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Modules 1 2 3 4 5 6 7 8 9 10 11


Teaching and Learning Methods in Higher Education

Reflect on the following as you work through this Module

  • Article 9. Innovative educational approaches: critical thinking and creativity

    b. Higher education institutions should educate students to become well informed and deeply motivated citizens, who can think critically, analyse problems of society, look for solutions to the problems of society, apply them and accept social responsibilities.

    c. To achieve these goals, it may be necessary to recast curricula, using new and appropriate methods, so as to go beyond cognitive mastery of disciplines. New pedagogical and didactical approaches should be accessible and promoted in order to facilitate the acquisition of skills, competencies and abilities for communication, creative and critical analysis, independent thinking and team work in multicultural contexts, where creativity also involves combining traditional or local knowledge and know-how with advanced science and technology. These recast curricula should take into account the gender dimension and the specific cultural, historic and economic context of each country. The teaching of human rights standards and education on the needs of communities in all parts of the world should be reflected in the curricula of all disciplines, particularly those preparing for entrepreneurship. Academic personnel should play a significant role in determining the curriculum.

    d. New methods of education will also imply new types of teaching-learning materials. These have to be coupled with new methods of testing that will promote not only powers of memory but also powers of comprehension, skills for practical work and creativity.



    It is common to hear higher education students say:

    "I am going for lectures"

    "I am going for practicals"

    "I am going for field work"

    "I am going for tutorials"

    No doubt, this gives a feel of the modes of instruction they receive. Lectures, practicals, fieldwork and tutorials are common methods of instruction in higher education. What are the strengths and weaknesses of these methods? Are there some other ways by which the development of knowledge, skills and attitudes can be better facilitated by the higher education teacher? Are there better methods and study skills that higher education students can apply for more effective learning? These are the major questions on focus in this module.

    In this module, we shall

    UNIT 1

    At the end of the unit, you should be able to define the following concepts:

    One thousand two hundred learners in auditorium A. There is a row; there is so much noise that people cannot hear each other speak .An elderly person comes in. We wonder who he is. Then the person goes towards the board. Everybody is silent. It is hot and the room is dark because the windows let little sunlight in. The dictation begins, one would think that this is high school; worse, primary school .This must certainly be the teacher. He speaks a lot without stopping. The learners who are at the back cannot hear well. One of them bursts out laughing. The contagious laughter spreads everywhere, from back to front of the auditorium. The teacher demands silence. The dictation resumes. The laughter spreads from another corner. Driven to distraction, the teacher picks up his things. He threatens; he then leaves the classroom, the way he came in.

    How is the rest of the academic year going to be if it starts this way?

    Auditorium B is smaller and better ventilated than auditorium A. There are two groups of eight hundred learners at each end of the room. It is very lively, very pleasant. They look very serious but they are comfortable. There is a gentleman of a certain age. The teacher perhaps. He speaks little, observes and sometimes asks questions.

    These are some of the scenarios that are played out in higher education. We shall keep these scenarios in mind as we begin with defining some of the key concepts in this module.

    Teaching and Learning

    Any teaching/educational system relies on three poles: knowledge, the learner and learning situation.

    BODY OF KNOWLEDGE (Epistemology)




    with his / her abilities These are human, material and

    (physical, moral, psychological, financial resources of the system

    intellectual etc)


    Figure 3.1: The three poles of the educational system



    Teaching can be defined as a set of processes and procedures used by the teacher for the purpose of making learning happen. Obanya (1998) sees it as the process of bringing about positive changes in a learner.


    The learning situation or the teaching environment is the set of resources available for implementing the teaching/learning process. These include human resources (lecturers, learners, administrators and support personnel); physical resources (e.g. classrooms, library, laboratory, and workshops); material resources (teaching material, audiovisual materials and others) financial materials (operational allowances, scholarships, training grants and others); and the political and social context (democracy versus dictatorship, peace versus war).


    Learning can be defined as an internal process which occurs in the learner . It is a relatively permanent change in the behaviour of a person (the learner). Research by cognitive psychologists (e.g. Brainard, 1997) shows that learning takes place in three stages: the motivation stage, the acquisition stage and the performance stage.

    The Motivation Phase

    The leaner receives a stimulus to learn. This provides the drive (kick start) for the learning process. He/she selects information from the environment, which is obtained by the sensory receptors.

