Effective Mathematics Teaching

Teaching mathematics can only be described as truly effective when it positively impacts student learning. We know that teaching practices can make a major difference in student outcomes, as well as what makes a difference in the classroom.

Research and evidence from the field of mathematics let us know, with a fair degree of certitude, how effective teachers of mathematics skillfully integrate a range of instructional approaches and resources to meet the diverse learning needs of their students. 

Effective teachers:

• Know the pedagogy that determines how their students successfully learn
• Know and understand the content and practices that students need to comprehend, as described in the Standards framework
• Know the students they teach as learners
• Challenge all students at their own level
• Encourage risk-taking
• Create purposeful learning experiences for students through the use of relevant and meaningful contexts

Effective teachers know how students learn

Effective teachers of mathematics know the pedagogy that determines how their students successfully learn. Such teachers recognize that in order for students to effectively use the mathematics they need to understand the concepts presented as well as become fluent with the skill taught. It is through the ongoing and increasingly complex application of concepts and skills that students become secure and competent in their use.

Effective teachers of mathematics are knowledgeable in the theory of learning their subject.

Dimensions of Effective Mathematics Practice

Effective Teachers:

• Know how students learn
• Know what students need to learn
• Know what their students already know
• Encourage risk-taking
• Create purposeful learning experiences
• Create a challenge

They recognize the importance of using concrete materials and visual representations to develop a deep understanding of the subject. They have a clear picture of the learning progression that best develops the knowledge base and skills of their students. They also have a broad palate of learning experiences they can use in the classroom, to meet the different learning needs of each student.

Effective teachers are able to look at student misconceptions, either in the classwork, through homework, or through assessments, and reteach the material using their understanding of the developmental nature of what becomes before or after the misconception. A deep understanding of the content enables teachers to directly address the specific misunderstandings that students may have.

Such teachers need to be continual learners. Effective pedagogy is the subject of ongoing research and development, and the way to teach and learn mathematics is never static.

Effective teachers know what students need to know

Effective teachers know and understand the content and practices of the mathematics Standards framework that students need to know. Such teachers have deep understanding of concepts and utilize multiple ways to represent and explain them. They are also fluent with the procedures and practices their students will need in order to succeed in mathematics.

The Common Core focus on career and college readiness requires that students be able to apply mathematics to complex problems in multiple contexts, both real and mathematical. As a consequence, this is also true for their teachers.

Effective teachers augment fluency procedure with:

• Deep conceptual understanding
• Knowledge of where and how to apply and use mathematics skills and concepts

Students need to use the above both in school and in real-world contexts.

Effective teachers know their students as learners

Knowing a student as a mathematics learner is complex. An effective mathematics teacher quickly builds a picture of their students by progressively providing opportunities to demonstrate what he/she is learning. This way, teachers update and deepen their understanding of individual students.

The effective teacher continuously uses this growing knowledge of students as learners to inform their instruction so they can better meet students’ needs.

Assessment in mathematics is primarily formative. It involves collecting information from a range of sources, in a variety of ways. This includes information on students’ strategies, understandings, attitudes, and prior knowledge and skills. Assessing a student involves making informed judgments about what the student knows. Hence, effective teachers not only monitor the performance of a student but also their ability to show their understanding of the content that has been taught. 

Effective teachers:

• Integrate assessment into instructional practice
• Acknowledge students’ prior learning and help them make connections between what they already know and what they are currently learning
• Gather information from a range of formal and informal sources using a variety of means, in particular, written and verbal, and analyze the information presented
• Use ongoing assessments to identify the learning needs of each student. This allows them to teach proactively, assisting students to meet articulated goals

Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both the curriculum and instruction should begin with problems, dilemmas, and questions for students.

– Hibert et al., 1996

… our best evidence… is that what happens in classrooms through quality teaching and through the quality of the learning environment generated by the teacher and the students, is the key variable in explaining up to 59%, or even more, of the variance in student scores.

– Ministry of Education, 2003

Effective teachers create a challenge

Each student learns best within their ‘Zone of Proximal Development.’ The effective teacher:

• Is able to identify and keep track of a student’s ‘Zone’ through ongoing formative assessment
• Designs instruction that enables each student to learn within their ‘Zone’
• Provides each student with challenges that meet their own level through the careful use of investigative tasks

When referring to “challenges,” it is meant that a student will need to experience some degree of struggle to achieve a learning goal. An effective teacher will challenge every student to consistently operate at the upper end of their ‘Zone of Proximal Development.’

Many strategies introduce challenges into a lesson, including the use of a rigorous open task, the use of questioning strategies (such as turn and talk) that involve all students, asking exploratory and generative questions, and consistently requiring students to pose, reflect on, and justify arguments.

Along with providing a challenge is the need to provide differing degrees of support for students. The greater the challenge, the greater the need for teacher support in a gradual release of responsibility from the teacher to the student.

Aims of teaching mathematics are to be framed in light of the educational values of the subject. Value is the spring-board of aim. We know that mathematics has wide applications in our daily life. It has great cultural and disciplinary values. Thus we may mention the aims of teaching mathematics as under:

Aims

1. To enable the students to solve mathematical problems of daily life. We have to select the content and methods of teaching so that the students are able to make use of their learning of mathematics in daily life.

2. To enable the students to understand the contribution of mathematics to the development of culture and civilization.

3. To develop thinking and reasoning power of the students.

4. To prepare a sound foundation needed for various vocations. Mathematics is needed in various professions such as those of engineers, bankers, scientists, accountants, statisticians etc.

5. To prepare the child for further learning in mathematics and the related fields. School mathematics should also aim at preparing him for higher learning in mathematics.

6. To develop in the child desirable habits and attitudes like habit of hard work, self-reliance, concentration and discovery.

7. To give the child an insight into the relationship of different topics and branches of the subject.

8. To enable the child to understand popular literature. He should be so prepared that he finds no handicap in understanding mathematical terms and concepts used in various journals, magazines, newspapers etc.

9. To teach the child the art of economic and creative living.

10. To develop in the child rational and scientific attitude towards life.

Objectives

Aims of teaching mathematics are genially scoped whereas objectives of the subject are specific goals leading ultimately to the general aims of the subject. The objectives of teaching mathematics in school can be described as under:

A. Knowledge Objectives

Through mathematics, a pupil acquires the knowledge of the following:

(i) He learns mathematical language, for example, mathematical symbols, formulae figures, diagrams, definitions etc.

(ii) He understands and uses mathematical concepts like concept o area, volume, number, direction etc.

(iii) He learns the fundamental mathematical ideas, processes, rules and relationships.

(iv) He understands the historical background of various topics a contribution of mathematicians.

(v) He understands the significance and use of the units of measurement.

B. Skill Objectives

Mathematics develops the following skills:

(i) The child learns to express thoughts clearly and accurately.

(ii) He learns to perform calculations orally.

