DEPARTMENT OF EDUCATION AHMADU BELLO UNIVERSITY, ZARIA.

COURSE TITLE:

PSYCHOLOGY OF TEACHING AND LEARNING OF MATHEMATICS

TOPIC:

THE TEACHING AND LEARNING OF DIFFICULT MATHEMATICS

CONCEPTS

BY

SULAYMAAN ABDULWAAHID OLANIYI M.ED/EDUC/07036/2010-11

LECTURER:

DR. MAMMAN MUSA

MARCH, 2012.

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ABSTRACT

This paper aims at studying the teaching and learning of difficult mathematics concepts. The study tries to identify some topics that

teachers found difficult to understand and teach, and which students also found difficult to comprehend. Researches on difficult concepts

and areas of primary and secondary revealed that the core topics where the problem of teaching and learning occur most in

mathematics are plane and solid shapes, measurement of plane and solid shape, polygon, geometrical ratio geometrical transformation,

latitude and longitude. The study further traces the factors that are responsible for the difficulties. The findings show that Parental,

Students and Teacher Attitudes; Teacher Factor; Students Mathematics Background and Teacher Incompetence are some of

these factors. In the light of these, this study suggested that the government should give a greater emphasis to in-service education to

teacher in other to improve their efficiency and performances etc

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INTRODUCTION

Mathematics is the body of knowledge centered on concepts such as quality,

structure, space and change and also the academic discipline that studies them. According

to Ojerinde (1999), mathematics is the communication system for those concepts of

shapes, size, quality and orders used to describe diverse phenomena both in physical and

economic situation. Harbor – Peter (2000) also defined mathematics as the science of

number and space.

Mathematics is undoubtedly an independent subject and as such, other subjects

depend on it. This is supported by Badmus (2002) who stated that mathematics is an

indispensable tool in many other disciplines which have been developed because of the

importance to modern life. Usman (2002) sees it as a subject that encroaches into all

aspects of human endeavors and it has been described as the life wire in the studies of

various disciplines. Such disciplines are physics, chemistry, biology, economics,

sociology, linguistics, computer science, information theory, cybernetics among others.

Mathematics as observed by Abiodun (1997) is the major tool available for

formulating theories in the sciences as well as in other field. It is used in explaining

observation and experiments in other field of inquiry. Anguele and Usman (2007)

observed mathematics as a science of the methods by which quantities sought are

deducible from others known or supported. Thus, anyone who neglects mathematics may

not be able to go far in the sciences and in fact other thing of the world.

Mathematics is undoubtedly plays a great role in the natural development which

are shown in many researches. For example Ukeje (1997) described the importance and

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the attention given to mathematics as stemming from the fact that without mathematics,

there is no science, without science, there is no modern technology and without modern

technology there is no modern society. This fact could therefore be extended to imply

that mathematics being such an important element in modern societal development

deserves some good attention.

In Nigeria, Mathematics is a compulsory subject for all students of primary and

secondary school. It position in the National Policy on Education is as a result of its role

towards technological and industrial developments. No nation can develop scientifically

and technological without proper foundation in school mathematics (Okafor, 2005).

Despite all the aforementioned important of mathematics to the society in general,

the performance of learner has been poor at all level of education. Right from the primary

school through to the secondary school, the story is still the same. Adeniyi (1988)

indicated that students’ achievement in mathematics both at primary and secondary levels

worsen as the years go. A study conducted by Nigeria Educational Research and

Development (NERDC, 1997) on the performance of students in public examination over

some years has confirmed students’ poor performance in physical sciences (Mathematics,

Physics and Chemistry).

Mathematics Educators have put up noble efforts aimed at identifying the major

problems associated with the teaching and learning of Mathematics in both primary and

secondary schools. The establishment of professionals’ bodies such as Mathematical

Association of Nigeria (MAN), the National Mathematical Centre (NMC) is a good

example in support of our national mathematics educational strides. By taking a look at

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the reports of the West African Examination Council chief examiners in the year 2006,

we could see that all these laudable effort are not been appreciated. See the table below.

3-Years Results of the senior secondary Mathematics in the May/June 2004-2006

WASSCE Examination in Nigeria.

