EFFECTS OF DIRECT INSTRUCTION AND WILSON’S LEARNING CYCLE ON REMEDIATION OF ERRORS COMMITTED IN LOGICAL REASONING BY FURTHER MATHEMATICS STUDENTS

Abstract

This study was designed to determine the effects of direct instruction and Wilson’s learning cycle on remediation of errors committed in logical reasoning by further mathematics students.  Seven research questions were formulated and four hypotheses were tested at the 0.05 level of significance. The study employed the quasi-experimental design. Specifically, non- equivalent control group, pretest-posttest design. Sixty five (65) SSII students (44 males, 21 females) from four co-educational public secondary schools in Northern Education Zone of Plateau State were used for the study. Thirty three (33) students were involved in direct instruction and thirty two (32) students were involved in Wilson learning cycle. Purposive sampling was used to select schools that met the criteria of the study out of which two schools were randomly sampled using flip of the coin to assigned two schools into Direct Instruction (DI) and two schools into Wilson’s Learning Cycle (WLC). All the groups completed the same unit covered within a period of five weeks. The groups were taught by their further mathematics teachers after they had been exposed to training by the researcher. One instrument was mainly used for this study, Logical Reasoning Test (LRT) and its Marking Guide which is in form of rubrics consisting of seven (7) different types of errors. The errors were symbolic error, comprehension error, transformation  error,  process  skill  error,  encoding  error,  careless  error  and  logical  error.  This instrument was validated and the reliability coefficient determined after it was trial-tested. The reliability coefficient was determined using Kendall’s coefficient of concordance and was found to be

0.96. Data obtained from the administration of the LRT and its Marking Guide was used for analysis using Frequency count and Percentages, Wilcoxon signed rank test and Mann-Whitney U test.  The findings  showed  that  teaching  Logical  Reasoning  (LR)  using  DI  and  WLC  reduces  the  errors committed by Further Mathematics (FM) students. WLC appears to reduce errors more than DI. All the errors were prominent among male except for comprehension error and logical error which were prominent among female when taught LR with DI. Similarly, all the errors where prominent among male except for logical error which was prominent among female when taught LR with WLC. Female students committed fewer errors than their male counterparts with both DI and WLC. Based on the findings, it was recommended that workshops, seminars and conferences should be organized for teachers, students, parents (during PTA) to implement the findings of this study in classrooms, at home, among others for better outcomes in FM.

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CHAPTER ONE

INTRODUCTION

Background of the Study

Mathematics simply referred to as science of space and numbers, has grown in scope and complexity to include formal arithmetic, geometry, statistics, algebra, trigonometry and calculus etc. Basically,  mathematics  is  the  foundation  of  science  and  technology.  Science  is  the  body  of knowledge obtained  from methods based upon observations and experiment. Technology is  the application of scientific knowledge to produce human and material objects for human comfort and existence. The functional role of mathematics to science and technology is multifaceted and multifarious that no area of science, technology and business enterprises escapes its application (Okereke, 2006). Ale and Adetula (2010) noted that without mathematics, there is no science, without science, there is no modern technology, and without modern technology there is no modern society. For a nation to develop scientifically, technologically and economically, the study of mathematics is essential.

The scientific and technological development of any nation depends largely on the mathematical knowledge of its citizens (Ekwueme & Ali, 2012). This has necessitated the Nigerian government to make mathematics one of the core subjects in both primary and senior secondary school curriculum (Federal Republic of Nigeria -FRN, 2004). Despite the importance accorded to mathematics in Nigeria’s quest for scientific and technological development, so many students still dislike, and fear mathematics, leading to mass failure and consistent abysmal performance in internal and external mathematics examination (Maduabum & Odili, 2006). This assertion by Maduabum and Odili is supported by analysis of Senior School Certificate Examination (SSCE) in Plateau State (see appendix F).