    The Acquisition Phase

    The information acquired is processed in the following manner:

    It enters the short -term memory from which it can be retrieved and exploited within a very short time. But the capacity of the short-term memory is very limited. The acquired information, following rehearsal, is stored in the long-term memory.



    Learning is influenced by the teacher - learner relationship. The roles of the teacher and the learner vary in this relationship. On the one hand, the teacher can be a mere transmitter of knowledge; the learner is entirely dependent on what the instructor says or does. He or she is then more of a "recipient" than a "learner". On the other hand, the teacher can play the role of a guide, or a facilitator. The learner is assisted in becoming autonomous, that is to say, in being able to plan his/her learning.


    Prégent (1990) defines a method of teaching as particular way of organising pedagogical activities knowingly implemented according to certain rules in order to make learners reach specified objectives.

    UNIT 2

    At the end of this unit, you will be able to:


    Promoting Teaching and Learning in Higher Education in the Natural Sciences

    Sam ‘Tunde Bajah

    Out of an exciting higher education intervention – Medium –Term Programme on Staff Development in Eastern and Southern African Universities in which I was lucky to participate, a handbook for University Lecturers with the title Teach Your Best was produced. In writing the Foreword to that book, Professor Gichanga, Vice-Chancellor, University of Nairobi underscored the need to revisit universities in terms of staff development and in terms of teaching and learning. He drew attention to the following issues:

    Throughout Africa, institutions of higher learning are in a state of crisis. Universities are bursting at the seams due to ever increasing student numbers. Currently, only a handful of lecturers have been professionally trained in the art of teaching. All over the world, it is now recognised that excellence in teaching must be nurtured. In the face of such serious problems, it is becoming more and more evident that the quality of staff is a crucial element in ensuring that universities retain their traditional mission of discovering, transmitting and preserving knowledge (Gichaga, 1993).

    While we are still battling with teaching and learning problems in higher education, we must not forget that in the natural sciences, discoveries are coming up at an unprecedented rate. The 21st century will see the world of the natural sciences take a giant leap – teaching and learning in an age that will become high-tech in terms of information must be conceived differently from what they are now.

    Input Variables/Quality of Teaching

    Teaching has oftentimes been described as a profession although some, for plausible reasons refer to it as an occupation. At all the levels below the university, teachers are not only trained (prepared) but also certified to teach. A professional teaching qualification is a passport to advancement in the profession. At least so it was if you aspire to becoming a Headmaster or a Principal of a Secondary School. The situation however in the Universities is different. There is an assumption that has become the norm also in the natural sciences- the possession of a Ph.D degree in that field of specialisation is all that an aspiring lecturer needs in order to take up a teaching position in any of the Department in the Natural Sciences in a university. If teachers teach (not cheat, a la Obanya), what do lecturers do? Teach or lecture? This raises the first question for our discussion.

    Question 1

    Should university lecturers be expected to possess a teaching qualification after a Ph.D? If yes, what sort of post-Ph.D training should be given?

    For effective teaching, not only should the teacher be properly prepared; there is also need to provide that teacher with adequate facilities. Evidence which come out of most of the African Universities is that teaching facilities are far less adequate both in terms of quantity and variability. In the natural sciences, the arena for teaching is the laboratory. There is enough empirical evidence to show that as the student population increases, both the number and quality of teaching laboratories are static. This raises a second question which addresses the state of the laboratories.


    Question 2

    What is the optimum number and quality of teaching laboratories needed for teaching the natural sciences? Should the same parameter be used for Physics/Chemistry/Biology?

    To determine the quality of teaching in a university, there is need not only to find out [assess] what is going on now but also to suggest how things can be improved. Whether university teachers like it or not, every time they stand before a group of students, they are being assessed by their students.

    Question 3

    What tips can we give to university teachers to make their teaching effective? What are the variables that need to be considered for good teaching?


    Input Variables/Quality of Learning

    One important ingredient of teaching is that it leads to effective learning. As observed in many universities, teachers provide through lectures, the guide plan for learners. University teachers provide a framework for learning. Indeed, that is why many university teachers still claim that they are expected to ‘lecture’ not ‘teach’. The onus for learning rests with the students. For students who in the secondary schools have been used to teachers teaching, coming into a university presents a different scenario. Some of the students are not ready for that dramatic change [Teaching ------------- Lecturing]. Statistics in most of the universities show that the age profile of the learners is dropping. Most are finding it extremely difficult to handle their learning all by themselves. Is this problem age-related?