Year Entries % pass with

credit

% with pass and

below

2004 844,371 34.52 65.42

2005 943,371 35.55 64.45

2006 1,045,406 39.92 60.08

Students in all grade levels find the learning of mathematics difficult. However,

according to Oruh (1989), no subject is easy to learn and in each particular subject some

branches of it present more difficulties to both teacher and the students. Mathematics

with its abstract symbolism, its logical structures, its wide application, has unique

learning problems (Usman, 2002). Researches on difficult concepts and area of primary

and secondary school mathematics revealed that the core topics where the problems of

teaching and learning occurs most in mathematics are plane and solid shapes,

measurement of plans and solid shapes, polygon, geometrical ratio, geometrical

transformation, latitude and longitude.

Ohuche and Obioma (1983) indicated that primary school teachers are mostly

incompetent in geometry, mensuration and statistics. This automatically implies that

students will have difficulties in learning those areas.

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Oyedeji (1992) in a study sought to identify topics in the Nigerian Primary

Education Mathematics Curriculum, the result of the study shows that 15 topics are

identified as areas of difficulties, 9 topics are identified as having average difficulties and

24 topics identified as easy to teach topics.

The implication of this result is that the issue of poor performance in WAEC and

other liked examination started from the formation of the knowledge. In addition

Fajemidagba (1986) indicates that both teacher and students found word problem the

most difficult topic in the mathematics curriculum.

The WAEC Chief Examiners’ Report (2004 and 2005) on the performances of

senior secondary school in mathematics has consistency shown a downward trend in

students’ academic achievement in the subject. In the reports the following were some of

the students’ weakness in mathematics: Surd, area of circle, sector and segments, bearing

and distances, circle geometry, construction, latitude and longitude, trigonometric ratio,

three dimensional problems involving inequalities.

Similar finding was done by Kajuru (2008) in his attempt to establish the fear and

poor performance of students. A random sample of 900 candidates whose scripts

(WAEC) were marked by three different assistant examiners assigned to him for this

purpose. Each year on hundred (100) scripts out of three hundred given to the research

assistant was selected and find out how many attempted number and numeration;

algebraic process, geometry and trigonometry; statistic and probability, in other to record

the level of participation and performance of the candidate in the subject. The finding

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shown that, the performances of students in trigonometry and geometry were below

average.

Kurumeh (2007), in a study on teaching difficult concept in mathematics,

observed difficult concepts in mathematics in three ways i.e. from the teaching (teachers),

from the learning process (students) and from the nature or way the topics are (subject).

The study listed the shortcoming from the teachers of mathematics, from the student

learning the subject and the nature of the subject itself.

A research conducted by Harbor Peter (1992) and one from Obioma (1992)

separately indicted that that students find the following topics difficult in further

mathematics: vector geometry and processing services, sets and logic, statistic and

dynamic, correlation, construction and geometric proof. a survey of primary school

teachers mastery of primary school mathematics content was also conducted by Harbor

Peter and Ojomaka (1991). The researchers developed a 98 items multiple choice mastery

test based on the federal government of Nigeria primary school mathematics curriculum.

The result indicated that the primary school teacher have no mastery of the primary

school mathematics content as the teachers’ means score on the five sections of the

primary school mathematics curriculum is far below the expected mean.

Mathematics and the Students’ Achievement in Sciences

The issue of difficulties in mathematics concepts has reflected even in science

subjects offered at the secondary school level. It is very clear that it is the level of

mathematics that determines the level of science subjects. This has been one of the reason

of the career week program that are usually organize for the junior secondary category

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students, who are moving to the senior category. In secondary school, the senior category

is strictly divided into two (2): science and non-science. One of the measures for

selecting science students is their good performance in mathematics in their junior West

African examination council.

Researches from various fields like biology, physics and chemistry education have

shown the difficulty, in teaching some area of these subjects, as a result of low level of

understanding mathematics concept either from the teacher or the students.

Finley, et al (1982) investigated both content difficulty and importance as

perceived by physics biology and chemistry teachers. The results of the study shows that

in biology, most of important but difficult topics were in the area of mendelians genetics,

which assume a good mathematical background for their understanding; the chemistry

teachers indicated that writing chemical equation and formula, the concept of mole,

atomic structure, the arrangement of electron in atoms and stoichiometry all which

requires good mathematical background were difficult and important. The above research

work shows that students who find mathematics easy to understand tend to turn toward

chemistry, and those who fund mathematics difficulty choose against chemistry

(Godman, 1977; Reney, 1974; Osibodu 1981). In physics four of the six difficult and

important areas were related to forces on object and accompanying motion etc.