The consistent mass failure and abysmal performances in mathematics has cast doubts on the country’s hope of attainment of scientific and technological development by vision 20:2020 (Usman

& Nwabueze, 2011). This situation of consistent poor achievement in general

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mathematicshas created some worries to researchers, stakeholders of mathematics, Nigeria government and all those involved in mathematics education.

Mathematics education researchers have identified some factors responsible for mass failure and consistent abysmal mathematics performance to include use of inappropriate and ineffective teaching approaches, students’ phobia, process errors committed by students when solving mathematical  problems,  students’  perception  of  mathematics  as  a  difficult  subject  reserved  for talented students among others (Maliki, Ngban & Ibu, 2009; Chiason, Okwu & Kurumeh, 2010; Akpan, 2014). The National Council for Curriculum Assessment (NCCA, 2005) and Sidhu (2006) asserted that many students view mathematics as difficult and mainly for talented students. However, Musa and Agwagah (2006) asserted that if students find the study of mathematics more appealing; perceive the subject matter as interesting, motivating, useful and relevant to their daily living then students’ performance can be improved. Ekue and Umukoro (2011) observed that students learn, retain and  understand  when what  they are  taught  is  linked  correctly and  meaningfully to  their experiences and when real life examples are used. This observation by Ekue and Umukoro is captured in the objectives of general mathematics thus:

    To generate interest in mathematics and to provide a solid foundation for everyday living;

    To foster the desire and ability to be accurate to a degree relevant to the problem at hand;

    To develop precise, logical and abstract thinking;

    To stimulate and encourage creativity and

    To provide necessary mathematical background for further education (Federal Ministry of

Education -FME, 2008).

These objectives of general mathematics are expected to be acquired by students through the use of effective and appropriate teaching methods by mathematics teachers but unfortunately the rate of failure of students in mathematics is an issue that worries mathematics educators. Few students that passed mathematics with good results in Senior School Certificate Examination

(SSCE) encounter difficulty in mathematics or mathematics related courses at tertiary level.     3

Evidence abounds that many students with good results in general mathematics do not do well in first year tertiary institution mathematics courses (Ekwueme & Ali, 2012; Bakke & Igharo, 2013, Akpan,

2014). Therefore, there is a need to bridge the gap between secondary school general mathematics and first year tertiary mathematics courses especially for Science, Engineering, and other courses that need mathematics at Tertiary Institutions. This led to the development of Further Mathematics (FM) curriculum (Federal Ministry of Education-FME, 1985) and thus, theintroduction of further mathematics at secondary school level.

The science of number, quantity and space studied in its own right as pure mathematics or as applied  to other disciplines such as  physics, engineering and  statistics  is referred to  as  further mathematics. It is study to develop the powers of logic, reasoning and problem solving beyond that of a single mathematics course. Further mathematics, just like general mathematics is a procedural subject that employed stages or processes in an attempt to solve problems.

In thepreparation of the final copy of FM curriculum, it was identified that components for Further Mathematics are Pure Mathematics, Mechanics and Statistics (Nigerian Educational Research and Development Council-NERDC, 2008). Topics/concepts to be taught under each of the identified areas were then carefully selected to suit the immediate demands of the country for technological advancement. The scope and depth of the topics/concepts to be taught at each level of the Senior Secondary School were carefully documented or stated.

The objectives of further mathematics curriculum developed focused on the following:

           It helps the students to develop conceptual and manipulative skills in mathematics so as to prepare them for further studies in mathematics and its application;

           it reflects continuity with those used in the Universities, Polytechnics, Federal Colleges of Education and Colleges of Science and Technology, so that graduates of the curriculum have nothing to unlearn on entering any of the above mentioned institutions;

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        It is meant for potential Mathematicians, Engineers and Scientist.

            Hence, it is recommended for students with high ability in general mathematics that will need to acquire a good foundation for future studies in mathematics or mathematics related courses(NERDC,2008).