    In a secondary school environment, class size is relatively small compared with what obtains in a university. Moreover, in the university, the learner is the manager of time unlike in situations [in secondary schools] where the time for learning is planned by someone other than the learner. At the university, there are far more variability in terms of the type of people one interacts with. Indeed, the psycho-social environment in for instance a natural science laboratory or on campus can be traumatic for learners who not we well prepared. The result of all the above is that the traumatised young learner in a university faced with a much higher level of cognitive learning content performs poorly. Statistics in the university in Nigeria show a rather high failure rate at the end of the first year in natural science courses.


    The discussion so far has raised more questions than providing answers. This is intentional as this presentation is meant to stimulate discussion so that at the end, we will come up with pragmatic suggestions/recommendations. Here are five suggestions to help our discussion.

    Building confidence in the university teacher

    A confident teacher is one who is well prepared for the job and invariably confident teachers are good teachers. The system should provide opportunities for those teachers who want to improve their professional competence. In the natural sciences, a good teacher must learn various techniques.

    Wait-time for learners

    The University System [especially in Nigeria] is moving out of phase with the old well laid down learning period. The semesters are now truncated; students are rushed and confused. Many students cannot confidently tell you when the next semester will effectively begin. The learners also need adequate time to plan their learning.

    Provision of adequate resources to promote teaching/learning

    Facilities in the natural sciences need to be adequate for individual use. The nature of the disciplines which make up natural sciences call for learner-material interaction strategy. Laboratories should be well equipped if we want to promote teaching and learning in the natural sciences.

    Need for learners to assess their teachers

    The University system is used to teachers assessing learners through cognitive tests. There are few instances where learners have been given the opportunity to assess the teaching of their lecturers. Information shared from the latter is known to promote teaching and learning, and this will be more so in the natural sciences.

    Teaching students how to learn

    Textbooks and lecture notes alone will not be adequate in sourcing information in the vast area of the natural sciences. As we prepare our learners for the 21st century, we must also teach them how to access information available mainly in the information super highway.


    In this brief discussion on promoting teaching and learning in the natural sciences in the context of higher education, we conclude with the following:

    Let me here share aspects of my contribution in the book ‘Teach Your Best’. These clips I presume will be relevant to our discussion on promoting teaching and learning in the natural sciences.

    The teacher as an Authority

    There are a number of areas you should consider when preparing your lecture.

    Area to Consider Questions to ask Yourself

    1. Identification of topic/objectives What do you want to teach?

    2. Audience For what group of students is the lecture meant?

    3. Location and Duration Where will the lecture be given and for how long?

    4. Subject Matter What is the scope of the lecture? How far, in terms of

  • depth and breadth, do you think you can go with the subject matter in the prescribed time?
  • 5. References Is there any fact you think you have to look up in a book,

  • journal or in your notebook? Are you confident with the facts you want to put across.
  • 6. Trouble Shooting ` Are there any areas in the subject

  • matter that you
  • anticipate students might find difficult to grasp? If so, do you want to double check these areas to be prepared for the questions.
  • 7. Instructional Media Do you want to use any aids during

  • the lecture? Have you
  • checked if they are available and in good working condition?
  • Areas to consider when preparing your lecture

    If you have carefully considered all these seven areas, you will probably enjoy giving the lecture as much as your students will enjoy listening to you



    The technique adopted in the delivery can make all the difference between a good and a bad lecture. Whatever the size of the lecture room, your voice must be clearly heard. You are the best judge of how to pitch your voice. If the room is too large for you to be heard, then you must use a microphone. You should adopt a conversational style of delivery and not keep your eyes glued to your lecture notes. Make eye contact with individual students, and scan the class as a whole. In a situation where students have few textbooks, lecture note mean a lot to them. Therefore when you have to write on the chalkboard, you must make sure that your writing is legible. Sometimes you may use an overhead transparency, In that case, you should make sure that the lettering on the transparencies is focussed sharply. Allow sufficient time for the students to take their notes before removing the transparency. As a lecturer, you must strive to take your students forward by advancing their knowledge from a known starting point. In pedagogical terms, that starting point is referred to as the academic entry point. Therefore, before you start lecturing, you should establish the academic entry point of your students.