Having identified the difficult mathematics concepts, it is very important to trace the

factors that gave room its difficulties. And these are what we shall be examining below.

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Factors Affecting the Teaching and Learning of Mathematics

The importance of mathematics as a subject that is a basic to all scientific and

technological development is of a universal acceptance to all in the field of education.

With such a universal acceptance of mathematics as a key subject for progress and

development, yet is the most disliked and failed subject among the students. Almost

every year students manifest poor performance both in internal and external examination.

Supporting this argument are the studies of Harbor (1989), Galadima (1998) and Ojo

(1990). Furthermore, Osibodu (1977) in a study noted a regular high rate of failure at the

school certificate level. What has account for this tread?

Many reasons have been suggested to be responsible for it. These include the

death of unqualified mathematics teachers (Sanni, 2004), ineffective teaching as a result

of faulty methodology(Ale, 1989), curricular inadequacies and lack of satisfactory

teacher motivations’ strategies (Obodo, 2003).And according to Penso (2002),(1) the

pupil (cognitive and affective characteristics); (2) the content( aspect of the content of the

lesson); (3) the teacher (the learning methods); and (4) the lesson (the learning

atmosphere are source of difficulties. Here, we shall look briefly into some of these

factors.

Factors Related to Attitudes

• Parental Attitude and Students’ Achievement in Mathematics

It is widely acknowledge that student’s performance in mathematics is generally

poor. However, when talking about poor performance, it is very important to take a look

at parental attitude. It is a matter of fact that, the student’s poor performance is not solely

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developed from the school, but also the home background contribute to it. Aina (2006)

opined that the student’s performance and attitudes in mathematics can happen through

any of the following.

(i) Parental encouragement or discouragement

(ii) What parents expect their child’s performance to be?

(iii) Parents own attitude towards the subjects.

Various studies have borne evidence to these facts. For examples in a survey by the

Ale (1984), some students categorically stated that their parents discouraged them by

saying mathematics is not an easy subject for their family.

These findings show that there is a need for parents to help their children through their

own positive altitude and high expectation from them in mathematics.

• Students’ Attitudes and their Achievement in Mathematics

The attitude of students towards the subject also account for their failure in the

subject. May students prefer to be absent or have irregular attendance among others?

Neale (1969) found out that attitude and achievement have a reciprocal effect in their

relationship in that attitude affect achievement and achievement affect attitudes.

Burstein (1992) in a comparative study of factor influences mathematics

achievement found out that there is a direct link between the students’ attitude towards

mathematics and students outcome. Yara (2006) also observed that students beliefs, and

attitudes have the potential to either facilitate or inhibit learning. The students who have

more positive attitude toward a subject tend to perform better in that subject. And that

must have been the reason for what was said by Gibbons, Kimmel and O’Shea (1997).

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Gibbons, Kimmel and O’Shea (1997) opined that: “Students attitudes about the value of

learning science (mathematics) may be considered as both an input and outcome variable

because their attitudes toward the subject can be related to educational achievement in

ways that reinforce higher or lower performance”.

This implies that attitude of a person towards a program or course determines that

outcome of that program or course.

• Teacher Attitude and Achievement in Mathematics

The influence of teacher attitude on the achievement cannot be overemphasized.

According to Wiseman (1970), “Teacher attitude is one of the most powerful forces…

affecting pupils’ progress”. These attitudes of students influenced by the teacher can be

by his methods of teaching or his personality. The studies carried out by Bolaji, (2005)

have shown that the teachers’ method of mathematics teaching and his personality greatly

accounted for the students’ positive or negative attitude towards mathematics. It is

therefore necessary for mathematics teacher to have a positive attitude towards

mathematics in order to help his students do the same.

Students, Background and Achievement in Mathematics

Another importance factor for consideration is the background of the students.

According to Adolptus (2011), poor foundation of pupils in their primary school level

affects their performances in their future level. Rabiu (2002) also pointed out that child’s

inability to attain a satisfactory understanding of the basic mathematical concept taught in

primary school may make assimilation of further concept difficult.

