceptable behaviors.

3. Chaining

Chaining is the sequential connection of two or more previously learned non-verbal stimulus-response actions. The examples of chaining are tying a shoe, opening a door, starting an automobile, throwing a ball, sharpening a pencil, and painting a ceiling.

In order for chaining to occur, the learner must have previously learned each stimulus-response link required in the chain. If each link has been learned, chaining can be facilitated by helping the learner establish the correct sequence of stimulus-response acts for the chain.

Most activities in mathematics which entail manipulation of physical devices such as rulers, compasses, and geometric models require chaining. Learning to bisect an angle with a straightedge and a compass requires proper sequencing and implementing of a set of previously learned stimulus-response type skills. Among these skills are the ability to use a compass to strike an arc and the ability to construct a straight line between two points.

4. Verbal Association

Verbal association is chaining of verbal stimuli; that is, the sequential connection of two or more previously learned verbal stimulus-response actions.

The mental processes involved in verbal association are very complex and not completely understood at present. Most researchers do agree that efficient verbal association requires the use of intervening mental links which act as codes and which can be either verbal, auditory, or visual images.

 These codes usually occur in the learner’s mind and will vary from learner to learner according to each person’s unique mental storehouse of codes. For example, one person may use the verbal mental code “ y is determined by x ” as a cue for the word function , another person may code function symbolically as “ y = f(x) ”, and someone else may visualize two sets of elements enclosed in circles with arrows extending from the elements of one set to the elements of the other set.

The most important use of the verbal association type of learning is in verbal dialogue. Good oratory and writing depend upon a vast store of memorized verbal associations in the mind of the orator or writer. To express ideas and rational arguments in mathematics it is necessary to have a large store of verbal association about mathematics.

5. Discrimination Learning

Discrimination learning is learning to differentiate among chains; that is, to recognize various physical and conceptual objects. There are two kinds of discrimination. They are single discrimination and multiple-discrimination. As students are learning various discriminations among chains, they may also be forming these stimulus-response chains at the same time. This somewhat disorganized learning situation can, and usually does, result in several phenomena of multiple discrimination learning (generalization, extinction, and interference).

1. Generalization is the tendency for the learner to classify a set of similar but distinct chains into a single category and fail to discriminate or differentiate among the chains.
2. If appropriate reinforcement is absent from the learning of a chain of stimuli and responses, extinction or elimination of that chain occurs.
3. Interference can be a problem in learning a foreign language such as French, which has many words similar in meaning and spelling to English words.

6. Concept Learning

Concept learning is learning to recognize common properties of concrete objects or events and responding to these objects or events as a class. In order for students to learn a concept, simpler types of prerequisite learning must have occurred. Acquisition of any specific concept must be accompanied by prerequisite stimulus-response chains, appropriate verbal associations, and multiple-discrimination of distinguishing characteristics. For example, the first step in acquiring the concept of circle might be learning to say the word circle as a self-generated stimulus-response connection, so that students can repeat the word.

Then students may learn to identify several different objects as circles by acquiring individual verbal association. Next, students may learn to discriminate between circles and other objects such as triangles and squares. It is also important for students to be exposed to circles in a wide variety of representative situations so that they learn to recognize circles which are imbedded in more complex objects. When the students are able to spontaneously identify circles in unfamiliar contexts, they have acquired the concept of circle.

7. Rule Learning

Rule learning is the ability to respond to an entire set of situations (stimuli) with a whole set of actions (responses). Rule learning appears to be the predominant type of learning to facilitate efficient and coherent human functioning. Our speech, writing, routine daily activities, and many of our behaviors are governed by rules which we have learned.

In order for people to communicate and interact, and for society to function in any form except anarchy, a huge and complex set of rules must be learned and observed by a large majority of people. Much of mathematics learning is rule learning.

            However without knowing the rule that can be represented by , we would not be able to generalize beyond those few specific multiplication problem which we have already attempted. In first, most people learn and use the rule that multiplication is commutative without being able to state it. In order to discuss this rule, it must be given either a verbal or a symbolic formulation such as “the order in which multiplication is done doesn’t make any difference in the answer,

This particular rule and rules in general, can be thought of as sets of relations among sets of concepts. Mathematics teachers need to be aware that being able to state a definition or write a rule on a sheet of paper is little indication of whether a student has learned the rule. If students are to learn a rule they must have previously learned the chains of concepts that constitute the rule. The conditions of rule learning begin by specifying the behavior expected of the learner in order to verify that the rule has been learned. A rule has been learned when the learner can appropriately and correctly apply the rule in a number of different situations. In his book The Conditions of Learning, Robert Gagne (1970) gives a five step instructional sequence for teaching rules:

 Step 1 : Inform the learner about the form of the performance to be expected when learning is completed.

 Step 2 : Question the learner in a way that requires the reinstatement (recall) of the previously learned concepts that make up the rule.

Step 3 : Use verbal statements (cues) that will lead the learner to put the rule together, as a chain of concepts, in the proper order.

Step 4 : By means of a question, ask the learner to “demonstrate” one of (sic) more concrete instances of the rule.