    Excerpted from:

    Bajah, S.T. (1998, September). Promoting teaching and learning in higher education in the natural sciences. Presented at the UNESCO Workshop on Teaching and Learning in Higher Education. University of Ibadan, Nigeria.

    Prepare a checklist using suggestions in Reading 4.1 on "The Teacher as Authority". Assess your last lecture using the checklist.


    Selection Criteria of Teaching and Learning Methods

    Methods of teaching and learning can be classified as follows according to the ability or inability to foster autonomous learning:

    These are large group methods e.g. lecture, and symposium.

    These are methods implemented in small groups or call for individual work by students.

    Let us now undertake a description of some of these methods.



    In a lecture, the teacher addresses learners without interruption. This method is used for large classes. It allows the teacher to use the whole of the teaching time. This method has limitations because it does not foster learning. As a matter of fact, the learner’s main task is to listen carefully. He or he is a listener, a little active; a little autonomous, since dependence is on what the teacher says and does. The opinions of the learner count very little.



    Small group methods include:


    The aim of the seminar is an in-depth exploration of a specialised topic. It consists of periodic (usually weekly) meetings of small groups of learners (sometimes between 10 and 15) and a teacher who acts as an expert or a moderator. The learners are to read one text (or texts) on a specialised topic. They write what will be the subject of the meeting (in a report form), give it in advance to their peers (one week before, for example). The discussion will focus on the arguments and conclusions of the participants. These meetings allow an in-depth look at one topic at time. They develop in the learner, abilities for synthesis, critical analysis and communication skills.


    Group discussion is a method that allows the learner to talk about his/her experiences, and to share ideas. It develops in the learner abilities for listening, comprehension, synthesis and critical analysis. During group discussion the fluent learner can dominate the discussion. The teacher should possess the qualities of a good moderator to maximise interest in the use of such a method in learning.


    The case study is a written record of a hypothetical or real-life problem. The case study must 1) present the learner with situations that are very much related to the ones the learner knows or will know and 2) lead to decisions like those that will have to be made in real life. The case study can allow the learner to seek information necessary for the study of the case.


    The practicals

    The practicals make it possible to combine theory and practice. The practicals give the learner an opportunity to go beyond the words which remain as abstract symbols. Practicals give the learner the opportunity to observe, to describe, to interpret, to solve problems, to manipulate, and to collate and report information.

    Computer – Assisted Learning

    Using this method, the computer presents the material to be learned in an interactive manner. It is a system that allows immediate feedback, and the establishment of a specific working pace.

    Individually Prescribed Teaching

    The originality of individual prescription teaching resides in the following main characteristics :

    Distance Teaching

    Distance teaching or tele-teaching is an advanced form of what was once called ‘’correspondence course’’. In a course based on such a teaching method, the learner works alone, extramurals, most of the time at home. Once registered for the course of his or her choice, the learner receives the course documents by mail. In most cases, a written guide indicates what work has to be done with those documents. Details will be given in module 7.

    Some Methods of Teaching Environmental Science Concepts

    Peter Okebukola and Michael Ahove


    This strategy assumes a higher gradient of teacher knowledge on the topic relative to that of the students. In the pure lecture mode, the teacher transmits information to the students who are actively engaged in note copying and "soaking in" the supposedly rich content from the teacher. Only scant opportunity is available to the students for questions and discussions.

    In the lecture/discussion mode, however, there is a preponderant two-way communication between the teacher and the students. A lower knowledge gradient is assumed as students, like the teacher, are expected to contribute to information building during the class session. For instance, in a lesson on water conservation, many students could be as aware as the teacher of industrial and domestic techniques for conserving water. Such knowledge could have been obtained through the media, especially television and radio and also from practice at home. Lecture/discussion on such a topic would, therefore, involve both teacher-talk and a lot of student-talk. There will be sharing and compiling of ideas and reconstructing and negotiation of meanings in a constructivist sense.



