University of Nigeria Instructional Materials f… · The physically exception child, such as the impaired vision, the impaired hearing, the imparied speech, the crippled, and the

University of Nigeria Virtual Library

Serial No

ISBN 978- 175- 438-9

Author 1 AGWAGAH, U.N.V.

Author 2 Author 3

Title Improving Instructional Materials for Mathematics Learning of Exceptional

Children Keywords

Description Improving Instructional Materials

for Mathematics Learning of Exceptional Children

Category

Science Education

Publisher Africana Fep

Publication Date 1999

Signature

AND UCATION

A Book of Reading in Honour of MSGR. PROF. F. C. OKAFOR

Edited by

Anthony AIC &

Beatrice A. Okeke

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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of Africana-FEP Publishers Limited. ISBN 978 175 438 9

Very Rev. Msgr. Prof. F.C. Okafor, Ph.D.

422 Philosophy and Education

Some of these teaching materials can be borrowed from Resgurce Centres, State Secretariate of mathematical Associations, and so on.

Use of Instructional Materials Teachers should use materials for relevant and useful purposes especially as they concern the concepts being taught. Activities involving the use of instructional materials should not be misused, underused or abused. children should be allowed full participation in such activities with the teacher acting as a guide or umpire. The use of instructional materials is often guided by the following considerations:

Storage and retrieval of the material Skill needed in using the material The educational benefits of using the materials Handling problems Pupils acceptability of the material Maintenancehpgrading problems from repeated use

Conclusion Mathematics is an important subject because it has multi-

- faceted applications. The primary school is the entry point of inculcating mathematics in pupils. Such inculcation is better grounded through teaching that uses real-life objects during instructions. Activities could be games, quizzes, puzzles, listening to audio-visual materials, handling of objects, construction of shapes etc. Activities and the requisite instructional materials need to be carefully planned and implemented based on certain considerations that were presented and discussed in this paper.

Improving Instructional Materials for Mathematics Learning of lEKceptional Children

U. N. V. Agwagah Sub-Department of Science Education

Universiv of Nigeria, Nsukka

Conceptual Definition of Exceptional Children The term exceptional children is an umbrella term which e3compasses many different groups of children and many different degrees of disability within each group. Cruichshank (1967:3), defined an exceptional child as "one who deviates intellectually, physically, socially, or emotionally so markedlj from what is considered to be normal growth and development that he cannot receive maximum benefit from a regular school programme and requires a special class or supplementary instruction and services". For Kirk (1972), an exceptional child is one who deviates from the normal or average child in mental, physical and social characteristics to such an extent that he requires a modification of school practices or special educational services in order to develop to his maximum capacity. With these definitions, the exceptional child is regarded as one who deviates from the normal growth in various dimensions of development, and who requires special attention in school. Thus, an exceptional child is one whose


needs and whose situation do not fit into the general framework of traditional mathematics education. The different types of exceptional children are; 9 The intellectually exceptional, such as the gifted and

creative child, the slow learner, the educable mentally re, arded, and the severally mentally retarded. The physically exception child, such as the impaired vision, the impaired hearing, the imparied speech, the crippled, and the brain injured. The emotionally exceptional child, such as the delinquents.

9 The multi-handicapped child, that is the child who has more than one defect given above (Chauhan, 1978: 470). The physically, and/or the emotionally exceptional child, or the multi-handicapped child may be regarded as the learning - handicapped or learning - disabled child.

According to Wilmot and Thornton (1989) learning - handicapped child may have any combination of memory deficits, visuaVauditory perception or discrimination deficits, abstract reasoning difficulties, or other difficulties that interfere with learning. This shows that the term exceptional children represent children on both the extremes of physical, social and academic achievements.

In the National Policy on Education (FGN, 1998), exceptional children are categorized into two groups. The first group refers to children who have learning difficulties because of different kinds of handicaps - blindness, partial - sightedness, deafness, hardness +f-hearing, mental retardation, social maladjustment, limb deformity or malformation, etc, due to circumstances of birth, inheritance, social position, mental and physical heath patterns or accident in later life. As a result, such children are unable to cope with the regular school class organization and methods. Another group of exceptional children are the specially gifted and talented children who are

Improving Instructional Materials ... 425

intellectually precocious and find themselves insufficiently challenged by the programmes of the regular school. In this paper, therefore, the issues related to exceptional children are considered for two distinct populations in the mathematics classroom - mathematically gifted children and learning - handicapped children.