The further mathematics curriculum adopted the approach that general mathematics curriculum and further mathematics curriculum should be taught as one continuous curriculum for those who are mathematically inclined. Thus, the curriculum includes all topics in secondary school general mathematics curriculum in  addition to  higher  level topics.  Coordinate Geometry and Operations Research (OR) are added for enrichment of the curriculum in line with current trends and for added value orientation with capital market and real life applications in mind (NERDC, 2008). Teachers are expected to go at a faster rate through the more elementary work and put particular emphasis on these parts on which further mathematics builds (NERDC, 2008: 1). They will also have more time to spend on higher level mathematics concepts which include  permutation and combination, binomial theorem, differential calculus, conic sections, projectiles, vector geometry,  logical reasoning among others Obioma; Harbor-Peters in Bakke and Igharo (2013). At the end of the course, the students will take the full set of papers for the Senior Secondary Certificate Examination (SSCE) general mathematics curriculum in addition to further mathematics papers. When necessary, a student can transfer from the further mathematics class to the Senior Secondary (SS) general mathematics class without difficulty (NERDC, 2008).

However,  since  the  introduction of further  mathematics  many  factors have  bedeviled  the implementation of the curriculum which in turn has resulted in poor performance of students in the subject. The factors responsible for poor performance in further mathematics include: Inability of mathematics teachers to teach some topics in further mathematics (Eteli, 2010); lack of instructional materials   to teach high level topics (Bakke  & Igharo, 2013); lack of mathematics background required to understand the topics by students among others (Olatoye, 2007). Besides,

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students commit a lot of errors while solving some high level mathematical problems in further mathematics.According to Ekwueme and Ali (2012), the error committed at any required stage of solving mathematical problems is referred to as process error.

Generally, an error means a simple lapse of care or concentration which almost everyone makes at least occasionally (Young & O’Shea, 1981). Mathematically, an error means the deviation from a correct solution of a problem. In this study, an error is regarded as a mistake in the process of solving or answering a mathematical problem algorithmically, procedurally or by any other method. Error could be found in wrongly answered problems which have flaws in the process that generated the answers (Young & O’Shea, 1981). Some research findings in mathematics related to errors could be applied to further mathematics. For instance, Aguele (2004) have indicated that factor such as the errors committed by students while carrying out mathematical operations cause poor performances. These errors according to Aguele are:  conceptual skill error (understanding of the problem as well as understanding specific items of symbols), process skills (performing the mathematical operations necessary for the task, random response, wrong operations, faulty algorithms, computation or no response), encoding (writing the answer in an acceptable form and sequentially too), motivation (would have correctly solved or answer the question or problem had the student tried), careless error (errors that may not be repeated), question form (making an error due to the way the problem was presented, may be ambiguous). Therefore, if the poor performances of students offering further mathematics must be improved upon, these errors or weaknesses should be identified and remedied.

Remediation could be seen as an activity carried out to improve skills or designed to help students with learning difficulties to improve their skills or knowledge.Remediation is the evaluation of students’ work to determine an appropriate way to correct errors and is one of the main tenets of remedial or corrective education for all students (Salvia & Ysseldyke, 2004). Further Mathematics is a procedural subject that involves step by step activities and the error

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committed  in  any  of the  steps  is  called  Process  Errors(Ekwueme  &  Ali,  2012).  In  this  study, remediation is taken as whatever the teacher does within the realm of teaching inside or outside the class to correct observed errors committed by further mathematics students in logical reasoning. This could be through correcting errors made in a given class work, test, assignment and verbal responses. The main function of remediation while teaching for instance, is to remove the effects of poor learning or lack of learning. This can prevent students from falling behind in their education which will enable them to move forward in pursuance of their academics. Educators typically analyze students’ mathematical errors with the intent to improve instruction and correct misconceptions (Mastropieri & Scruggs, 2002). Identification and remediation of students’ logical reasoning errors has the potential to improve instructional planning and, ultimately, students’ performances (Riccomini, 2005). The difficulties experienced by some students might be as a result of strategies employed by some teachers while teaching, especially teaching of higher level mathematical topics like Logical Reasoning (LR).