(iii) He develops the ability to organize and interpret the given data

(iv) He learns to reach accurate conclusions by accurate and logic reasoning.

(v) He learns to analyse problems and discover fundamental relationships.

(vi) He develops speed and accuracy in solving problems.

(vii) He develops the skill to draw accurate geometrical figures,

(viii) He develops the ability to use mathematical apparatuses a tools skillfully.

C. Appreciation Objectives

The child learns to appreciate:

(i) The contribution of mathematics to the development of various subjects and occupations.

(ii) The role played by mathematics in modern life.

(iii) The mathematical type of thought serves as a model for scientific thinking in other fields.

(iv) The rigour and power of mathematical processes and accrue of results.

(v) The cultural value of mathematics.

(vi) The value of mathematics as a leisure time activity.

D. Attitude Objectives

Mathematics helps in the development of the following attitudes:

(i) The child develops the attitude of systematically pursuing a task to completion.

(ii) He develops a heuristic attitude. He tries to make independent discoveries. He develops the habit of logical reasoning.

(iii) He is brief and precise in expressing statements and results,

(iv)He develops the habit of verification.

(v) He develops power concentration and independent thinking. 

(vi) He develops the habit of self-reliance.

We have discussed the aims and objectives of teaching mathematics in general. The teacher should carefully choose the objectives regarding a particular topic. The nature of students will also be kept in view.

AIMS OF TEACHING MATHEMATICS(In Detail)

1.      Practical Aim (Utilitarian Aim)

One cannot do without the use of fundamental process of the subject mathematics in daily life. Any person ignorant of mathematics will be at the mercy of others and will be easily cheated. A person from labour class, a businessman, an industrialist, a banker to the highest class of the society utilizes the knowledge of mathematics in one form or the other. Whoever earns and spends uses mathematics and there cannot be anybody who lives without earning and spending.

Counting, subtraction, multiplication, division, weighing, selling, buying, etc., will have got an immense practical value in life. The knowledge and skill in these processes can be provided in an effective and systematic manner only by teaching mathematics in schools. In many occupations like accountancy, banking, tailoring, carpentry, taxation, insurance, etc., which fulfills the needs of man can be carried out by the use of mathematics. These agencies depend on mathematics for their successful functioning. It has become the basis for the world’s entire business and commercial system. 

Ignorance of mathematics in the masses is a formidable obstacle in the way of country’s progress. Individual resources add up to form national resources. An individual who is ignorant of calculation is often ruined himself and also causes for the national loss by wasting his time, energy, and money. There are family budget, school budget, factory budget, national budget, etc., which owe the fundamentals of mathematics. Natural phenomena like rising and setting of the moon and the suns change of seasons, speed of rotation of planets, etc., need time specification.

            Mathematics will continue to occupy a prominent place in man’s life. In all activities of life like arranging a party, admitting a child to school, celebrating a marriage, purchasing or selling a property, etc, mathematical considerations are uppermost in human mind. In order to create a system in life we have to fix timings, prices, rates, percentages, exchanges, commissions, discounts, profit and loss, areas, volumes, etc. In the absence of these fixations life in the present complex society revert back into confusion and chaos. 

The number imparts the system through our life. In this complex world passing through scientific and technological age the practical value of mathematics is going to be increasingly felt and recognized.

The following are the practical aims of teaching mathematics.

1. To enable the students to have clear ideas about the number concept.

2. To give the individual an understanding of ideas and operations in number and quantity needed in daily life.

3. To enable the individual to have a clear comprehension of the way the number is applied to all measures but most particularly to those frequently used concepts such as length, volume, area, weight, temperature, speed etc.

4. To enable the individual to become proficient in the four fundamental operations of addition, subtraction, multiplication and divisions.

5. To provide the basis of mathematical skills and processes which will be needed for vocational purposes.

6. To enable the learner to acquire and develop mathematical skills and attitude to meet the demands of (i) daily life (ii) future mathematical work and (iii) work in the related fields of knowledge.

7. To enable the students to make appropriate approximations.

8. To enable the learner to understand the concept of ratio and scale drawing, read and interpret graphs, diagrams and tables.

9. To enable the individual to apply his mathematics to a wide range of problems that occur in daily life.

2.      Disciplinary Aim

            The principal value of mathematics arises from the fact that it exercises the reasoning power more and climbs from memory, less from any other school subjects. It disciplines the mind and develops reasoning power. Locke is of the opinion that ‘mathematics is a way to settle in the mind a habit of reasoning. A person who had studied mathematics is capable of using his power of reasoning in an independent way’. Its truths are definite and exact. The learner has to argue the correctness or incorrectness of a statement. The reasoning in the mathematical world is of special kind processing characteristics that make it especially suited for training the minds of the pupils. 

It can be studied under following heads –

1.            Characteristics of Simplicity: In this subject teaching and learning advances by degrees from simple to complex. It teaches that definite facts are always expressed in a simple language and definite facts are always easily understandable.

2.            Characteristics of Accuracy: Accurate reasoning, thinking and judgment are essential for the study of mathematics. The pupils learn the value and appreciation of accuracy and adopt it as a principle of life. It is in the nature of the subject that it cannot be learned through the vagueness of thoughts and arguments. Accuracy, exactness, and precision compose the beauty of mathematics. He learns to influence and command others by accuracy.

3.            Certainly of Results: There is no possibility of a difference of opinion between the teacher and the pupil. It is possible for the learner to remove his difficulties by self-effort and to be sure of the removal. He develops faith in self-effort, which is the secret of success in life.

4.            Originality of Thinking: Most work in mathematics demands original thinking. Reproduction and cramming of ideas of others are not very much appreciated. The pupil can safely depend on the memory in other subjects but in the mathematics without original thinking and intelligent reasoning there cannot be satisfactory progress. This practice in originality enables the situation with confidence in his future life.

5.              The similarity to Reasoning of Life: Clear and exact thinking is as important in daily life as in the mathematical study. Before starting with the solution of a problem the pupil has to grasp the whole meaning similarly in daily life while undertaking a task, one must have a firm grip on the situation. This habit of thinking will get transferred to the problem of daily life.

6.   Verification of Results: This gives a sense of achievement, confidence, and pleasure. This verification of results also likely to inculcate the habit of self-criticism and self-evaluation.

7. Concentration of Mind: Every problem of education and life demands concentration.

Mathematics cannot be learned without whole-hearted concentration. Hamilton says, ‘the study of mathematics cures the vice of mental distraction and cultivates the habit of continuous attention’.

In every field, new ideas and new methods are being introduced rapidly. In this, ever-advancing society the important thinking is not only to learn facts but also to know how to learn facts. This is the discipline of the mind. Mathematics has a vast scope of application. Mathematics has the ability to apply knowledge to new situations and acquire the power to think effectively so that intellectual power of the learner is strengthened.

