Odisha State Board CHSE Odisha Class 11 Foundations of Education Solutions Unit 4 Method of Teaching Maths Questions and Answers.

## CHSE Odisha 11th Class Foundations of Education Unit 4 Method of Teaching Maths Questions and Answers

Long Questions With Answers

Question 1.

Discuss the Aims and Objectives of teaching Mathematics.

Answer:

The knowledge of mathematics is very essential for everybody. The most important aims and objectives of mathematics are discussed below.

To develop the thinking and reasoning, power of the child:

The power of thinking and reasoning is very much essential for an. individual to lead a disciplined and well-adjusted life. These powers can be developed by knowing mathematics.

To provide a suitable discipline to the mind:

Mathematics knowledge makes the mind of the learners disciplined which is essential for leading a healthy social life.

To develop an art of living:

Mathematics prepares children for economic, purposeful, productive, creative, and constructive life. The children learn an act of effective living.

To acquaint the learners with cultures:

Mathematics is the backbone of culture. So by studying mathematics an individual becomes acquainted with his own culture. So cultural development is possible.

To prepare the pupil for various professions:

The children are prepared to enter into various professions of engineering, cashiers, statisticians, accountants, auditors, bankers, etc.

To prepare the students for various higher educational centers:

Mathematics forms the basis of many educational courses and as engineering physical science etc.

To develop the habits of concentration, self-reliance, and discovery:

The habits of concentration, self-reliance, and power to discover new things, new laws, and principles in students are created by mathematics.

To create a love for hard work:

Mathematics as a subject needs consistent hard work. This has helped the student to undertake hard work for a longer period.

Question 2.

Discuss briefly the inductive-deductive method of teaching mathematics. Bring a difference between the inductive and deductive methods.

Answer:

“Inductive-deductive method is the combination of two separate method-inductive and deductive methods.

Inductive method:-

The inductive method is based on induction. Induction is proving a universal truth or theorem by showing that if it is true of any particular case, it is also true for the next case in the same serial order. In this method, we proceed from particular to general, from concrete cases to abstract cases, and from specific to general formulas. In adopting this method, the students are required not to accept the already, discovered formula without knowing the formula by adopting inductive reasoning.

Example No.1:-

The students may be asked to construct a few triangles of various sizes and shapes. They may be asked to measure and sum the angle in each case. Then the sum will come to be the same in all cases. i.e, the sum in all cases will come to be two right angles. Hence, the students may conclude through induction that the sum of these angles of a triangle is equal to two right angles.

Example No.2:- Suppose we find out the simple interest of Rs. 400/- in four years at 5% per annum. It will be equal to Rs.80/-

Or, S.I=\(\frac{400 \times 5 \times 4}{100}\)=80.00

Similarly, the simple interest Rs. 500/- in a year at 6% per annum will be 90.00.

Or, S.I=\(\frac{500 \times 6 \times 3}{100}\)=90.00

From the above, two examples the students can evolve a rule that,

Simple Interest=

S.I=\(\frac{\text { PRT }}{100}\)

Deductive Method :

The deduction is the chief generalized form. In this method, one follows deductive reasoning which is just the opposite of inductive reasoning. Abstract ideas are preceded by concrete experience. The students memorize the different formulas and then apply them to solve a particular problem.

Examples- If the teacher wants to teach the calculation of simple interest in the class the formula for calculating interest to the students.

i.e. S.L =\(\frac{\text { PRT }}{100}\)

Question 2.

Explain with examples the analytic and synthetic methods of teaching mathematics. What are the merits and demerits?

Answer:

Analytic Method:

Analytic means breaking up the problem in such a manner that it ultimately gets connected with some known. The method proceeds from known to known. The analysis is the process of unfolding the problem to know the hidden aspects. We have to begin with what is to be found out and then proceed to further steps and possibilities that may concern the unknown with the known, the desired result is found out.

Merits of Analytic Method:

The Analytic Method has the following merits.

It is a logical method that leaves no doubt and it convinces the learner. The steps are developed in a general manner. Each step has a reason and justification. It facilitates understanding and creates an urge to discover facts. As the students face questions is what a statement is into simple elements they grapple with the problem confidently and intelligently. He gains competencies and skills.

Demerits:

- It is a lengthy method.
- It is very difficult to acquire efficiency and speed.
- It may not be applicable to all topics equally.

Example

If \(\frac{a}{b}=\frac{c}{d}\) prove that \(\frac{a c-2 b^2}{b}=\frac{c^2-2 b d}{d}\)

∴\(\frac{a c-2 b}{b}=\frac{c^2-2 b d}{d}\)

By cross multiplication

acd – 2b^{2}d = be^{2} – 2b^{2}d

Cancellation of the common quality -2b²d from both sides can further he canceled.

acd = bc^{2 }will be true

If this is if ad = bc arranged in a more systematic form, ad = bc will be true.

\(\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}\) which is given is thus true.

So, we can say that,

\(\frac{\mathrm{ac}-\mathrm{b}^2}{\mathrm{~d}}=\frac{\mathrm{c}^2-2 \mathrm{bd}}{\mathrm{d}}\) is also true.

Synthetic Method :

The synthetic method is just the opposite of the analytic method. One has to proceed from known to unknown in this method. Synthesis implies the placing together of the parts to get the solution. One has to start from what is known as given and proceed toward the unknown part of the problem, thus, the unknown information becomes known and free. In practice, synthesis is complementary to analysis.

Merits :

- This is a logical method.
- It is short and elegant.
- It glorifies memory.

Demerits :

It leaves a long number of doubts in minds of readers and offers no explanation for them. As the reader gets no satisfactory explanation for his doubts while solving the problem, he will be perplexed when faced with a need problem. He may not recall the steps of synthesis. There is no provision for a complete understanding of the method. Discovery and thinking have no place in this method. Memory work and homework are too heavy.