    Project Method

    The project method classically involves breaking down a topic e.g. pollution, into integral components or sub-topics such as air pollution, water pollution, land pollution and noise pollution. Student groups are then assigned the sub-topics to carry out investigations and produce reports e.g. on causes, effects, and prevention of the assigned type of pollution for presentation to, and discussion by the entire class. The role of the teacher is to provide guidance when required and to monitor the progress made by each group. Each student group is free to adopt whatever methodology it deems appropriate for tackling the task.



    Concept mapping

    Concept mapping is a model of instructional strategy developed by Novak and his associates in 1972. It is a metalearning technique for assisting learners to organise information about science concepts in a meaningful manner inorder to facilitate meaningful learning. It is based on the premise that concepts do not exist in isolation but interrelate with others to make meaning. Organising new concepts/information into a form that shows these interrelationships helps learners make mental connections.

    The strategy was developed from Ausubel's (1968) assimilation theory of cognitive learning based on the idea that new concept meanings were acquired through assimilation into existing concept propositional frameworks. Ausubel and his associates had the task of how to present these frameworks. Thus given the additional ideas from Ausubel's theory that "the cognitive structure is organised hierarchically, and that most new learning occurs through derivative or correlative subsumption of new concept meanings under existing concept/propositional ideas" (Novak, 1977), they developed the idea of hierarchical representation of concept propositional framework which was later described as "cognitive maps" or "concept maps" (Novak, 1979).

    Concept maps are diagrams indicating interrelationships among concepts as representation of meanings or ideational frameworks specific to a domain of knowledge (Novak, 1990b). The maps can be applied to any subject matter and to any level within the subject. Maps generated by a learner report his or her conceptual organisation of the topic. They are intended to represent meaningful relationships between concepts in the form of propositions. Propositions are two or more concept labels linked by words in a semantic unit. Concept map in its simplest form would be just two concepts connected by a "linking word" and forming a proposition. For example "leaves are green" would represent a simple concept map forming a valid proposition between the concepts "leaves and "green." Apart from a small number of concepts which children learn through the discovery learning process, most meanings are learned through a combination of propositions which is acquired and in which the concept is embedded. Although concrete empirical propositions may facilitate concept learning, the regularity represented by the concept label is given additional meaning through propositional statements including the concept. Thus "tomato is red," tomato is a fruit, "tomato is a berry", "tomato is edible and so on leads to increasing meaning and precision of meaning for the concept tomato.

    Concept maps are therefore schematic devices to represent a set of concept meanings embedded in a framework of propositions. They work to make evident to both students and teachers the small number of key ideas they must focus upon for any specific learning task. They can also provide a kind of visual road map for a "journey" we are about to begin and some of the pathways we may take to connect meanings of concepts in propositions. After completing a learning task, concept maps provide a schematic summary of what has been learned. It has been recommended that concept maps should be hierarchical since meaningful learning proceeds most easily when new concepts or concept meanings are subsumed under broader more inclusive concepts, that is, more general, more inclusive concepts should be at the top of the map, with progressively more specific, less inclusive concepts arranged subordinately.

    Concept maps can be constructed by students from texts or after class discussions/lecture. It involves listing the main ideas/concepts and words and arranging these in a hierarchy. The most general, abstract and most inclusive (superordinate) concepts are lower down in the hierarchy. This array of concepts is connected by lines or arrows carrying labels in a propositional or prepositional form. At the terminus of each branch may be found examples of the terminal concept. A finished concept map is analogous to a road map with every concept depending on others for meaning.

    Thus, in a concept-mapping exercise (Okebukola,1990), students:

  • 1. note the keywords/concepts, phrases or ideas that are used during the lesson or read in a text;

    2. arrange the concepts and main ideas in a hierarchy from the most general most inclusive and abstract (superordinate) to the most specific and concrete (subordinate);

    3. raw circles or eclipses around the concepts;

    4. connect the concepts (in circle) by means of lines or arrows accompanied by linking words so that each branch of map can be read from the top down;

    5. provide examples, if possible, at the terminus of each branch; and

    6. cross-link hierarchies or branches of the map where appropriate.

  • Concepts are generally isolated by circles and connecting and labelled with linking words which describe how the connected concept are related to each other. Two connected concepts make a prepositional linkage or statement about how some piece of the world looks or works. Cross links are prepositional linkages that connect different segment of the concept hierarchy. Cross links are particularly powerful connections which form web of relevant conceptions, probably enhancing anchorage and stability in the cognitive structure.