     The teaching of mathematics intends to realize the following disciplinary aims,

1. To provide opportunities that enable the learners to exercise and discipline mental faculties.

2. To help the learner in the intelligent use of reasoning power.

3. To develop constructive imagination and inventive faculties.

4. To develop the character through systematic and orderly habits.

5. To help the learner to be original and creative in thinking.

6. To help the individual to become self-reliant and independent.

3.      Cultural Aim

            Cultural aim helps the pupils to grow in the cultured situation. The greatness of Indian culture is once reflected through the glory of Indian mathematics of olden days. Similarly having come to know the progress of Egyptians and Greeks in mathematics, one can be aware of their progress in culture and civilization. Mathematics does not only acquaint us with the culture and civilization but it also helps in its preservation, promotion, and transmission to the coming generation. Further after knowing what our ancestors have done we as the students of mathematics bring new idea in the body of mathematical knowledge. And thus increase our cultural heritage. 

Culture is not possible unless there is proper development of the power of reasoning and judgment. The aim of teaching of mathematics is to develop cultured citizen who can discharge their obligations to society effectively and successfully. Our entire present civilization depends on intellectual penetration and utilization of nature has its real foundation in mathematical science. As the education commission report (1964-66) was conscious of this need when it wrote ‘one of the outstanding characteristics of scientific culture is the quantification of mathematics. Therefore it assures a prominent position in modern education.

Proper foundation in the knowledge of the subject should be laid at the school’. Mathematics played a major role in bringing man to the advanced stage of development. The prosperity and the cultural advancement of man depend on the advancement in mathematics. That is why Hogben says, ‘mathematics is the mirror of civilization’. Different laws of science and scientific instruments are based on the exact mathematical concept. For example, astronomy and physics are the most exact science and their exactness is the outcome of the usefulness of mathematics.

Mathematics is the backbone of our civilization. What we have in our modern culture and civilization owes its depth to science and technology, which in turn, depends upon the progress in mathematics. Moreover in various cultural arts like poetry, drawing, painting, music, architecture and design making, mathematics is playing a vital role and therefore it can be safely said that mathematics is intimately linked with the culture and civilization.

The cultural aims can be summarized as follows,

1. To enable the student to appreciate the part played by mathematics in the culture of the past and that it continues to play in the present world.

2. To enable the student to appreciate the role played by mathematics in preserving and transmitting our cultural traditions.

3. To enable him to appreciate various cultural arts like drawing, design making, painting, poetry, music, sculpture and architecture.

4. To provide through mathematics ideas, aesthetic and intellectual enjoyment and satisfaction and to give an opportunity for creative expression.

5. To help the student explore creative fields such as art and architecture.

6. To make the learner aware of the strength and virtues of the culture he has inherited.

7. To develop in the individual an aesthetic awareness of mathematical shapes and patterns in nature as well as the products of our civilization.

4.   Social Aim:

The important social aims of teaching mathematics are as under,

1. To develop in the individual an awareness of the mathematical principles and operations which will enable the individual to understand and participate in the general, social and economic life of his community.

2. To enable the student to understand how the methods of mathematics such as scientific, intuitive, deductive and inventive are used to investigate, interpret and to make decision in human affairs.

3. To help the pupil acquire social and moral values to lead a fruitful life in the society.

4. To help the pupil in the formation of social laws and social order needed for social harmony.

5. To provide the pupils scientific and technological knowledge necessary for adjusting to the rapidly changing society and social life.

6. To help the learner appreciate how mathematics contributes to his understanding of natural phenomena.

7. To help the pupil interpret social and economic phenomena.

Objectives of Teaching Mathematics – National Policy of Education (1986)

 At the end of high school stage, a pupil should be able to –

· Acquire knowledge and understanding of the terms, concepts, principles, processes, symbols and mastery of computational and other fundamental processes that are required daily like and for higher learning in mathematics.

· Develop skills of drawing, measuring, estimating, and demonstrating.

· Apply mathematical knowledge and skills to solve problems that occur in daily life as well as problems related to higher learning in mathematics or allied areas.

· Develop the ability to think, reason, analyze, and articulate logically.

· Appreciate the power and beauty of mathematics.

· Show an interest in mathematics by participating in mathematical competitions and engaging in its learning, etc.

· Develop reverence and respect towards great mathematicians, particularly towards great Indian mathematicians for their contributions to the field of mathematical knowledge.

· Develop necessary skills to work with modern technological devices such as calculations, computers, etc.

Objectives of Teaching Mathematics – New Curriculum Document (2000)

The learners-

· Consolidate the mathematical knowledge and skills acquired at the upper primary stage.

· Acquire knowledge and understanding of the terms, symbols, concepts, principles, process, proofs, etc.

· Develop a mastery of basic algebraic skills.

· Develop drawing skills.

· Apply mathematical knowledge and skills to solve real mathematical problems by developing abilities to analyze, to see interrelationship involved, to think and reason.

· Develop the ability to articulate logically.

· Develop awareness of the need for national unity, national integration, protection of the environment, observance of small family, norms, removal of social barriers, and elimination of sex biases.

· Develop necessary skills to work with modern technological devices such as calculators, computers, etc.

· Develop an interest in mathematics and participate in mathematical competitions and other mathematical club activities in the school.

· Develop an appreciation for mathematics as a problem-solving tool in various fields for its beautiful structures and patterns, etc.

· Develop reverence and respect towards great mathematicians, particularly towards the Indian mathematicians for their contributions to the field of mathematics.

METHODS OF TEACHING MATHEMATICS

Mathematics has always been the most important subject in the school curriculum. Traditional mathematics teaching has been found to be unsatisfactory. During recent years the demand has grown to make mathematics teaching more imaginative, creative, and interesting for pupils. Clearly, the demands made on the mathematics teacher are almost unlimited. The teacher must have a specialized understanding of the foundations of mathematical thinking and learning. He/She should also possess skills to put together the whole structure of mathematics in the minds of his/her students. He, like a master technician, should decide what kind of learning is worth what; realize and make use of motivation and individual differences in learning. He//She should be able to translate his1 her training into practice. 

Finally, he/she should plan or design the instruction so that an individualized discovery-oriented (or problem-solving) learning is fostered. A few of the current trends in the methods and media used in mathematics instruction are mentioned here. These include the basic features of more recent ideas which are the gift of educationists and psychologists. It is expected that teachers would try to fit them into their practical scheme of teaching. 

1.Lecture Method

In this method, knowledge is delivered through a speech. This is the oldest and most important teaching method because it is always remained a part of all other instructional methodologies. In this method, a teacher takes part as an active participant and students are at the receiving end most of the time. 

That is why; it is a teacher centred approach. This is also referred to as direct instruction, training model, active teaching and explicit instruction. Lecture method is not only used for teaching theoretical concepts but it is also helpful for giving training of complex skills and procedures.