Blueprint on Education for the Exceptional Children in Nigeria A blueprint is a pattern from which we build (Pounders, 1981). It is a vision, pattern, plan, scheme, design or specification. A blueprint involves an advanced consideration of such issues as resources, methods and personnel, etc needed to achieve a particular complex task. It makes one hold a mental picture of What is expected. With this mental picture or pattern, it is easy to evaluate the outcome of the task.

Within government circle, a blueprint could mean a policy. According to Weeks (1986), a policy is a formalised set of procedures designed to guide behaviour, to maintain consistency in behaviour or alter it with a view to achieving specific goals. Since a large number of children fall in the category of exceptional children, provision must be made for their education, otherwise their potentialities will remain undeveloped, resulting in great wastage of human resources. Hence, in accordance with the prevailing government policy to provide equal educational opportunities for all children, irrespective of their caste, creed and religion, it is stated in the National Policy on Education (FGN, l998:39), that "special education shall be provided for the two categories of exceptional children. Special mention was even made in section 8, paragraph 84, item a, about gifted children, that "the Federal and state ministry of education shall,


in collaboration with appropriate bodies, provide special programmes for gifted children. I t is also stated in the policy that all necessary facilities that would ensure easy access to education shall be provided (Section 8, para. 84, item C). Thus, the policy aimed at providing adequate education for all special cases or exceptional children, and a diversified and appropriate cumculum for all the beneficiaries. Provision of diversified and appropriate curriculum would also mean the provision of appropriate instructional techniques and instructional materials.

Meaning of Instructional Materials Instructional materials are any forms of materials that can aid or speed up the process of learning. They are the necessary extra ingredient that makes teaching and learning mathematics a pleasant and satisfymg experience (Johnson, 1971). According to Ogomaka (in press), instructional materials are, tangible objects employed by the teacher or an organization for the purposes of giving or facilitating instruction.

In the educational sense, instructional materials are all those tangible objects used to facilitate teaching and learning. They are pre-selected objects Srmght into a teaching-learning situation to provide auditory, olfactory, tactile or visual stimuli/imagery relevant to and/or necessary for achieving the objectives of the teaching - learning situation. Instructional materials give to a lesson the breadth and depth that would be difficult to obtain in any other way. They link thought processes and reality, they relate past experiences to a new situation. Hence, they help to make w i t i o n s from one idea to another.

The creative teacher uses instructional materials to add meaning and interest to verbal instruction. However, in using instructional materials, the teacher must not fail to take


students from the concrete representation to the concept behind it.

Instructional materials/teaching aids available for the teachinulearning of mathematics fall into three broad categories. They are: literature, audiovisual aids and models, and manipulative materials (Johnson, 1971). Literature includes: books, pamphlets and journal articles, periodicals, magazines, newspapers, etc. Audiovisual aids include; films, filmstrips, videotapes, and overhead projectors, as well as such traditional equipment as flannel boards and cutouts, charts and chalkboards. With the overhead projector, the teacher can project his writing on the screen, use previously prepared transparencies, or use a variety of the commercial transparencies that are available. Models and manipulative materials include; demonstration aids such as, the number line, the hundred board (a board with 100 peg holes arranged 10 by lo), an area demonstration device; laboratory devices such as $lace 61uqboa;ds, number sticks, numeration blocks>pin, dials, area aids, geometry boards, measuring instruments; computation devices such as abacus, calculators, computers; learning games; enrichment materials such as puzzles,-tricks, toys; kits for laboratory experiments in mapping, curve stitching, probability, and geometry; geometric s hgpes, models, graphs and drawings. Odili (1992), -listed the following instructional mat&ls as theXGtfrequently mentionedin the mathemati-Me-@o-ds courses of. N.C.E. teacher- education institutions in Nigeria; real objects; pictures, charts and graphs; display boards such as chalkboard, bulletin board, flannel board; dramaticx&pression; games, simulation and role playing; slides, filmstrips and transparencies; motion pictures and related projection equipment; records and record -players; tape and tape recorders; radio; television and video tape recordings; reading machines and programmed instnrcdon; learning kits; hand-outs; textbooks, etc.