    Rather than just connecting general concepts to specific concepts, cross links tend to connect different sub-domains of conceptual structures. Linkages that are made only vertically would be more likely to be forgotten than those made both vertically and laterally. Vertical connections are somewhat more specific instances of concepts, whereas cross-links relate together concepts in different domains of hierarchy. Unlike rote learning in which series of propositions are memorised and not related to each other, with concept mapping new concepts and propositions are connected into a whole existing relevant framework.




  • Ausubel, D.P. (1963): The Psychology of Meaningful Verbal Learning New York, Grune & Stratton.

    Novak J.D. (1990): Concept Maps and Vee Diagrams: Two Cognitive tool to facilitate Meaningful Learning, Instructional Science, 19, pp. 29 - 52.

    Novak J.D. & Gowin D.B. (1984): Learning How to Learn, New York, Cambridge University Press.

    Okebukola, P.A.O. (1990). Attaining meaningful learning of concepts in genetics and ecology: A test of the efficacy of the concept mapping heuristic. Journal of Research in Science Teaching, 27(5), 493 - 504.

    Okebukola, P.A. & Jegede, O.J. (1989): Students' anxiety towards and perception of difficulty of selected topic in biology under the concept mapping heuristic, Research in Science and Technological Education, 7, pp. 84 - 92.

    Schmid, R.F. & Telero G. (1990). Concept mapping as an instructional strategy for High School Biology, Journal of Educational Research, 84, pp. 78 - 85.

    Wandersee, J.H. (1990). Concept mapping and the cartography of cognition. Journal of Research in Science Teaching, 27(10), 923 - 936.

  • Excerpted from:

    Okebukola, P.A.O. & Ahove, M.A.N. (1997). Strategies for Environmental Education. Ibadan: STAN

    Critically review the methods described in Reading 3.2. What other methods do you put to good use in your class?




    Promoting Teaching and Learning of Mathematics in Higher Education



    Let us now look at the task of the Mathematics Lecturer (teacher?) who is facing his/her first year students fresh from the secondary school. In doing this, we base our presentation on the need to strengthen students’ entry behaviour on which teaching and learning will be built in the institution. We shall adopt the strategy of giving the teacher some tips for effective teaching of mathematics to the neophyte undergraduates or freshmen.

    1. Dispel a Myth: These freshmen usually come with a myth that Mathematics is a meaningless, abstract collection of figures, symbols and letters for the purpose of gymnastic manipulations to obtain some predetermined right answers. This myth must be dispelled at once. One effective way of doing this is to express, as much as possible, mathematical symbols and expressions in words and relate them to practical life. For example, it is astonishing that many freshmen can easily write the expression for Pythagoras’ theorem, but when asked to express it in words and relate it to practical life, they are found wanting. The same thing applies to the three fundamental equations of kinematics:
    2. V = U + at

      S = ut + (1/2)at 2

      v2 = u2 + 2as

      There is an abysmally low number of freshmen who can express these equations in words and relate them to practical situations although they can recite them using their secondary school rote memory experience. The higher education teacher must therefore teach the neophyte undergraduate how to state mathematical expressions in meaningful words and get them to see their relationship with life.

    3. Explain Letters and Symbols: Take the case of letters and symbols in Algebra, Trigonometry and Calculus. The students are used to the letters of the English alphabet – a, b, c, … x, y, z – in Algebra, but do they know that these letters represent unknown quantities which may be found through logical operations? They also know some other "strange" letters such as " , $ , ( , 2 , and T from their elementary trigonometry in the secondary school. But hardly do they know the source or names of these letters, mainly because their teachers at the lower level also did not know (what a vicious cycle!). The lecturer should dispel the myth by letting the learners know that these are letters of the Greek Alphabet which we use when there may be confusion in repeated uses of the letters of the English Alphabet. The first letter of the Greek Alphabet is " (alpha) and the last letter is T (omega). (Remember "You are the Alpha and Omega, the beginning and the end…). Some others are $ (beta), ( (gamma), ) (delta), 2 (theta), n (phi)…
    4. These case of Integration in Calculus is another situation which usually seems mythical to students when based on an abstract background. Let us take it that the freshmen have a knowledge of elementary Differential Calculus in which they have been taught how quantities and magnitudes can be studied in details by breaking them down into very small pieces represented by ) x. In the process of Integration we want to collect all these pieces back to make a whole (explain the meaning of "integrate"). Hence we talk of the "sum of all the small parts ) x", i.e., "sum of all ) x.