A. Merits and Demerits of Lecture Method

Lecture method has also some merits which are as follows.

i. This is the most convenient and easy method. 

ii. This is the fastest way to deliver knowledge so when the syllabus is so heavy then it becomes necessary.

iii. When strength of a class is very high then it becomes more important. 

iv. This is so economical because there is no equipment involved in it and only one teacher can teach so many students.

v. This is very helpful to introduce new concepts. 

vi. This can be used to raise the interest level of the students while applying any other teaching method

This method has also some demerits which are listed below-

viii. This is a teacher centred approach so students cannot play an active role. 

ix. This method does not develop reasoning and thinking ability in the students. 

x. Sometimes lectures become boring because there is no activity involved in it. 

xi. In this method, teacher-student relationship is not developed in proper way. 

xii. This method is relatively more useful in higher classes. 

xiii. It becomes essential to enhance writing and communication skills by the teachers. 

B. Application of Lecture Method in Mathematics at Secondary Level

As no practical work is involved in this method, so it can only be used to clarify the basic concepts of each unit given in the textbooks of mathematics. It is applicable to teach all branches of mathematics including sets, logarithms, algebra, matrices, statistics,

geometry and trigonometry.

 Mathematical problems related to these branches cannot be solved by this method but the procedures and methods to solve them can be explained in a very good manner. The historical perspective of these branches and their relevance to the real life can also be described by this method.

2. Inductive Method

This method is also called scientific method in which we proceed from known to unknown, from specific to general and from example to rule or formula. In this method based on induction, students are presented some similar examples or problems related to one particular domain. Then students try to establish a formula, rule, law or principal by observing them. If a generalised result is true for those similar examples or problems then it would also be true for all other such kind of examples (Sidhu, 1995).

A. Merits and Demerits of Inductive Method

This method has also some merits and demerits. Its merits are as follows.

i. This method is useful to introduce a new mathematical concept along with a formula or rule.

ii. Students who like the inductive approach can infer the more complicated rules or  formulas (Felder, 1993).

iii. This is a student centred approach because students play an active role in it. 

iv. As the students may establish laws and principles by themselves so it gives them  confidence.

v. This method helps to motivate the students to think logically and make the learning environment more interesting.

vi. This is based on reasoning and experimentation. 

vii. This is quite suitable for primary and secondary level classes. 

viii. Students easily remember the laws or principles which they prove by themselves. 

This method has some demerits as well. 

i. It is quite time consuming and laborious as well. 

ii. To establish a law or principle is not the complete process of learning. Students have to practise a lot to understand the concept fully.

iii. Sometimes a formula or rule proved with the help of some similar examples does  not applicable in other similar cases.

iv. Only experienced teachers can use this method in a right way. 

v. This method does not help in developing problem solving ability in the students. 

B. Application of Inductive Method in Mathematics at Secondary Level

Inductive method is used to establish laws, principals, formulas and methods instead of solving mathematical problems. Therefore it can be used in all branches of mathematics but establishing laws or formulas at the secondary level is only involved in algebra, matrices and to some extent geometry.

3. Deductive Method

This method is totally different from inductive method. In this method, we proceed from general to specific and from a rule to an example. Already constructed formulas, rules, methods or principles are taught to the students and they apply them to solve the

problems (Sidhu, 1995). In this teaching approach, we can also prove a theorem with the help of undefined terms, defined terms, axioms and postulates. Then with the help of that theorem along with different rules and principles, we can derive other theorems as well

(Singh, 2007).

A. Merits and Demerits of Deductive Method

 There are some merits and demerits of this method as well (Sekhar, 2006). Some merits are listed below.

i. This method is very easy and short. 

ii. To remember a formula or rule is not very difficult so this method is blessing for those students who cannot remember complicated procedures (Brigham & Matins,1999).

iii. Teachers can complete the syllabus easily by this method. 

iv. This method helps to enhance the computational ability of the students. 

v. It is helpful to teach those concepts in which derivation of rules or methods is not involved.

vi. With the help of this method, we can prove different theorems using already defined formulas or principles.

 This method has the following demerits.

i. It becomes very difficult for students when they have to remember so many rules and formulas.

ii. This method does not help in improving reasoning ability in the students. 

iii. It is not effective at lower level classes. 

iv. This method is not constructivist. If a student forgets a rule or principle then he or she cannot reconstruct that easily (Sidhu, 1995).

v. This method does not encourage discovery learning. 

vi. It cannot make students creative. 

vii. Students may be doubtful about the reason of using one particular formula. 

B. Application of Deductive Method in Mathematics at Secondary Level

 Deductive method is the highly used method in mathematics. It is used to solve those problems in which complicated procedures are not involved and they can be solved by applying different kinds of already established laws, methods, formulas and principles

directly. Such kind of problems can be found in all units of syllabus of mathematics at secondary level including sets, logarithms, algebra, matrices, variation, statistics, geometry and trigonometry.

4. Heuristic Method

The word heuristic was drawn from a Greek word “heurisco” which means “I find out”. Heuristic method is based on child’s psychology who always wants to discover something by himself or herself. That is why it is also known as discovery method. Sometimes a teacher only focuses on delivering lectures through speech in which students do not actively participate and get bored most of the time. But in the heuristic method, students are encouraged to reach the solution by

constructing the knowledge themselves.

 Teacher only facilitates them by raising relevant questions. That is why it is also called inquiry method. As students discover the solution under the guidance of a teacher so it is also known as guided discovery method or programmed instruction. 

So many researches have proved that heuristic or discovery method is more effective in teaching mathematics than expository approach.

A. Merits and Demerits of Heuristic Method

This method has merits and demerits as well . Its merits

are as follows.

i. It is a student centred approach. 

ii. It gives confidence to the students because they discover the solution by  themselves.

iii. It makes students creative. 

iv. It develops reasoning and thinking abilities in the students. 

v. It clears concepts in a better way. 

vi. Continuously inquiring the students keeps them active and they do not get bored. 

 Demerits of this method are given below. 

i. This method is quite time consuming. 

ii. It is essential for all teachers to be properly skilled with this method otherwise it is very difficult for them to apply this in the classroom.

iii. If any student has less aptitude towards discovery then it becomes very difficult for him or her to learn something through this method.

iv. It is only applicable if strength of a class is low.

v. If a teacher fails to give proper guidance to the students then they may get  discouraged.

vi. This method is not suitable for teaching all kinds of mathematical problems. 

vii. Sometimes a teacher fails to ask proper questions during the discovery process so that distracts students.

viii. With the help of this method, lengthy syllabus cannot be finished in time. 

B. Application of Heuristic Method in Mathematics at Secondary Level

Heuristic method can be used to teach all branches of mathematics. It is helpful when students are not master to solve problems related to one particular concept and they need guidance. When students get master of different methods and formulas then they are

encouraged for deductive or problem solving methods to solve the same problems.

V. Analytic Method

 In this method, we analyse the problem first by breaking up the problem in small segments and then move towards solution. It is also called descriptive method. It leads us from the unknown part of the problem to something already known or given in the

problem statement. This method emphasises on why we are applying different kinds of operations and what is the relationship between the required solution and other portions of the problem.