Philosophical Reflection on Mathematics and its Pedagogical Implications for Instructional Materials One Platonic philosophy, as presented in Burnet (1960), is that which identified the forms with numbers. This doctrine, which is attributed to Plato, is that numbers are not addible to one another. According to Plato, we speak of adding two and two to make four as if there were many twos. But it is clear that there is only one number two, the form of two or the dyad. Thus, when we say two and two is four, we mean that two units of a given kind added to two units of the same kind are equal to four units of that kind. We do not mean that the number two added to the number two is the number four, for the number two does not consist of two units nor does the number four consist of four units. Each number is universal, and every universal is one and unique. I t follows therefore that the "two and two which make four", should be distinguished from the "two and two pebbles which make four pebbles, for instance, this distinction can only be made when concrete materials are used in the learning process. For this reason, instructional materials such as pebbles, bottle covers. stick etc, are used to represent and distinct the units when teaching the primary school pupils.

Also, for Plato, the objects of mathematics differed from the things of sense in being eternal and immovable and from the forms in being many, whereas each form is one and unique (Burnet, 1960). Plato illustrated this with the mathematicians' diagrams and models. According to him, the models or diagrams of say circles, triangles, etc, are only approximations to the ideal mathematical figures. They are unique forms in a pre-eminent sense.

Another argument of Plato is called the 'ielational argument" (Brent, 1978). He argued that in mathematics we use terms that admit of very precise definition, for example, the


concepts 'half and 'double'. Yet a practical experiment reveals that we cannot apply such precisely definable mathematical concepts to our everyday commonsense world. For instance, suppose we bring together three bank notes: a $450 note, a $4100 note, and a $4200 note and ask whether it is half or double. The answer is clearly that it is both; $4100 is half of $4200 note but double the #50 note. We thus have an object that is both half and double at the same time, so we are unable to say that our everyday commonsense $4100 note is unambiguously anything. Most everyday commonsense experience is inevitably like this. Thus, for Plato, mathematical concepts do not apply to the world of commonsense experience. This implies that whatever mathematics concept is taught, should be kept in a realm of reality. Students will solve problems better if they are kept in the realm of commonsense

gather than in the unreal world of formal rules. The mathematics we teach must start from real and interesting situations for students to learn. Instructional materials play important roles in this regard, especially in the education of exceptional children.

The above views relate to the pragmatist philosophy, sometimes known as Experimentdism and at other times called instrumentalism. (Okeke, 1995). The pragmatists believe that teachingAearning must involve action. It must involve doing things by the pupils. Thus, the pragmatists reject and jettison abstraction, and speculation. They emphasize observable data or phenomena and turn towards facts, towards action, and towards power (Okafor, 1984). Pragmatist education method therefore is child-centred and affords pupils the opportunity to actively participate and explore in teachingAearning situation, Provision of a wide variety of instructional materials will provide learners the opportunity of doing things, and being actively involved, and will enable them to discover things for themselves.


According to Okeke (1995:1), "the child-centred method of teaching fits the gifted.. . ." This is because it gives the pupils the opportunity to reconstruct and reorganize their experience, and this makes for originality and enhances creative potentials of pupils. Thus, it is emphasized that for gifted children to be trained in the method of critical thinking that fosters creativity, they should be provided with a wide variety of sources and guidance that will enable them discover things for themselves. For the gifted children, the presence of instructional materials create stimulating learning and school environments which are prerequisites for inculcating and encouraging the spirit of inquiry, wanting to know, and problem solving, which are traits of giftedness. Thus, adequate provision and use of instructional materials will encourage gifted children to experiment and explore within their learning environment.

The active learning approaches are also advocated for the learning -handicapped children. Wilmot and Thornton (1 989), presented a report of how Terry, a seventh grade, learning - disabled pupil with memory and visual perception deficits was able "to remember that the volume of a pyramid is one third the volume of a prism, as a result of his participation in a water- pouring experiment" involving pyramids and prisms of like

- - height and bases. Also, a study by Kuhfitting (1974), on the Relative ~ffectiveness' of Concrete Aids in Discovery Learning, showed that low ability subjects benefited from concrete aids more than high ability subjects in mastering abstract skills.