Logical reasoning is a concept in further mathematics that is concerned with reasoning and thinking, and  is  also  the  ability to  draw suitable  inferences and conclusions after  a persuasive argument  (Egbe,  Odili  &  Ugbebor,  2010).The  study of  Logical  Reasoning  (LR)  can  make  an individual know the difference between persuasions through various psychological techniques and those based on rational arguments and supporting evidences (Bosede, 1992). The inclusion of this topic (logical reasoning) in further mathematics curriculum is aimed at building the students’ intelligence by making them think objectively and understand modeling of the world. According to Egbeke cited in Bakke and Igharo (2013),  refers to the skills the student or individual can avoid, attempt to mold or shape one’s beliefs, buying habits, political decisions and social behaviours as one comes across hidden persuaders, political or religious leaders and advertisers through their flowery rhetoric subtle and beguiling advertisement. Hence, there is  need for proper learning of logical reasoning. Furthermore,the FM curriculum pointed out that in studying logical

reasoning students are expected to recognize true and false statements; give examples of negation and contrapositive of simple statements; identify the antecedents and consequents of simple statements; identify and write the conditional statements of simple statements; form compound statements from simple statements; introduce logical connectives of statements using symbols (NERDC, 2008). For example,  in  teaching  ‘compound  statement’  which  is  a  statement  that  consists  of two  or  more statements or sub statements: ‘Nworgu is an intelligent and courageous man’, consists of the sub statements: ‘Nworgu is an intelligent man’ and ‘Nworgu is a courageous man’, the connective for ‘or’ is ‘v’, connective for ‘and’ is ‘^’.

As noble as the above objectives in logical reasoning, the students’performances in LR are not encouraging.For instance,  in the  last  ten years,  according  to  West  African Examination Council (WAEC) Chief Examiners’ Report (WAEC, 2003, 2005, 2007, 2013), very few students of further mathematics attempted questions in logical reasoning. Similarly, National Examination Council (NECO) Chief Examiners’ Report noted that less than five percent of students that sat for examination attempted questions in LR (NECO, 2011, 2014). Besides, those that attempted questions committed errors that led to outright failure. A lot of efforts have been made to address this problem but the rate of  failure  has  turned  to  be  a  visual  cycle.  A  lot  of teaching  methods  have  been  identified  as remediationmethods. The methods include; laboratory method, guided discovery method, procedural method, direct instruction (DI) method, Wilson’s Learning Cycle (WLC) method among others.

Direct  instruction is  a  method that  uses instruction (the act  or process)  in order to  give consistent classroom routines to boost student skill (mastery) in reading, spelling, language arts, and mathematics (Moore, 2012). In this study, Direct instruction is considered as a method that uses instruction (the act or process) in order to give consistent classroom routines to boost student(s) skill (mastery) in Logical Reasoning. Moore, further pointed that direct instruction

involves five stages which include: Orientation, presentation, structured practice, guided practice and independent practice.

In the case of orientation, teachers activate students’ relevant prior knowledge and experiences and help them to connect it to the new knowledge they will gain from the lesson. Teachers’ also familiarize learners with the focus of the lesson using student-friendly language, explain the lesson’s purpose by telling students what they are expected to be “able to do”(Moore, 2012).

Presentation is  the  second  and the explicit  phase of direct  instruction,  in which teachers identify a specific strategy for students, then model exactly where, how, and why to apply the strategy to get meaning from the content under consideration. If the teaching objective involves a strategy such as comparing ideas, teachers might use a graphic organizer as part of their modeling, stimulate them to think, speak aloud frequently as they complete the organizer. If the objective involves helping students grasp an important content-area, concept from a nonfiction selection, teachers may identify its characteristics, along with examples and non-examples, definitions, and rules. Throughout this and other phases of direct instruction, teachers check frequently for understanding of all students and provide immediate corrective feedback when needed. The most effective presentations in direct instruction include both verbal and visual explanations (Joyce & Weil, 2000).