A. Merits and Demerits of Analytic Method.

Analytic method has also some merits and demerits (Sidhu, 1995; Sekhar, 2006).

Merits of this method are as follows.

i. This is a pure logical method so there is always less chance of doubts. 

ii. Discovering the solution is an essential part of this method so it enhances logical  thinking and reasoning ability of the students (Agarwal, 1992).

iii. Students always play an active role in this method. 

iv. Students do not need to memorise any set procedure to solve a problem. 

v. It encourages scientific attitude. 

 This method has the following demerits.

i. This method is quite lengthy and time consuming (Agarwal, 1992). 

ii. This is not suitable for all kinds of problems. 

iii. Only skilled teachers can apply this method. 

iv. This is not suitable if the syllabus is so lengthy. 

B. Application of Analytic Method in Mathematics at Secondary Level

 Because of discovery approach, only such kind of problems can be taught with the help of this method in which we have to prove something. At secondary level, such problems can only be found in the units of algebra, geometry, ratio and proportion (variation).

VI. Synthetic Method

 This method is completely opposite to the analytic method as we proceed from the given or known elements in the problems to the desired solution or unknown. In this method, we synthesise or put together separate elements or small portions given in the problems to draw a series of conclusions until the unknown or desired result is found (Sidhu, 1995). 

This method is quite simple and led by analytic method. Process of analysis in analytic method clears the basics of any concept. On the other hand, synthetic method is based on already learnt concepts. 

Therefore it is quite necessary to go through the analytic method to become master of specific mathematical concepts then synthetic method can be used to solve the problems more quickly. In this method, students are not  bound to give reason for each and every step while solving a mathematical problem.

That is why it cannot be preferred alone to derive mathematical proofs (Butler & Wren, 1965).

A. Merits and Demerits of Synthetic Method

 This method has also some merits  as given below.

i. Synthetic method is short and brief. 

ii. It is quick because of deductive reasoning. 

iii. It sharpens the memory of students. 

iv. Teachers can finish the lengthy course in time through it. 

v. It provides opportunity to the students to practise mathematical formulas or  procedures.

 Demerits of this method are as follows.

i. It is not student centred. 

ii. It does not develop reasoning ability in the students. 

iii. Students have to remember so many steps without reasoning. 

iv. It does not employ heuristic approach. 

v. If a student forgets any mathematical proof then it is very difficult to recall it step by step.

vi. It does not clarify the concepts completely. 

vii. It is neither psychological nor scientific in nature. 

B. Application of Synthetic Method in Mathematics at Secondary Level

 Just like analytic method, this method can be used for such problems in which we have to prove something. It is also useful to find out something unknown with the help of given conditions in the problem statement. These problems can be found in the units of algebra, ratio and proportion (variation) and geometry at secondary level.

vii. Problem Solving Method

 Instructional methodologies should improve reasoning ability in the students. In this way, they become capable to find out the solutions of different kinds of problems not only during the studies but in their daily routine matters as well. Every child has the curiosity to explore the things and this psychology of the children can be utilised in a

better way through problem solving method. It is the most important instructional methodology for mathematics, the most famous psychologists, also gave the top priority to this method.

 In this method, students are given such problems which cannot be solved easily or their solutions are not obvious.

 A student tries to reach the goals or solutions through the set of events or procedures. Gagné (1970) calls these events or procedures as lower order capabilities in which formulas, rules and concepts are used from which a student is already familiar. According to him, what the student learns is called a higher order principle which is the result of lower order capabilities.

A. Merits and Demerits of Problem Solving Method

 Problem solving method has also some merits and demerits. There are the following merits of this method (Taplin, 1995; Singh, 2007).

i. This method is scientific in nature. 

ii. It is student centred. 

iii. It is helpful to enhance the reasoning ability of the students. 

iv. Students are provided opportunity to apply their previous knowledge through  problem solving.

v. Students learn how to face totally new situation by solving different kinds of  questions.

vi. Teacher can assess the abilities of his or her students easily. 

vii. This method improves logical thinking in the students which leads towards  creativity.

 There are some demerits of this method as well (Sidhu, 1995; Singh, 2007).

i. This method is quite time consuming. 

ii. This is usually not recommended for lower classes. 

iii. Textbooks do not provide enough help to apply this method because such books are usually written in a traditional way.

iv. Logical thinking is involved in this method therefore physical kind of activities are totally neglected.

B. Application of Problem Solving Method in Mathematics at Secondary Level

 This method is used to solve those complicated problems which cannot be solved with the help of single law or formula. Usually word problems are solved with it. At secondary level, such kind of problems can be found in the units of algebra, trigonometry, ratio and proportion (variation).

viii. Laboratory Method

Mathematics is different from the subjects involving readings thus practical work  is its major part. Laboratory method has the capacity to deal with practical work in mathematics. It is a method of “learning by doing”. That is why, different kinds of tools and equipment’s are used in it to perform practical work which includes drawing of different shapes, taking measurements of geometrical figures and making of charts and graphs. Students go through different experiments in laboratory or classroom and learn

by observing and calculating themselves. During this process, they get opportunity to draw conclusions and generalise different laws and formulas. Therefore, this method can be said an extended form of inductive method (Sidhu, 1995).

 The role of a teacher in this method is to supervise the whole process and give proper instructions to the students at each step. He or she should keep some points in mind to make this method successful (Singh, 2007).

i. Necessary equipments related to the laboratory work should be arranged in  advance.

ii. Teacher should continuously observe the practical work of every student and guide him or her accordingly.

iii. Every student should be encouraged throughout the practical work. 

iv. All necessary concepts should be cleared before starting experimental work. 

 If number of the students is high and required equipment is not enough then students can be divided into small groups.

A. Merits and Demerits of Laboratory Method

 This method has also some merits and demerits (Sekhar, 2006). 

Merits of this method are as follows.

i. It is student centred method. 

ii. Students play an active role so they do not get bored. 

iii. It is based on discovery approach. 

iv. Knowledge gained through practical work is long lasting. 

v. As students establish laws and formulas by themselves so they gain confidence. 

vi. Practical utilisation of mathematics is realised by the students. 

vii. When students work in the groups then their learning becomes fast because of sharing information and ideas.

viii. The teacher-student relationship gets strengthened. 

 Laboratory method has the following demerits.

i. It is very lengthy process. 

ii. It is restricted to those topics only in which practical work is involved. 

iii. It is very difficult for so many schools to spend a lot of money on  tools and equipments involved in this method.

iv. Teachers have to practise a lot before applying this method in the classroom or  laboratory.

v. Students cannot practise this method to establish laws or principles independently. 

vi. It is more effective in lower level classes as compare to secondary level. 