Step 5 : (Optional, but useful for later instruction): By a suitable question, require the learner to make a verbal statement of the rule.

8. 8. Problem-Solving

 Problem-solving  is a higher order and more complex type learning than rule-learning, and rule acquisition is prerequisite to problem-solving. Problem solving involves selecting and chaining sets of rules in a manner unique to the learner which results in the establishment of a higher order set of rules which was previously unknown to the learner.

Real-world problem solving usually involves five steps, they are:

i Presentation of the problem in a general form

ii Restatement of the problem into an operational definition

iii Formulation of alternative hypothesis and procedures which may be appropriate means of attacking the problem

iv Testing hypothesis and carrying out procedures to obtain a solution or a set of alternative solutions

v  Deciding which possible solution is most appropriate or verifying that a single solution is correct.

A learning hierarchy for problem-solving or rule-learning is a structure containing a sequence of subordinate and prerequisite abilities which a student must master before he or she can learn the higher order task. Gagne describes learning as observable changes in people’s behavior, and his learning hierarchies are composed of abilities which can be observed or measured. According to Gagne, if a person has learned, then that person can carry out some activity that he or she could not do previously. Since most activities in mathematics require definable and observable prerequisite learning, mathematics topics lend themselves to hierarchical analyses. When specifying a learning hierarchy for a mathematical skill, it is usually not necessary to consider all of subordinate skills.

Constructing a learning hierarchy for a mathematical topic is more than merely listing the steps in learning the rule or solving the problem. Preparing a list of steps is a good starting point; however the distinguishing characteristic of a learning hierarchy is an up-side-down tree diagram of subordinate and super-ordinate abilities which can be demonstrated by students or measured by teachers.

1.7.3    The theory also have nine instructional events which are corresponding to the cognitive processes which are:

Gaining attention (reception), by cognitive reception gagne mean that the Mathematics teacher should bring real life materials that relate to the topic in suggest that things like world globe, world map and also models that will better the teaching of the concept longitude and latitude.

Informing learners about the objective of the study (expectancy), by this Robert gagne emphasize that the objectives of the study should be stated clearly so that the students will know what they are expected to know.

Stimulating recall of prior knowledge learning (retrieval), teachers should create connection between what they are about to know and what they have already know in the past.

Presenting the stimulus (selective perception),  give definitions necessary for the topic.

Providing learning guidance (sematic encoding), Robert state that this is a stage were the teacher gives details explaination solution to the students and also give them good guidance.

Eliciting performance (responding),this is where the mathematics teacher evaluate his students to know the level of their achievement.

Providing feedback (renforcement), give students back their scripts after assessment processes had taken place and also motivate the students for future learning.

Assessing performance, teachers should always mark students scripts.

Enhancing retention and transfer of knowledge(generalization), this events should satisfied and gives the necessary condition for learning of longitude and latitude and also serve as a basis for designing instructions and selecting appropriate models for teaching longitude and latitude.(Gillbert,2018)while Gagne theoretical frame work covers all the aspects of learning.

The theory focus on intellectual skills which has been applied to designed instruction for all the domains of learning (Gagne and Driscon 1988).

1.8        SCOPE OF THE STUDY.

            The study will focus on the effects of mathematical models on students interest and performance in longitude and latitude among SS3 students in Jos- South LGA of Plateau state.The researcher will make use of the findings and take decision on the entire jos-south LGA of plateau state. Because the LGA contain schools which are both private and public containing different categories of students, they operate on the same curriculum and are supervised by the same area  inspectorate office and are registered schools.

            For the purpose of this research work the researcher will consider a topic longitude and latitude because mathematics curriculum contain so many topics and this topic is one of the most challenging  topic in WAEC and NECO examination that several research report have shown that students always perform poorly. longitude and latitude is a very large topic in content, therefore the researcher will choose a sub-topic distance along the prime meridians and along great circle, location of points on the earth surface, angle between two points on the earth surface, radius of the earth, and calculation of the shortest distance and speed for this research findings because they are areas of difficulties students normally have especially in Jos-south LGA of plateau state. This research will only consider SS3 level because the topic is taught in SS3 according to reviewed mathematics curriculum of Nigeria.The study was also limited to only Jos-South because of financial problems encountered by the researcher.

1.9   OPERATIONAL DIFINATION OF TERMS

MATHEMATICS: is a subject taught in primary, secondary and tertiary institution of learning which aids  individual or group of people to be able to reason, think and calculate, relate to the world around them and take  valid decision in the society.

INTEREST: are motivational feeling that student have towards longitude and latitude.

PERFORMANCE: Is the average achievement of students grade in teacher made achievement test on longitude and latitude.

LONGTITUDE: Are angular distance along the equator measured in degrees it has a range of 0 to 180 east or west of the Greenwich meridian.

LATITUDE: Are angular distance of a point north or south of the equator measured in degrees from 0⁰-90⁰ and are parallel to the equator.

MATHEMATICAL MODELS: This are improvised material that the teacher design to teach the concept of longitude and latitude.