The structured practice phase of direct instruction is the third and calls for teachers to begin the process of handing over to students the strategy or concept that they have modeled. Using new but related materials, teachers apply the steps of a strategy or the dimensions of a concept, involving students in ways in which they cannot fail. For example, students use graphic organizers, sentence frames,  or  other  structured  supports  that  organize  the  successful  use  of  the  strategy  in  direct instruction.

Guided practice is the fourth phase of direct instruction and it helps student to move toward independence. In this phase, teachers give students increasing responsibility for applying a strategy or concept in order to learn new material. Teachers use structured response techniques to ensure that every student participates in the class work and to check the accuracy of students’ responses in order to provide immediate corrective feedback, if necessary. The teacher withdraws support gradually and only when students show that they can work on their own.

Independent  Practice  is  the  final  phase  of  direct  instruction.  At  this  level,  students independently practice work with a strategy or concept by applying their new knowledge in unfamiliar situations. During this phase, students’ main responsibility centres on completing academic tasks on their own, althou.gh teachers still have to monitor what they do and respond to their efforts. The five phases (stages) allow teachers to scaffold instruction, gradually shifting and releasing responsibility for completing a task from themselves to students (Joyce & Weil, 2000).

The second method of teaching used in this study is the Wilson’s Learning Cycle (WLC). This is a method that exposes the students to motivational activities requiring physical experience and interaction as a basis to acquiring knowledge (Oduwale & Odiase, 1996). Oduwale and Odiase further pointed out that WLC involves five stages: Initiation, abstracting, schematizing, consolidation and transfer.

Initiation is the first stage where a diagnostic test allows for the strengths and weakness of students  to  be  identified  is  carried  out.  This  is  necessary  in  order  to  effectively  carry out  the remediation process. This is done before the actual commencement of the remedial teaching (remediation) by administering a test (pre-test). It is not a daily activity hence; the day to day lesson actively starts with students using concrete materials to help them solve problems intuitively. This stage also is congruent with the exploration phase of the Science Curriculum Improvement Study (SCIS). The exploration phase in the SCIS gives students experiences with the

concepts to be developed before they are discussed, read aloud and named (Abraham & Renner,

1986).

Abstracting is the second stage and is concerned with moving the learning from the concrete to a more abstract level including the use of symbols that is from whole to parts. During this stage, the concept is namedand presented to the students in classroom discussion (teacher interact with students) and  activities  of students  are  encouraged  (teacher  motivate  students)  to  describe  or  explain  the activities learned.

Schematizing (Modeling), is the third stage which involves helping the students fit new rule into their mental schemes and thus link it to the related ideas. During this stage, a scheme or a model which can be used and re-used to answer such and related problem is developed. In other words say formula,  diagrams,  symbols  if  any  is  abstracted  during  the  modeling  stage.  Several  problems (questions) using formula, diagrams, symbols are solved cooperatively with the students.

Consolidation (Practice), this fourth stage is intended to make the students gain mastery of the skills they have leant. Students are provided with opportunities for meaningful practice so that the learner will be able to retrieve and use concepts in an automatic and accurate way (Oduwale & Odiase,

1996).

Transfer which is the last stage of WLC comes after the consolidation stage, where the teacher  identifies students with difficulties. These categories of students are grouped together to transfer or share ideas. If possible the teacher is expected to point out the main areas of worries. It does not necessarily mean going over the entire lesson that has been taught but uses different examples to demonstrate the model (Scheme) to the students.

The Direct instruction and Wilson’s Learning Cycle are similar in the stages involved but differences also exist. A diagnostic test to identify the strengths and weaknesses of students is not always  necessary before the commencement of the new topic in DI but is more often in WLC; Direct instruction encourages independent practice with classwork, assignments given and

feedback from student while WLC encourages cooperative work; DI encourages teacher to start from simple to complex while WLC encourages teacher to start from whole to parts; students motivate themselves (intrinsic motivation) in DI while WLC motivation of students is often done by the teacher (extrinsic motivation). The teacher may interact with the students by questions, use familiar examples, corrections and encouragement to stimulate responses etc.