B. Application of Laboratory Method in Mathematics at Secondary Level

 This method is mostly used for practical geometry. At the secondary level, it can also be used to establish or verify the laws and theorems in sets and trigonometry. These laws and theorems are usually proved through inductive method but laboratory method can be used at alternative basis to create interest among the students.

iX. Project Method

 This method is also based on the philosophy of “learning by doing”. It was devised  by famous educationist Prof. Dr. William H. Kilpatrick who defined this method as “whole-hearted purposeful activity” (Kilpatrick, 1918). In this method, students are 

engaged in such kind of projects in which they get opportunity to apply their theoretical knowledge and learn practically. In these projects, students work in natural environment outside or within the boundary of school. During this process, they face different

mathematical kind of problems in real life and then try to solve them with previously gained knowledge. Projects may be allocated at individual level but usually students are divided in the small groups to accomplish them (Sidhu, 1995).

 Project method provides cooperative learning in which not only students share the ideas and knowledge but they also get motivated to complete the tasks as soon as possible. Famous educationist John Dewey (1916) emphasised on social interaction of the

learners for the first time then Herbert Thelen (1954, 1960) also gave importance to cooperative learning in small groups.

A. Merits and Demerits of Project Method

 There are some merits and demerits of this method (Sekhar, 2006). 

Its merits are as follows.

i. It is totally student centred method. 

ii. It helps students to correlate the mathematical knowledge with real life problems. 

iii. It is a social activity that helps to promote friendly environment among students. 

iv. Students share their ideas and experiences with each other. 

v. It gives confidence to the students. 

vi. Students learn so many other things during projects in real life scenarios. 

vii. Students remain active and enjoy throughout the project. 

 Project method has the following demerits.

i. It is quite time consuming. 

ii. It is costly because so many equipments are involved in it. 

iii. Because of excessive practical work, students cannot give much attention to  practise the mathematical operations.

iv. Usually textbooks are not designed according to this method. 

v. It is very difficult to complete the syllabus in time with the help of this method  especially when strength of a class is very high.

B. Application of Project Method in Mathematics at Secondary Level

 This method is not used to teach one particular concept of mathematics. When students get master of different areas of mathematics like algebra, geometry or trigonometry with the help of other teaching methods then project method provides opportunity to them to apply their already learnt knowledge in real life scenarios.

TECHNIQUES IN TEACHING MATHEMATICS 

Whatever method a mathematics teacher may adopt for teaching, the daily mathematics lesson tends to follow a fairly standard pattern. Each lesson builds upon the lessons 3 4 previously taught. Hence, reinforcement by adequate practice or drill of previously learnt skills becomes an important task. 

Similarly, for gaining mastery of the new skills learnt, Approaches and proper assignment some-work need to be planned. As a rule, oral recitation and written Techniques of Teaching work both form vital components of any lesson. Some of the techniques in teaching mathematics are discussed here. 

1.Drill and Practice 

Drill is one of the most essential ways (or methods) of learning in mathematics. The controlling purpose of all teaching activity is to reduce necessary learning to habit. Gaining mastery requires acquisition of habits, hence drill practice plays an important role in acquiring mastery. By and large, practice lessons are of three types. 

The first category of lessons for mastery is of basic subject matter, e.g., multiplication tables, addition combinations, fractional equivalents of decimals and percentages, factorization, construction in geometry, etc. These include subject matter which must be thoroughly mastered so that speed and accuracy is ensured on which future learning can be based. The second category includes lessons for the mastery of procedures. 

In mathematics one has to adhere to a systematic arrangement of steps, follow correct algorithms to scrutinize and check the correctness of each step, label appropriately parts in a diagram, sort out data, translate problems into symbolic form, practice short cuts, etc. 

The third category consists of lessons which strive to develop the power of thinking and reasoning, and increase the concentration and interest of the learner. Such lessons include quizzes, puzzles and historical material which does not form part of a regular lesson. 

Although, a certain amount of formal drill is inevitable, preference should be given to functional or meaningful drill. Meaningful drill implies prior understanding of content and its appropriate application. This drill is purposeful and is determined by need as well as by use. 

An effective drill lesson should be organized keeping in view the following considerations: 

1. Drill should follow learning and understanding of basics. It should not encourage rote memorization without understanding the subject matter. 

2. Drill should be varied. Some routine procedures make learning monotonous and uninteresting. 

3. Drill should be individualized and rewarding to each pupil. Each child should see its purpose and utility. 

4. Drill periods should be short and the learner’s achievement should be frequently tested. 

5. Drill should not be planned merely to keep pupils “busy”. It should be based upon thought-provoking situations to avoid the repetition of any process mechanically. 

6. Drill may also provide diagnostic information about pupils. 

2 Oral and Written Work 

Typical mathematical situations which people meet outside the school demand oral computation or involve a visual impression not completely, describe in words. A good deal of socio-economic information requires a quick response to a quantitative data. It is, therefore, desirable to develop a high rate of performance consistent with accuracy. 

Oral work helps each child work at the optimum rate which insures maximum accuracy for him. In any lesson, both oral and written (especially when modern practice or work books are used), work should mingle freely. 

Oral work provides a rapid drill designed to habituate a fundamental process, a mode of thinking, or a set of facts. It helps in completing more work in any given period. Generally, material for oral work should not be read from a textbook, Spontaneity in grasping the data, and organization of thought in a limited time, are important aspects to test the pupil’s responses. 

However, when a teacher finds it necessary to inspect check work done by each child or Approaches to Teaching to give children practice in independent work, written work becomes a necessity of Mathematics.

 Throughout wrong work accuracy in computation, legibility of figures and symbols, speed consisted-with accuracy, proper algorithm, i.e., logical and sequential arrangement of steps in the solution, neatness of work and correctness of results should be kept in mind. Written work can also be kept as a collective record which may help in assessing the pupil’s progress over a period. 

3 Play Way Technique 

The most recent technique in the teaching of mathematics in the Play Way technique or teaching through games. A game is a planned activity which the students undertake with the guidance of the teacher. Although only some concepts can be taught through games, the most use of games like the quiz, the puzzle, the guessing game, etc., is the drill or oral practice various mathematical concepts. For example, here we demonstrate a quiz for class VIII. 

Divide the class into two groups and ask questions one by one from each team on, say, factorization. Then, ask a question on the factor theorem (which has not been taught). If the students are unable to answer, they would be motivated to learn the new concept and continue with the game. Thus, Play Way techniques can be used for concept formation and motivation too. 

4 Assignment and Home Work 

Every mathematics teacher assigns home work. The usual argument in favour of home work is that it provides additional time for practice and develops the habit of self-study and self-reliance. It is assumed that classroom time is only for teaching and not for practice. 

However, home work presents a number of problems – the study is unsupervised, it encourages the use of cheap notes and guides, etc. Often the unsupervised home work develops undesirable study habits. Home work, which is an extension of or supplementary to class work and which does not take away much of the free time of the child, is considered legitimate. Any home work assigned to pupils should be corrected and kept as a cumulative record in notebooks rather than on loose sheets of paper. In recent years the concept of home work is being replaced by differentiated assignments which are adjusted to the individual progress of each child and which encourage each one to do higher own learning. 