Example:

Let us take the same example.

\(\frac{a}{b}=\frac{c}{d}\) then prove that \(\frac{a-2 b^2}{b}=\frac{c^2-2 b d}{d}\)

We have to start with the given or known fact \(\frac{a}{b}=\frac{c}{d}\)

∴\(\frac{2 b}{c}\) be subtracted from both sides

∴\(\frac{a}{b}=\frac{2 b}{c}=\frac{c}{d}=\frac{2 b}{c}\)

Or, \(\frac{a c-2 b^2}{b c}=\frac{c^2-2 b d}{c d}\)

\(\frac{a-2 b^2}{b}=\frac{c^2-2 b d}{d}\)(Cancelling from both sides)

Thus identity is proved.

Question 3.

Explain the problem-solving methods of teaching mathematics. What are the merits and demerits?

Answer:

The problem-solving method aims at presenting and repurposing the existence of problems in the teaching-learning situation. A problem is a sort of difficulty which has to be overcome to reach the goal. It may be a purely mental difficulty. The problem-solving methods aim at presenting the knowledge to be learned in the form of a problem.

It begins with a problematic situation and consists of continuous meaningful well-integrated activity. Mathematics is a subject of problems. Its teaching and learning depend on solving innumerable problems. Efficiency and ability in solving problems is a guarantee in learning this subject. The procedure of problem-solving is (almost like the project method. It can be taken the form of an inductive deductive method.

Steps to the situation :

Sensing the problem, interpreting, defining, and delimiting the problem. Gathering data in a systematic manner, organizing and evaluating the data, formulating tentative solutions, arriving at the true and correct solution, and verifying the results. It is a research-like method that involves scientific thinking as a process of learning.

How it is employed :

Suppose finding the volume of a cylinder is a problem before the class. Its formula is to be developed on the basis of the earlier formula for the volume of a thing while analyzing the problem it gets connected with the previous knowledge that the volume of any regular solid can be found by multiplying the area of its base with the height of the object.

The area of the base of the cylinder is found by the only known formula a new the results are checked. The solution to the problem and the result comes from the students. The teacher remains in the background and directs or guides the students from the position.

Merits of problem-solving method:

This method satisfies the laws of teaching. It involves reflective thinking. So it stimulates thinking and learning through self-effort, reasoning, and critical judgment in the students. It develops qualities of imitative and self-dependence in the students. It is a stimulating method, The problem is a challenge.

Once it is properly recognized it acts as a great motivating force and directs the students, attention, and activity. It serves individual differences. A student can solve any number of problems in a specific and make progress accordingly. It is especially suitable for mathematics which is a subject of problems. It develops desirable study habits in the students.

Limitations:

The process is purely literary. It only needs a mental solution. Life problems -need some physical activity also. All problems cannot be solved by this method. The method does not suit the students in lower classes. Teachers, the burden becomes heavy. Textbooks written in the traditional style do not help in the use of this method. There is an absence of suitable books for reference and guidance.

Question 4.

Discuss the steps in lesson planning.

Answer:

J.F. Herbert has suggested six important steps in planning a lesson. After his name, those steps are called “Herbartion” steps.

These six steps are:

- Preparation
- Introduction
- Presentation
- Recapitulation (comprehension)
- Summarisation
- Application

Preparation:

The teacher has to prepare himself and the students for the lesson. He has to formulate the objectives, select the content matters from the textbook, select the teaching aids and prepare the lesson accordingly.

Introduction:

The main purpose of the introduction is to motivate the pupils. The teacher has to test the previous knowledge of the students by asking some questions. Then the teacher can know the background knowledge is to be linked with the previous knowledge through the introduction. A teacher can introduce a reason by various means such as:-

- asking question
- showing pictures and models
- citing an example
- dramatization
- quoting a dialogue

Presentation:

It is the most important step in the lesson. During this step, the teacher presents some new ideas to the pupils. Questioning discussion, demonstration of aids, active pupil participation, and blackboard work are some of the essential elements of the presentation. The objectives of the lesson determine the nature of the presentation.

Recapitulation:

The teacher should ascertain to what extent the students have understood the topic taught by him. To test their understanding and comprehension the teacher has to put some questions. On this topic, after the presentation is over, this will also help the teacher to know whether his teaching is effective or not.

Summarisation :

The teacher has to associate and generalize the subject matter taught in the lesson in forming a blackboard. Summary, a formula or a rule ‘or a skeleton chart of the important learning points. The step completes the presentation by providing the gist of the topic.

Application :

At this step, the students make use to acquire, knowledge in familiar situations. It tests the validity of the generalization, rule principles or formula arrived at by the pupils at the end of the topic. Through the application, the new knowledge acquired by the students is retained in their minds for a longer period.

Question 5.

Six aims and objectives of Mathematics:

Answer:

It develops the power of thinking and reasoning. It helps the child to solve mathematical problems. It develops the self-confidence and habit of concentration. To help the child to develop the power of expression, and appreciation. It enables the child to go through the transaction of coins. It helps the child to lead a career as an accountant, auditor, engineer, and scientist.

Question 6.

Analytic method:

Answer:

- It proceeds from unknown to known.
- It is a process of thinking.
- It demands exploration.
- It is a method for. thinkers and discoverers.
- It develops originality.
- It is informal, psychological, and based on heuristic lines.

Question 7.

Aids used in teaching mathematics:

Answer:

Visual aids:

Samples, umbrellas, stick-rounded things, dolls, toys, balloons, plates blackboard, models, etc.

Audio Aids:

Radio, gramophone, tape recorders, etc.

Audio-visual aids:

T.V. and films. Through these aids, subjects are taught by experts. These are effective aids in teaching mathematics.