Researchers have shown that a method of teaching which is effective in teaching a particular topic may not be effective in another topic. Burner in Dalu (2010) stated that, no single method is suitable for all topics, situations and time. For instance, in a situation whereby the teacher intends to teach students ‘construction’, the teacher is expected to do that when all or majority of the students are having mathematical sets. The choice of the methods is in line also with the suggestion given by WAEC Chief Examiners’ Report (2013), that students should be given frequent exercises to enhance skill development and application of theory. One of the ways this can be done is by comparing methods of teaching. Any good teaching method is supposed to carry the learners that is, male and female along. Hence, the researcher examined the effects of Direct Instruction (DI) and Wilson’s Learning  Cycle  (WLC)  on  remediation  of  errors  committed  in  logical  reasoning  by  further mathematics students in relation to gender.

Gender refers to socially constructed roles and socially learned behaviours and expectations associated with males and females (Oakley, 1996; Okeke, 2000 and UNESCO, 2000). Males and females  are  biologically  different  although all  cultures  interpret  and  elaborate  their  biologically inherent differences into a set of social expectations about what behaviours and activities are appropriate for them and what rights, resources and even power they possess. In Nigeria, gender gaps occur in treatment of males and females which put females in a corner that has deterred their progress and achievement in schools (Nwagbara, 2009).The following gender-related problems in curriculum implementation have also been identified by Nwagbara (2009); discriminatory attitude of parents, gender stereotyping, cultural and religious factors, gender biased curriculum in favour

of males, and high dropout rates in Nigerian schools by females. These result in low performance in the classroom on the part of the females which often result in psychological and emotional battering.

These attributes of further mathematics literacy of student are inculcated using an effective teacher who can use appropriate activity-oriented teaching methods. This calls for the researcher to experiment on the use of direct instruction and Wilson’s learning cycle with regards to gender. Reports of studies in Africa revealed that there is generally low level of admissions in science and technology- based courses compared with admission into arts and social sciences programs especially in Nigeria (Bakke & Igharo, 2013). This is because female tends to commit more errors especially in these areas and more importantly, it is obvious that there is a wide disparity in enrolment and academic achievement of males and females in some areas of specialization (Alade, 2006). Alade,  further observed female students tend to drift or be guided towards areas of studies regarded as feminine and thus shy away from scientific and technological fields like further mathematics. In a similar view, Oluokun (2002) had earlier observed that certain career such as engineering which uses mathematical applications are said to be suitable to men while catering or secretarial studies are deemed suitable for women since they commit less errors in these areas. So the gap in enrolment and achievement at all levels of education is largely due to gender role expectations and gender stereotyping. Worried by the gender disparity in enrolment in colleges of education, the National Commission for Colleges of Education (NCCE) at various times in their research reports on gender issues in colleges of education revealed that the areas of female students concentration are the arts, education, social sciences, and vocational education. The specific subject areas where they significantly outnumbered the males are music, French, Nigerian Languages, home economics and secretarial studies for the females are rated high in terms of achievement which is as a result of less error commission.

Certainly, it is common to observe that enrollment rate of male in further mathematics in Nigerian secondary schools are higher than that of female. Logical reasoning deals with reading and comprehension using symbols and often females are reported to perform better than males (Education for All-EFA Global Monitoring Report, 2012). From 2011-2015, the percentage of male students that pass further mathematics is higher than that of female in Nigeria (Education Resource Centre-ERC Jos, October, 2015). Itis important to note that some studies favored males while others favored females. This shows that there is no human strength without some weaknesses. More needs to be done to ensure that mix up of male and female have equitable access to educational opportunities and achieve equal educational outcomes especially in further mathematics for scientific and technological advancement. Hence, the need to investigate the effects of DI and WLC on remediation of errors committed in LR by FM students.