Planning the assignment represents one of the most important phases of teaching. It is that part of instructional activity which is devoted to

 (i) organizing a task to be done, and 

(ii) fitting to the task an appropriate procedure for accomplishment. It assumes that the most effective learning is the product of self-imposed pupil activity. 

A few principles governing a good assignment are listed here: 

1. The assignment should be clear and definite. It should be brief but fairly explanatory to enable each child to understand the task assigned. 

2. It should anticipate difficulties in the work to be done, and suggest ways to overcome . them. 

3. It should connect the new lesson to past experiences and correlate the topic with all related subject matter. 

4. It should be interesting, motivating and thought provoking. 

5. The activities suggested for the assignment should be varied and adapted to the needs and interests of the students. 

Dr. Lorene Fox, in his article on “Home Work is What We Make It” has suggested the following criteria for good home work: 

1. Is the home work challenging to the students? 

2. Does it grow out of or is it related to the everyday life and interests of the students?

3. Does it encourage individual choice and creativity? 

4. Does it foster the habits of working together, planning and execution through democratic ways? 

5. Does it encourage discovery and the use of a variety of sources and ways of learning? 

6. Is it well adjusted to the available time frame of the child and inculcates good study habits? 

5 Unit Planning and Lesson Planning 

An important aspect of teaching relates to planning and conducting daily lessons. While most teachers base their daily teaching on the subject matter presented in the prescribed textbook, there are many who extend the subject matter to include vital experiences which have their source in the need or interest of the learner. The subject matter is organized into units to provide for as many types of functional activities as possible. 

A unit is a long-range plan to direct the instructional plan. A unit takes care of the logical unity of the subject matter and the psychological considerations of the learner – his needs,’interests and ability to learn. 

There are many advantages in the unit organization of content: 

a) The teacher directs the instruction programme and pupils carry it out with the cooperation of teachers and other pupils. 

b) Units cut across subject matter lines, thereby providing for a better correlation between different branches in mathematics and with other subject areas. Thus, learning is more integrated rather than fragmented. 

c) A variety of activities experiences provided for meeting individual differences in learning. Learning is not forced upon the child. 

d) Drill becomes functional and problem-solving skills more effective if developed in meaningful situations. Critical thinking is, thus, stimulated. 

A unit generally consists of three parts: 

(1) the purposes or objectives, 

(2) learning experiences to carry out these purposes, and 

(3) evaluation tests to find out how well the purposes have been achieved. 

1. Purposes are stated in terms of the understanding or the ability of the learners. The desired attitudes and appreciations which the teacher wants to see developed are also listed as outcomes. 

2. The learning experiences or activities are such that they contribute to the growth of the child and help move towards the stated purposes. 

Since learning is individual, it is desirable to suggest a wide variety of activities suitable for both group and individual work. 

These may be: 

a) Preparatory, which orient and motivate pupils for purposeful activity or test preliminary abilities of pupils. 

b) Developmental, which enable pupils to gain desired skills, abilities, attitudes, and understandings. These activities involve discussion, problem-solving experiences, construction and other forms of creative experiences and field trips. 

c) Culminating, which will be a sort of lion-conventional assignment, organizing exhibits, preparing reports, individual record-books, short plays or a review of Mathematics test. 

3. Evaluation includes plan to determine whether growth has taken place, how far it has gone, and in what direction. Lesson planning relates to the organization of a forty-minute period for teaching. Since mathematics is a sequential subject, each day’s lesson is a necessary foundation for understanding a subsequent lesson. A well-planned lesson gives a teacher a sense of security, keeps on the right track, checks waste of time and ensures a smooth transition from one part of the subject matter to the next. It is advisable that the teacher draw up a schedule of lessons he/she plans to teach in advance, preferably in the beginning of each week. 

The lesson plan should contain the following six parts: 

(1) Aims or objectives, 

(2) background material or previous knowledge, 

(3) introductory or motivational activities, 

(4) developmental activities, 

(5) summary, and 

(6) application. 

1. The aim should be stated in behavioural terms. The statements should be simple, clear and in direct present tense. 

2. The teacher must take note of previous learning required as a foundation for the lesson to be taught. This should clarify possible sources of errors or confusion and make sure pupils understand all that is necessary. 

3. Motivation should be ensured through short, simp1.e application problems. 

4. During the development of the lesson, the teacher should take the role of guide and avoid dominating the class. He/she should ensure pupil participation, make use of proper questioning techniques, elicit from pupils full and accurate statements, employ correct vocabulary, present a sufficient number and variety of situations, provoke students to formulate generalizations. When pupils submit their work, the teacher should see its correctness, point out errors, if any, and suggest remedial work, if necessary. 

6.Materials and Teaching Aids 

For many teachers, the only means of communicating with their pupils is through textbooks and blackboard work. The textbook has been the major source for providing explanatory material, directions for processes and procedures, a set of solved model examples, diagrammatic representations of quantitative relationships, practice exercises and model test papers. A good textbook saves the time and labour of the teachers. It also makes unnecessary the writing of exercises and problems dictated by the teacher. The usefulness of a textbook is increased if it includes suitable illustrations and diagrams. 

Too often difficulty arises because of the limitations of language or of readability in a textbook. A goo& textbook also provides exercises that call for oral rather than written respcrises. The mechanical requisites of paper, print size, format, and binding must measure up to approved standards in a good textbook. These days textbooks are supplemeiited by work-books and practice books which provide well distributed practice exercises arranged according to their difficulty level. The most common aid of the teacher is the chalkboard. Almost every classroom in every school contains one or more chalkboards. 

Class furniture is arranged so that the students usually sit facing the board. Pupil’s work can also be placed on the board and their errors can be easily corrected. Materials can be prepared at home on large sheets of paper which can be taped to the board. Colour is easily applied to the chalkboard to emphasize key ideas. These days a wide variety of learning aids are included in teaching to clarify concepts, their applications and uses. These include:

a) Concrete and semi-concrete materials including measuring instruments. Approaches and Techniques of Teaching Some of these materials can be easily collected from community sources, and some may be purchased. 

b) Excursions and exhibits. Contact with real life situations through tours and trips enables pupils to grasp the role of mathematics in life. Children find collections of pictures, clippings, posters and charts enjoyable and educationally profitable. 

c) Construction activities. Construction activities affored an excellent opportunity for learning by doing. 

d) Multisensory-aids, that is motion pictures, film strips and slides. 

Audio-visual aids help in communication in a manner that is stimulating, expedient, enjoyable and profitable to both the teacher and the learner. While selecting any aid the teacher should raise the following questions: 

1. Is the aid appropriate? That is, can it represent tlie idea or concept in a more satisfactory manner? Is it adapted to the age level? 