      Since Mathematics uses symbols to summarize sentences and phrases, we can use "s" to represent "sum of all" and hence write s) x. But this will be confused with any other "s" later in our work, so we shall modify the "s" by elongating it to stand as a special symbol looking like this: I . This is called the sign of integration, so we write I ) x. But as ) x becomes very small, we write the usual symbol dx to obtain the familiar expressions I dx, I xdx, I x2dx, …. As the case may be.

      Our intention here is to emphasize that for a strong foundation, it is always necessary for the teacher to present new topics in a simple and friendly manner by patiently explaining apparently strange and intimidating symbols and bringing up the frame of reference of the students to accommodate the new language to be used in the new topic. It is to be remembered that accommodation as a reorganization of the cognitive map is a more complex learning process than assimilation by which the "new" input finds a ready anchor.

    5. Relate to Technology: Furthermore, it is necessary, wherever possible, to point out the application of mathematics to practical situations, especially in the development of technology. The study of Electromagnetic Waves is now part of higher mathematics. How does this relate to the transmission of radiation in broadcasting and in communication in space travels? The quantum theory and Schroedinger’s equations are now part of higher mathematics. How do they relate to transmission of electrical energy in microchips which are used in the construction of sophisticated computers? In a lighter mood, it has been conjectured that Maxwell’s Electromagnetic Equations are what God wrote in the firmament, and "there was light"
    6. Emphasise Process and De-emphasise Answers: Most students of Mathematics are of the opinion that the purpose of learning Mathematics is to obtain "the" right answers to problems. Answers per se are important but certainly not the crucial point in learning Mathematics. Indeed, some questions in Higher Mathematics may not require a definite figure as an answer. Cases such as "prove that…", "show that…" do not require numerical answers. Of greater importance is the teaching of the processes of handling Mathematics as a series of logical operations. Hence, the teaching method to be adopted predominantly is what Nagel (1966) calls the "Deductive-Nomological Pattern of Explanation." As Oguntonade (1971) pointed out, "in an explanation of this type, the phenomenon to be explained, (called the explanandum phenomenon) is shown to be a necessary or logical consequence of a set of premises (the explanans) which consists of at least one universal law (L) and some instances of the universal law(s) drawn from the explanandum phenomenon. In its simplest form, this is the "if x, then y, provided z" pattern of explanation.
    7. Work copious examples: Since Mathematics is not a leisure-reading subject, but a subject to be practised, the teacher must take the lead by systematically working copious and varied examples for the students.
    8. Get the Students to Practice: The teacher should also adopt the Practice and Drill Method of teaching by getting the students to work copious examples in class and as home assignments. The question of student population explosion arises here. However, it is the practice at Higher Education level to get teaching assistants to assist the lecturer in supervising and grading students’ work.
    9. Take Advantage of Modern Technology: The teacher should update and use his knowledge of modern technology in the teaching of small and large Mathematics classes. The closed circuit television (CCTV), the Programmed Instruction Technique and the use of the Computer, especially the Compact Disc (CD) are some of the cases in view.

    The question which arises now is: Do these strategies and others like them guarantee learning by the students? Not necessary, but they are essential pre-requisites to facilitate learning.


    It is presumed that every professionally qualified teacher knows that the processes of teaching are different from those of learning. However, apart from actually teaching the contents of Mathematics, it is becoming increasingly necessary for the teacher to teach students how to learn. But in the final analysis the student must take advantage of the teacher’s efforts and learn on his own. For this purpose, we offer the following tips which are by no means exhaustive.