Statement of the Problem

Research evidence  over  the  years  has  indicated  poor  performances  in  mathematics  from internal and external examinations and also in further mathematics (FM). However, the performance of students in further mathematics has not been encouraging and this has been traced to the types of errors committed while solving problems in further mathematics as shown in appendix F. Recent literatures have suggested the use of some innovative approaches in teaching mathematical concepts that can serve as a remediating strategy to identify and remedy the errors committed by students while solving problems in further mathematics.

Some of these methods are laboratory method, problem solving method, discovery method, expository method, direct instruction; Wilson’s learning cycle among others. Meanwhile, the inclusion of logical reasoning in further mathematics in secondary school curriculum is aimed at  building students’ intelligence by making them think objectively to understand modeling of the world. There is no single method that is suitable for all topics, situation and time. Also, there is no human strength without some weaknesses. Therefore, the researcher intends to experiment on the

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use of Direct Instruction and Wilson’s Learning Cycle on remediation of errors committed in logical reasoning and to determine how male and female would cope using these methods. Even though each of these methods has five stages that have some elements of commonality but different strategies for the purpose of remediation. Hence, the problem of this study is: Would the use of direct instruction and Wilson’s learning cycle be effective in remediation of errors committed by further mathematics students in logical reasoning? Similarly, would the use of direct instruction and Wilson’s learning cycle reduce gender disparity in errors committed by male and female further mathematics students while solving problems in logical reasoning?

Purpose of the Study

The main purpose of the study was to determine the effects of Direct Instruction (DI) and Wilson’s Learning Cycle (WLC) on remediation of errors committed by Further Mathematics (FM) students in Logical Reasoning (LR). Specifically the study sought to determine:

1. Types of errors frequently committed by Further Mathematics (FM) students in logical reasoning (LR);

2. Effect of Direct Instruction (DI) on remediation of the errors committed by FM students in LR;

3. Effect of Wilson’s Learning Cycle (WLC) on remediation of the errors committed by FM students in LR;

4.        Influence of gender on remediation of the errors committed by FM students in LR when taught using direct instruction;

5.        Influence of gender on remediation of the errors committed by FM students in LR when taught using Wilson’s Learning Cycle (WLC)?

6.        The mean ranksof the difference in errors committed by FM students in LR after remediation with direct instruction and Wilson Learning Cycle?

7.        The mean ranksof the difference in errors committed by male and female FM students in LR

after remediation with both methods?

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Significance of the Study

The study has both theoretical and practical significances. The theoretical significance is to the existing theories of learning like Piaget’s cognitive theory of formal operational stage and theory of negative expertise Minsky (1983). This theory based on Piaget(1896-1980), the child makes typical errors of thinking that  are characteristics of age originating from mental growth, which we  should try to understand. Furthermore, that besides knowing these typical errors, we should also try to find out why the child makes them and how to help the child overcomes them. However, from literature typical errors were identified and based on these identified errors and why the child makes them that we can help the child to overcome them. Aguele (2004) pointed out that these errors emanated from poor background in mathematics and students’ perception of mathematics as a difficult subject reserve for only talented students. Hence the researcher intends to use the direct instruction and Wilson’s learning on remediation of errors committed in logical reasoning by further mathematics students. At this age of 15 and 17 years, the young SS II students can start to think more abstractly. In this stage, it is characterized by the ability to manipulate abstract as well as concrete objects, ideas and events. Logical reasoning is chosen for this purpose since majority of students fall within 15 to 17 years.