2. How expensive is the aid? 

3. Is the aid attractive? 

4. Is the aid easy to handle? 

7 Laboratory Approach to Teaching of Mathematics 

We are familiar with a laboratory as used in teaching science subjects. Recently, this idea has been extended to the teaching of mathematics. One reason why learning mathematics is considered difficult is the verbalistic quality of teaching. Verbalism is the use of words without emphasis on meaning and practice. It is not uncommon to find pupils who have learnt various concepts, formulae and theorems without a proper understanding of their meaning and use. In a laboratory pupils learn by doing. They participate in experiments, manipulate materials and models, use different instruments and are thus able to give meaning to verbalism. 

Much of the mathematics comes to life for the child in a laboratory situation. The laboratory provides an atmosphere in which problems are worked out in simulated life situations. Wall charts, models, mathematical instruments, film slides and video tapes and a lot of manipulative material should be provided in the laboratory. 

Various materials can be assembled from cheap and easily available things by the teachers and pupils: 

In a laboratory situation: 

a) Learning is child-centred, not teacher-dominated. The pupils carry out the activity. . The teacher acts as a guide or helper. 

b) The work is related to life situation and has significance for the learner. !

 c) Among pupils more interest is created since they work in concrete situations rather than in the abstract. 

d) Many community resources are utilized as the subject matter is organized into functional activities.

Ten approaches to the teaching of mathematics

1. Plan and provide a balanced experience that incorporates the exploration, acquisition, consolidation and application of knowledge and skills, with opportunities to use, extend and test ideas, thinking and reasoning.
   
2. Share the excitement of learning mathematics and capture children’s imagination by showing them the unusual or unexpected; give children examples of numbers or shapes that have special or surprising properties; show children how mathematics can be used creatively to represent, measure, predict and extrapolate to other situations.
3. Model for children how to explore mathematics and look for patterns, rules and properties; direct and steer children’s learning by providing examples that enable them to observe and identify the rules and laws and deduce for themselves when they apply; help children to describe, replicate and use patterns and properties; ensure that they meet both general applications of the rules and exceptions.
4. Give children opportunity to consolidate their learning; introduce frequent and regular periods of practice that are short, sharp and focused on children securing, with the necessary accuracy and precision, the mathematical knowledge, understanding and skills they have learned; ensure that they recognise how their learning builds on previous learning and help them to see connections; ensure that they feel appropriately supported and challenged by the work they are set.
5. Engage with children’s thinking; give them sufficient time for dialogue and discussion and space to think about their ideas, methods and mathematical representations of the real world; focus on underlying concepts and processes with prompting and probing questions.
6. Demonstrate and promote the correct use of mathematical vocabulary and the interpretation and use of symbols, images, diagrams and models as tools to support thinking, problem solving, reasoning and communication.
7. Provide children with the well-directed opportunity to use and apply what they have learned to solve routine and non-routine problems; highlight any properties or patterns they identify or create and make connections to other work they have done; draw on their ideas and model approaches and strategies children can use to support a line of enquiry or to interpret or explain their results and methods, using their own approaches and strategies.
8. Teach children how to evaluate solutions and analyse methods, deciding if they are appropriate and successful; help children to understand why some methods are more efficient than others; provide opportunities to compare and measure objects and identify the extent to which shapes and calculations are similar or different; develop children’s understanding and language of equivalence and deduction to support reasoning and explanation.
9. Periodically identify the knowledge, skills and understanding children acquire; pause and take stock to review children’s learning with them; highlight the strategies and processes upon which they are able to draw; provide opportunities that allow children to make connections and show how ideas in mathematics relate, and how their learning can be applied to new aspects of mathematics.
10. Model with children how they identify, manage and review their own learning; highlight the learning skills they have acquired and used and draw out how these might be applied across the curriculum.

QUALITIES OF A GOOD MATHEMATICS TEACHER

Being a good teacher is extremely important, and a good teacher is someone who a student will remember and cherish for the rest of their lives. 

But what really makes a good teacher? There are a lot of things to consider when answering this question. Below we will discuss the top 10 qualities of a good teacher that we believe are most important when it comes to quality teaching and really creating that strong student-teacher relationship.

Sound Knowledge of Mathematics

Every great math teacher has an extensive understanding of mathematics. They undergo a thorough training process in a recognized college or university where they acquire the knowledge and skills they need to teach learners effectively.

 This includes knowledge of geometry, statistics, algebra, arithmetic and calculus. The knowledge they get from these institutions gives them the confidence to explain clearly all the mathematical concepts to their students and solve equations easily. Great teachers do not consult answers at the back of the teacher’s guide booklet. They have all the answers at their fingertips and can help students solve problems instantly.

Engaging

Successful math teachers do not force students to follow their approach. Neither do they assume that they know everything to the point they ignore any form of correction. Instead, they act as facilitators, allowing the students to offer suggestions and solve problems differently on their own. They allow room for collective discussions so that everyone in the class arrives at the same conclusion. 

In case a student is unable to solve a problem, they do not allow him to quit. They work with such students and give them the motivation to identify where they went wrong and keep working on the problem until they find the right answer. They also provide the right guidance and support where necessary.

Good Motivator

Great math teachers know that students have different interests. They thus come up with flexible programs that identify with the students’ source of motivation. For instance, they may motivate students who want to become accountants by giving them mathematical problems related to their ambitions. The same applies to those who want to become engineers, scientists, doctors and any other relevant profession.

 They also talk with their students on a regular basis to help them acquire the right problem-solving skills applicable in the professions they want to enter into. By doing this, they prevent the students from losing interest and disengaging from studying mathematics.

Constantly Learning

Great math teachers know that they are not perfect. That is why they constantly read new materials to update their knowledge base. They also enroll for supplementary courses in their areas of specialty to better themselves and gain more confidence in the classroom. 

This may involve unlearning the outdated algorithms and mathematical terms and learning new ones. Once a great teacher learns about a new mathematical concept, he lets all the students know about it, leading to effective learning and better grades.

Caring

Great math teachers focus not only on the content being taught but also on the students. They have a caring attitude towards their students and are always ready to support those who are having problems. These problems could be due to emotional distress, learning difficulties or illnesses.

They are able to recognize when a student is having a bad day or needs some encouragement. They also understand that sometimes unforeseen problems can derail students from completing their homework on time. 

In such cases, they offer second chances when necessary and take some extra time to help such students catch up with the rest.

Being a great teacher does not necessarily mean you have to record high mean grades in your classes or comply with the school regulations. 

The items mentioned above are the key characteristics of a great math teacher you should have if you want to succeed in your career.

Professional qualities and competencies of mathematics teacher

• Faculty Meetings.  
• Students Projects.  
• Mathematics Club.  
• Social Events.  
• School Publications.  
• Excursions.  
• Educational Fairs. 
• Educational Quiz etc.