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Teacher Factor

A major problem facing the teaching and learning of mathematics today is lack of

enough teachers to handle the subject particularly at the secondary school level. The issue

is now becoming worsened as in the number of school most especially private schools

increase every now and then. This shortage of mathematics teachers led to borrowing

teachers who are specialists in others disciplines like chemistry, physicist, economics,

engineering etc to teach mathematics. In many cases such borrowed teachers are not

professional teacher i.e. they are not trained teachers.

Hence they lack both the adequate knowledge and the teaching methods required

to teach mathematics effectively. Usually such teachers avoid topics which they find

difficult for them to teach or sometimes they brush through the surface of those topics so

as to record them as being taught. As a result of the use of this kind of teacher, students

develop negative attitude toward mathematics and eventually they become very poor in

the subject. This is why Fakuade (1973) states that “A shortage of suitably qualified

teaching staff will results in poor teaching which invariably will result in a production of

another generation of poor students.

Teachers’ Competence and Students’ Achievement in Mathematics

The National policy on Education section 56 federal government of Nigeria (1998)

recognizes the importance of teachers in the achievement of educational goal at any level

by stating that “no educational system can rise above the quality of its teacher”. The

teacher is the key factor in determining the quality of education given in schools. In

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support of this view Stinnelt (1965) states that “one of the basic truths in education is that

the quality of education depends largely upon the quality of the teacher”.

In a study: problem of mathematics teaching, Besso et al (1985) reported that most

mathematics teachers, at the secondary school level are deficient in most of the secondary

school mathematics content. This report is also supported by the study of (Ali, 1989;

Igbokure, (1997). This problem is attributed to the inadequate quality of training received

by the teachers, since most of the course he was made to receive in his tertiary education

is not the ordinary level mathematics. Definitely, this should be the reason for his

incompetence in teaching what he himself is not well groomed.

Factor Related to Over-population in Mathematics Classroom

Mathematics being a core subject both at primary and secondary school level faces

a lot of challenges, one of which is over-population. Schools today are overcrowded in

classroom in which we find about 70 to 80 students or pupils per class. Unfortunately,

mathematical learning and teaching requires giving individual attention to learners, as

one learner’s problems to comprehend may be different to another.

Other Factors

A number of other factor which present problems in the teaching and learning of

mathematics include cognitive demand for mathematics learning, the structure of

mathematics, learning facilities types of learners et cetera.

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How the Teaching and Learning of Difficult Mathematics Concepts be Solved

According to Ali (2011)

“As far as teachers’ role in promoting students in-depth learning is

concerned, first of all, it is highly relevant to consider as what teachers

need to know and be able to do in order to promote deeper understanding

of the subject- matter knowledge in mathematics. This inevitably places

demand on teachers’ knowledge of subject matter, pedagogical content

knowledge, knowledge of the learner, knowledge of the curriculum, better

understanding of new classroom management strategies….”

The best solution to the problem on ground “Teaching and learning of difficult

mathematics concept are those points noted by Ali (2011).These are also supported by the

recommendations given by Adolphus in his studies on problem of teaching and learning

of geometry in secondary schools in River state, Nigeria. A few of these points shall be

explained under this section, but let first of all discourse the term teaching and its general

principles for better understanding of the profession.

Teaching

Many definitions have been given to teaching. Curzon (1990) defines teaching as

“A system of activities intended to induce learning, comprising the deliberate and

methodical creation and control of those conditions in which learning does occur.

According to Kochhar (1985):

“Teaching is an intricate, exacting, challenging job. It is not a chemical

process. That is, teaching is more than standing before a class and

applying a few specific techniques. It is not merely presenting textbook

information and then testing the student’s ability to repeat it: there is no

magic formula for transforming knowledge from the teacher’s mind to align

the pupil’s”.

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From the two definitions one can infer that a teaching job is not like a business

whereby at the end the day your main target is profit. The main concern of a teacher is

with the development of the whole child intellectually, emotionally, socially, spiritually

and in some degree, physically. This is why most other professionals who deal with

children cannot be compared with teaching.