However, although there are strong theoretical bases for the use of instructional materials in the teaching/learning process, especially for the exceptional children, and in mathematics learning, there seems to be some problems in the procurement and use of such materials.

Improving Instructional Materials ... 43 1

Problems of Using Instructional Materials in the Teaching/Learning of Mathematics There are some major problems in the procurement and use of instructional materials for the teaching and learning of the exceptional children in mathematics. One of such problems is the teachers' level of teaching skills. Teachers are frequently criticized for their level of mathematical knowledge and teaching skills (Hyde, 1989), especially at the primary school level of education. Many teachers lack knowledge of how to organize the mathematics instruction, and what materials to provide for instruction, especially for the exceptional children.

Related to the above problem, is the issue of teacher's qualification. Many teachers are not qualified mathematics i teachers, particularly in the teaching of exceptional children. I Chauhan (1978), pointed out that the teacher for gifted I chgdren must have some special characteristics, so also the teacher for learning - handicapped children (Cruickshank, 1967), and many of the teachers lack such special characteristics, and are not competent to teach the exceptional children. As a result, the teachers who are qualified are in short supply (Ojo, 1986; Sunday, 1994). Moreover, some teachers who are qualified and experienced are "out-dated, old- fashioned and out of tune with the latest developments in learning theories and modem effective teaching techniques" -

(Ojo, 1986). For instance, many qualified and experienced mathematics teachers are not knowledgeable in information technology and are not computer literate. Hence, they are not able to apply computer-aided instruction in mathematics, especially in the teaching of exceptional children.

Another major problem in the use of instructional materials is the unavailability of appropriate and relevant instructional materials. For instance, many of the mathematics textbooks available are not designed and written for the exceptional


children and there are no teachers' guide. Quadir and Nock (2000), reported that special libraries are not accessible and available for disabled users. This supports Sunday (1994) report that some schools do not have library buildings, and those who have, do not have adequate mathematics textbooks within their libraries.

One other serious problem of instructional materials is lack of funds. Schools are not well funded by government to enable them procure instructional materials. Also, governments and all those concerned with the management of schools are not able to provide teachers with the barest minimum of instructional materials and equipments. They treat the supply of instructional materials and equipments with levity (Ojo, 1986). But, the business of education is expensive and good education must cost money.

Improving Instructional Materials for Matltematics Learning of Exceptional Children* An element of the mathematics curriculum which is necessary in the learning of mathematics for exceptional children is the issue of relevant instructional materials: equipments, reading texts or printed materials, audio-visual and other gadgets, films, video discs, calculators, computer wares, etc. Although local materials are needed and are to be developed, but commercial ones are indispensable. Some of the desired materials are quite similar for both the mathematically gifted and learning - handicapped children, but for different reasons; while other materials are decidedly opposite, for both groups, and even for different types of disabilities. For instance, instructional materials in printed form are not suitable for the visually impaired children. The reading materials for this group of children are in Braille writing or talking books (Quadir, and Nock, 2000). Moreover, Quadir and Nock, stated that the children also need a transducer which is a device or system by which a material intended to be perceived through one sense, e.g, the sense of vision is transformed in

improving lnstruclional Materials ... 433

such a way that it can be perceived through another sense, hearing or touch. On the other hand, the physically disabled, for instance, can use materials in printed form.

However, one of the chanenges of mathematics education for exceptional children is keeping up with new developments. In this era of information technology, there is need for incorporating the technology in the mathematics education of exceptional children.

Technology is derived from the term 'Techne', coined by the Ancient Greeks to define the complexity and diversity of human activity. According to Webster's Ninth New Collegiate Dictionq in Lewis (1993:4), technology is "a scientific method of achieving a practical purpose". Many people think of ,

technology as the computer, but it involves more than that. The types of technology that can be used for the teaching

and learning of exceptional children, can be divided into two clas'ses. These are:

The generic technologies designed for the general public, such as computers that are not adapted in any way, calculators, tape recorders, videocassette recorders and players, etc. Assistive technology device - that is, technologies designed to accommodate the needs of persons with disabilities (Lewis, 1993:4). According to Lewis, an assistive technology device is "any item, piece of equipment, or product system, whether acquired commercially off the shelf, modified, or customized, that is used to increase, .

maintain, or improve functional capabilities of individuals with disabilities". He explained further that some assistive technologies are adaptations of generic devices. For example, the talking calculators, for persons with vision disabilities works the same way as an off-the-shelf hand- held calculator, except that it also speaks when a key is pressed or a computation is completed.