    1. Be familiar with the Scope of Work: The scope of the work required to be done every semester must be known to the student so that he plans how to learn and cover the work done during the semester.
    2. Speak Mathematics: The student should try to state every mathematical expression in his own words as much as possible. In addition, he should practice describing events, objects and issues of everyday occurrence in mathematical language. In this way, Mathematics will become part of the daily life of the student.
    3. Practice Mathematics: The student should learn that Mathematics is not a reading subject. It must be practised. Hence, the student must make it a point of duty to practice many problems everyday. While answers are important; they are not the main goal of practising. The processes are much more important than the answers. In addition, group practice among students is very useful in fostering independent practice later on. Furthermore, the student should persevere when a single problem seems to consume a lot of his time. The beauty of finally overcoming a knotty problem is that many like it can then be solved in quick succession later on because the "trick" has been discovered.
    4. Seek Lecturer’s Assistance: Whenever necessary the student should seek individual attention of the lecturer to assist him in solving a problem. But this should not be made the main strategy for private learning. The student should not say to the lecturer, "I don’t understand the problem, please solve it for me". Rather he should say, "This is how far I have tried in my attempt to solve this problem and I do not know how to proceed from here despite my attempts. Please, assist o unravel the knotty point".
    5. Use different Textbooks: For variety in practice of problems, the student should read the approaches offered by different authors and solve problems in their textbooks. This gives the student the confidence that he is a mathematician anywhere, judging by available standards, and not just a mathematician by virtue of sticking to a particular author’s ideas.
    6. Re-practice: After some weeks, the student should return to the sections and problems which he had practised and re-practice. This is one very useful way of self-test to ensure that Mathematics is not just being piece-meal or in an ad hoc manner.


    The only way in which the teacher can find out if the student has actually learnt what he is supposed to learn is to set tests for the student. Teacher-made tests in mathematics are usually subject to the violation of known norms of good testing methods and lecturers at higher education level tend to adopt the strategy of copying textbook items and setting tests in a hurry. This should not be so. Mathematics in particular, demands strict fairness to the student in testing what has been taught. The teacher is strongly advised to

    1. map out the area to be covered by the test
    2. identify the objectives of the test
    3. prepare a valid blue-print for the test
    4. respond to the test himself before administering it to the students
    5. season the test for at least one week and modify it on the basis of revised strategy in planning, and weighting of the items
    6. prepare a full-scale marking scheme showing the processes expected and the scores judiciously attached to each step in the processes
    7. produce and administer the test to the students under the conditions required by the regulations
    8. score the responses strictly in accordance with the marking scheme
    9. take and keep a proper record of the performances in each test
    10. give feedback to the students, to himself, to the institution and to the curriculum developers for various purposes which need not be discussed here.

    Excerpted from:

    Oguntonade, C.O. (1998, September). Promoting Teaching and Learning of Mathematics in Higher Education. Presented at the UNESCO Workshop on Teaching and Learning in Higher Education, University of Ibadan, Nigeria.

    What lessons have you learned from Reading 3.3 that you can apply to your discipline/subject area?

    1. Do teachers need to understand the nature and syntactical structure of Mathematics to be able to teach it effective?
    2. What are the practical strategies to be adopted by teachers of Mathematics at the tertiary level, particularly in view of the poor background or the students at the point of entry and their concept of Mathematics as a myth?

    UNIT 3

    At the end of this unit, you will able to:

    Strategies for Promoting Teaching and Learning

    In order to promote learning, the teacher will need to comply with the following prerequisites:

    One of the best ways to improve one’s performance constantly is to use a performance chart. In our case, this performance chart will be about the pedagogical methods used. It will be like the car dashboard, which gives critical information on the condition of the car, represented in our case by the state of teaching methods used.

    In order to control the quality of the methods used, the teacher will have to identify:

    Finally, no method is absolutely efficient. The efficiency of a method depends on the nature of the learner, the number of learners, the subject being taught, the teacher’s personality, the material and physical conditions etc.


    The Workshop – An Example Of A Small Groups Teaching Method

    Workshop Planning

    A workshop requires advance planning and organisation. Let us consider some of the planning and organisational issues.

    Several months before


    On the eve of the workshop

    The day of the workshop

    After the workshop




    In this Module, we examined some methods of teaching and learning in higher education. We looked at the "traditional method" of lecturing, and also small-group methods. The merits and demerits of the methods were discussed. An example of the use of the workshop was used as a way of demonstrating how to plan for and use any of the methods for teaching. It was emphasised that there is no best method for all occasions. It behoves the higher education teacher to identify the method that best facilitates learning for his or her particular groups of learners.



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