General Principles of Teaching (Mathematics)

For effective teaching of mathematics, a good mathematics teacher should be familiar

with some of the maxims of teaching. These, according to Stinnelt(1965), include:

1. Proceed from the known to the unknown ;

2. Proceed from the simple to the complex ;

3. Proceed from the analysis to synthesis ;

4. Proceed from whole to part;

5. Proceed from concrete to abstract;

6. Proceed from the particular to the general;

7. Proceed from empirical to rational;

8. Proceed from the actual to the representation;

9. Proceed from psychological to logical.

These above statement if are strictly followed by mathematics teachers, there will

be no reasons for difficulties in impacting the knowledge but most importantly, a good

teacher should also know pedagogical content knowledge of mathematic, attending

professional training in other to update his method of teaching, knowing the type of

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students: slow, normal and talented or gifted learner, and many more as stated by Ali

(2011).

Pedagogical Content Knowledge of Mathematics

The common belief in the society is that if a mathematics teacher knows mathematics

very well, he or she is the best person to teach mathematics. But what about- knowing

how to teach mathematics? Fernnema and Frank (1992) determined the components of

mathematics teachers’ knowledge as:

1. Knowledge of mathematics

- Content knowledge

- The nature of mathematics

- The mental organization of teacher knowledge

2. Knowledge of mathematics representation

3. Knowledge of students; knowledge of student’s cognition

4. Knowledge of teaching and decision making.

Fernnema and Frank (1992) argue that if a teacher has a conceptual understanding

of mathematics (knowledge of mathematics), this influences classroom instruction in a

positive way; therefore it is important as teachers to have mathematics content

knowledge.

They also emphasize the important of knowledge mathematical representation,

because if teachers have no knowledge of translating mathematical problems, into a form

that enables learners to relate them to what they already know, they will not learn with

understanding.

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Knowledge of students’ cognition is seen as one of the important components of

teachers’ knowledge, because, according to Fernnema and Frank (1992), learning is

based on what happens in the classroom, and thus not only what students do, but also the

learning environment is important for learning. The last component of teacher’s

knowledge is “knowledge of teaching and decision making’. Teachers” beliefs,

knowledge, judgment and thought, have an effect on the decisions they make which

influence their plans and action in the classroom (Fernnema and Frank, 1992)

According to An, Kulm, and Wu (2004) pedagogical content knowledge has three

components:

1. Knowledge of content

2. Knowledge of curriculum

3. Knowledge of teaching

Below is the network of pedagogical content knowledge according to An, Kulm and Wu,

2004; 147.]

Beliefs

Pedagogical contents

knowledge

knowledge

Building on

students’

mathematics idea

Promoting

students thinking

in mathematics

Engaging

students in

mathematics

learning

Addressing

students’

misconceptio

n

Teaching

Knowing students’

thinking

Content

Students’ learning

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In-Service Training

In other to promote effective teaching both at primary and secondary school level,

there is the need for professional training. According to Aina (2006), in-service training

enables teachers to keep abreast of development in teaching strategies, and motivational

devices towards helping students, to learn mathematics. Usman (2002) opined that the

more qualified better trained the teachers are, the greater success is recorded in the

educational objectives for a particular course. These programs include workshops and

seminars which are occasionally arranged for the subject teacher. Teachers are free to

bring their problems and difficulties for free and Frank discussion. Ezeife (1991) pointed

out that one of the objectives of in-service training or refresher courses for teachers is to

give them opportunities to up-date themselves in education generally and particularly in

their various special fields, thus improving their efficiency and professional performance.

Conclusion and Recommendation

From the previous finding on the teaching and learning of difficult mathematics

concepts the factors that are responsible for it have been considered but in details in this

study. It is therefore in the light of these that this study has suggested and discussed the

most appropriate solution to the problems which are in-service training, knowing the

principle of teaching etc.

Therefore, it is recommended that government should give greater emphasis to in-

service education for teachers in other to improve their efficiency and professional

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performances. In addition to that, government should give a project to its agencies and

professional bodies like National Mathematical Centre to provide teachers’ guidelines, in

teaching those identified concepts on difficult area in both primary and secondary school

level. State government should urge all school under them to send the list of the

mathematics teachers for better routine of staff for workshops and seminars.

Government needs to employ more qualified mathematics teachers and also need

to build more classes in both state primary and secondary school.

The Mathematics Teacher needs to take extra time in knowing the mathematics

curriculum content and should try to improve their teaching by knowing the pedagogical

content knowledge and the teachers should also motivate their students towards the

learning of the mathematics.

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