In the case of Theory of Negative Expertise, which was theorized by Minsky in 1983 have it that for many students and teachers errors are associated with negative feelings since it demoralizes students’ enthusiasm. Despite the fact that errors are the best teachers and learners”,   teachers and students hardly take advantage of errors in class. This theory based on this study showed that as students and teachers continue studying logical reasoning they become experts since practice makes perfect. Students should not feel bad as they commit errors for is capable of making them learned more and mastery is attend for better achievement. The practical significance is to the following: Teachers, Students, National Mathematical Centre, Parents and Guardians, Government, Researchersand Curriculum planners.

while

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Teachers could point  out typical errors committed by further  mathematics  students

teaching to take caution. The teachers can try to identify the errors and carry out re-teaching and remediation in the class.The students could be aware of such errors and take caution while writing assignments, tests, exams which may leads to students committing less errors and consequently high achievements in the results of further mathematics;could help to verify the effectiveness of the two teaching methods in remediation of the different error(s) categories.

The study could help curriculum planners. If curriculum planners must continue to provide adequate and relevant Further Mathematics (FM) curriculum, there would be need for a database concerning instructional procedures for effectively teaching the content of such curriculum or syllabus. The curriculum planners and authors of further mathematics textbooks may advocate the methods of teaching if effective on remediation of errors in the curriculum.

National Mathematical Centre (NMC), Abuja may benefit from this study as their primary objective is to improve the teaching and learning of mathematical values. They may organize Workshops, Seminarsand Conferences to educate the teachers to carry out remediation of errors during instruction in further mathematics classes.

Parents and Guardian that are learned can assist on the remediation of errors especially in the area the student is identified to have problem(s). They may do that by employing qualified mathematicians to teach their children at home.

Since government is aware that education is the bedrock of any societal development, government can provide facilities, pump money to the education sector to motivate the teachers, organize workshops, seminarsand conferences for better achievements in further mathematics. It could be useful to Researchers as a source of literature.

Scope of the Study

The study will focus on the effects of Direct Instruction (DI) and Wilson’s Learning Cycle

(WLC) on remediation of errors committed in logical reasoning by Further Mathematics Senior

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Secondary II Students with respect to gender in Plateau State. The topics intended to be covered is the  latest  curriculum of further  mathematics (FM) developed  in 2008  by  Federal Ministry of Education  Published  by  Nigerian  Educational  Research  and  Development  Council  (NERDC); which contained the following topics on Logical Reasoning: Statements, Negation, Conditional Statements, Compound Statement, Disjunction, Conjunction, Equivalent Statement, Tautology and Contradiction, Laws of the Algebra of Logical  Statement. This gives a total of nine (9) sub-topics (NERDC, 2008:10,18).

Research Questions

The following research questions were considered to guide the study:

1.         What are the frequency and mean oferrors committed by Further Mathematics (FM) students in logical reasoning (LR)?

2.       What is the effect of Direct Instruction (DI) on remediation of the errors committed by FM

students in LR?

3         What is the effect of Wilson’s Learning Cycle (WLC) on remediation of the errors committed by FM students in LR?

4.       What is the influence of gender on remediation of the errors committed by FM students in LR

when taught using direct instruction?

5.       What is the influence of gender on remediation of the errors committed by FM students in LR

when taught using Wilson’s Learning Cycle (WLC)?

6.        What are the mean ranksof the difference in errors committed by FM students in LR after remediation with direct instruction and Wilson Learning Cycle?

7.        What are the mean ranksof the difference in errors committed by male and female FM students in LR after remediation with both methods?

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Hypotheses

The followingnull hypotheses are formulated andtested at 0.05 level of significance.

H01:    There is no significant difference between the mean errors committed by male and female Further Mathematics (FM) students when taught Logical Reasoning (LR) with Direct Instruction (DI).

Ho2: There is no significant difference between the mean errors committed by male and female Further Mathematics (FM) students when taught Logical Reasoning (LR) with Wilson’s Learning Cycle (WLC).

HO3: There is no significant difference between the mean errors committed by Further Mathematics (FM) students when taught Logical Reasoning (LR) with Direct Instruction (DI) and those taught with Wilson’s Learning Cycle (WLC).

HO4: There is no significant difference between the mean errors committed by male and female Further Mathematics (FM) students when taught Logical Reasoning (LR) with Direct Instruction (DI) and Wilson’s Learning Cycle (WLC).

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