The importance of computer in ma~hematics, especially in the teaching and learning of exceptional children has been widely acclaimed. According to Lewis, computers play many important roles in. special education classrooms. These roles include, Compr~ter-assisted instruction. In this case, computers deliver instruction and/or Frovide students with opportunities for practicing new skills or information. In addition, computers are used as tools for accomplishing tasks such as writing, drawing, etc. Teachers use computers to manage information, assist in assessment, or create their own computer - based lessons. The following are also some of the advantages of computer for mathematics instruction:

Drill and practice to master basic skills Development of writing skills Problem solving Understanding abstract mathematics concepts Simulation in mathematics Manipulation of data Individualized learning activities and record keeping

Although both groups of mathematically gifted and learning - handicapped children benefit from the use of computers and other technologies, there are particular benefits for learning- handicapped children. Lewis (1993), presented a conceptualization model of these benefits as the ABC model. This model implies that technology car1

Augment abilities Bypass or Compensate for disabilities

An example of technology that augment the abilities of students is the magnification system which can enlarge books and other print materials so that students with impaired vision can see them. An example of technology that bypass disabilities is the

Improving Instructional Malerials ... 435

computer which can be operated by voice commands or by eye gaze, for students who cannot use their hands to type. An example of technology that compensates for disabilities is computer programme with spelling assistance which helps students with learning disabilities who are poor in spelling to correct their spelling mistakes. Thus, computers and other technologies can work magic with some stl~dents [Male, 1988), especially with exceptional children.

Another element for mathematics learning of exceptional children is the mathematics laboratory. A mathematics laboratory is the place or room for the students to do all those things the teacher would like them to do if only there were time (Sweet, 1971). It is a special room in which mathematical experiments are carried out, just like in any other science laboratory.

According to Mckillip et a1 (1978), the mathematics laboratory is a place where children have experiences that help

4 them learn mathematics. Most often, these experiences involve concrete or semi-concrete materials and real physical situations. So, virtually all laboratory lessons require concrete materials to help children to learn some mathematics they did not previously know, or with the materials, children can use the mathematics they already know to solve real or realistic problems use these materials. Thus, fram the laboratory experiences, children learn how to apply the mathematics they have learned. Sweet (1971), stated that the laboratory approach is equally appealing to the slow learner and the capable student. In other words, it is equally appealing to both the learning - handicapped and the gifted children.

Some of the materials and equipment to be contained which should be available in a mathematics laboratory include: a movie projector, an opaque projector, an overhead projector, a wood working table with tools, mechanical drawing instruments, a transit, a sextant, a plane table, display cases,


storage cabinets, a protractors, rulers, posterboard, construction paper, wax papers, 3 - by - 5 cards, casein glue, drinking straws, toothpicks, needles and thread, stapler, an angle mirror, tables of logarithms and square roots, several one - hundred - foot tapes, wooden mallets, oak stakes, books, journals, periodicals, films, video, calculators, computers, TV, OHPs, slides, etc.

Moreover, projects constitute the most important part of the mathematics laboratory (Sweet, 1971). Consequently, good resource materials are essential. Ridge and Renzulli (19811, strongly advocated the use of projectors as products of learning. The projects can be independent or small -group activities and should, if possible, arise from real problems and from students interest in a particular branch of mathematics. Projects can be used for both the learning - handicapped and the gifted children. With projects of any type, these students still need motivation, encouragLi,at, arid direction from the teacher (short, 1985).

Examples of mathematics projects range from a complex computer programme to a display created for a science fair to an extensive research paper. With projects, students can be made accountable for demonstrating their growth in mathematical understanding by using objects to illustrate or model a concept, procedure, or problem solution, and using

- objects to show that a given response is correct. Instructional materials for learning mathematics, can also

be improved through improvisation. It is not always that we have the real or the conventional equipment and materials. Even when they are available, they are not always affordable. Hence, in the face of scarce financial resources, since schools are not adequately funded by governments, teachers and schools are called upon to improvise instructional materials.

