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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
18
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.
19
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