Improving Instructional Materials.. 437

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curriculun~. London: George Allen and Unwin. Burnet, J (1960). Greek Plzilosoplzy, Thales to Pluto.

Macmillan and Co. Ltd. Chauhan, S. S (1978). Ad\-anced Educational Psychology.

New Delhi: Vikas Publishing House PVT Ltd. Cruickshank, W.M (ed) (1967). Educarior~ of exceptional

clzildrerl alld youth. London: Sta2le Press. Federal Republic of Nigeria (1998). ,r\i'a?ional po l iq or1

education. Lagos: NERDC Press. Hyde, A.A. (1989). Staff development: Directions and

realities. I n TI-ahon, P.R.8- Shutte, A.P (eds). ;v'elt, dil-ectiorls for ele1~1eilrtr7? scl~nol matlle!71atics. i'irginia: Xdtional Council of Teachers of Mlarhemarics.

Johnson, DA(1971) . Why use instructional materials in the mathematics classroom? In Aichelle, D.B.& Reys, R,E.(eds) .Readings in secondary S C I I O O I mathematics. Boston: Prindle, Weber and Schmidt, Inc..

. . . Kirk, S.A. (1972). E d u c d n g Exceptional Childten. Boston: Houghton MiffIm Co. . . . . . .

Kuhfitting, P.K.F.(W74). .The rehive effectiveness of . . ' .

' : concrete aids in ' discovery . . . le&ing: . S c h d . ' Science . and ' . . * . . . . ~&hematics. ui?oy (2k.. : . . ' :-- . . .

- , Lewis, R.B. (1 993). Specioi education technology. Los' - Angeles, ~difornia:. Brooks/Code Publishing Company.,

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Odili, G.A. (1992). The traking of N.C.E. Mathematics teachers: A focus on instructional materials. Abacus. 22 (I), 98 - 105.

Ogomaka, P.M.C (in press). Instructional materials for effective mathematics education at the secondary school level.

Ojo, J.O. (1986). Improving Mathematics teaching in our school. Abacus. 17 (I), 164 - 1?7.

Okafor, F.C. (1974). Philosophy ofeducation and third world perspective. Enugu: Star Publishing Company.

Okeke, BA. (1995). Teaching the gifted: The pragmatist dimension. Paper Presented at The National C~nference organized by the Faculty of Education, Nnamdi Azikiwe University, Awka 24th - 261h.0ct

Pounders, M. (1981). A carpenter is a worker who builds. Unity Magazine, Sept. 16 - 23.

Quadir, RO.& Nock, G.I.(2000). Special libraries and information services for disabled user. Paper Presented at the 2nd International Conference of the National Association of Women Academics at Abuja, sth - 10'. Sept.

Ridge, H.L& Renzulli, J.S. (1981). Teaching mathematics to the talented and gifted. In Glennon, V.J.(ea). The nlathematical education of exceptional children and youth. Reston, Va: National Council of Teachers of Mathematics.

Short, D. (1985). From the other side of the desk: How a gifted student wants to be taught. Gified/Creative/ Talented. 37,19-21.

Sunday, F.E. (1994) Government policy on the teaching of mathematics in Nigerian secondary schools. Paper Presented at the 31" Annual Conference of the Mathematical Association of Nigeria (MAN), held at the University of Ugo, 28th March - 1" April.

Sweet, R (1971), Do math labs. just happen? In Aichelle, D.B.& Reys, R.E. (eds) . Readings in secondary school mathematics. Boston: Prindle, Weber and Schmidt, Inc.


Weeks, J.R. (1986) Population: An introduction to concept and issues. 3'* ed. Belmont, .California: Wadsworth Pub. Co.

Wilm.ot, B. & Thornton, C.A. (1 989). Mathematics teaching and learning: Meeting the needs of special learners In Tragton, P.R.& Shul te, A.P. (eds). New Directions For Elementary School Mathematics. Roanoke, Virginia: National Council of Teachers of Mathematics.

Documents

University of Nigeria Instructional Materials f… · The physically exception child, such as the impaired vision, the impaired hearing, the imparied speech, the crippled, and the