Notes: Error Analysis | Mathematics & Pedagogy Paper 2 for CTET & TET Exams - CTET & State TET PDF Download

No measurement of a physical quantity can be entirely accurate. It is important to know, therefore, just how much the measured value is likely to deviate from the unknown, true, value of the quantity. The art of estimating these deviations should probably be called uncertainty analysis, but for historical reasons is referred as error analysis.

Classification of Errors

Some scholars have classified the errors of measurement and evaluation on the basis of their form. These are of following types 

1. Personal Errors: This type includes measurement errors committed by the measurer and evaluator. It has been seen that physical and mental state of the measurer, his personal viewpoint, his liking and prejudices have direct or indirect bearing on his measurement process. Such errors are generally seen in marking the essay type answers. Such errors can be corrected by making the tests objective.
2. Variable Errors: This type includes measurement errors occurring due to administration of measuring object, person or activity, measuring quality (variable), measuring tool or method. In educational measurement, errors are committed due to abnormal physical or mental state of the student or students, unclarity of measuring variable, unsuitability of measuring tool or method and their inadequate administration. For their eradication, it is necessary to deal with the students sympathetically, understand the measuring qualities clearly, select most suitable measuring method and its adequate administration.
3. Constant Errors: This type includes measuring errors in the form of measuring tool or method and errors in their use. Because the marks obtained by all the students are affected by such errors, so they are called constant errors. Such errors can be eradicated by making the measuring method more valid.
4. Interpretative Errors: The errors occurring in analysis of measurement results are called interpretative errors. Such errors can be done away with by development of suitable norms and correct statistical calculations.
5. Accidental Errors: By accidental errors we mean those errors which occur accidentally. These are of three types 

• Student Oriented Errors: The errors resulting due to the student falling ill, becoming agitated or terrified suddenly are called student oriented errors. These can be eradicated by making the student normal. 
• Test Oriented Errors: Such errors arise due to the test not being valid, reliable and objective. Such errors can be removed by making the test, valid, reliable and objective. 
• Errors in Marking Errors: in marking arise due to marking by the examiners with different viewpoints. Such errors are often committed in the marking of essay type answers. Such questions can be made objective. The training of evaluators is also necessary. 

6. Systematic Errors: By systematic errors we mean the errors committed by person in the process of measurement and evaluation. These errors are also of three types 

• Errors due to Ignorance: Errors due to ignorance are committed by the students on not seeing the instructions or manual properly. Such errors can be removed if the students work carefully. 
• Errors due to Forgetfulness: Sometimes the students erroneously write 0.01 in place of 0.1, or write 3.7 in place of 7.3, etc. Such errors can be controlled by carefulness. 
• Errors due to Biasedness: The errors arising due to prejudice or liking of the measurer are included in it. Such errors can be removed by removing the opportunities of subjectivity and making the measurement tool or method objective.

Teaching Methods

The process of interpreting the world of knowledge to pupil’s mind is called the ‘‘method of teaching’’. The world of knowledge includes the knowledge, interest, attitude, skill etc. i.e. all the three domains—cognitive, affective and psychomotor.

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Lecture Method

• This is a teacher centred method. In this method the teacher is an active participant and the child is a passive learner. This is not a psychological method. In this method the teacher speaks or delivers a lecture on a particular topic and the children listen. It is one way traffic because the teacher gives ideas and the children receive them. Some merits of lecture method is given below:
  • This is an easy, attractive and brief method. 
  • It is useful to give fact based knowledge and historical development of Mathematics. 
  • It is more useful for higher classes. 
  • It is a method of presenting the word, picture of ideas and experiences.
• There are some demerits of lecture method is given below 
  • This is a teacher centred method. So this is against psychological principles. 
  • The learner remains passive and inactive. 
  • It does not provides opportunities to develop various mental abilities like—reasoning, logical thinking, mathematical training etc. 
  • In this method, there is no provision for practical and creative work. 
  • It is not necessary that the children are attentive and understanding all, what the speaker/teacher is saying.

Inductive Method

• Inductive method of teaching and learning Mathematics is based on induction. 
• Induction means proving a universal truth or theorem by showing that if it is true in any particular case, it will be true in the next case in the same serial order. 
• This is a method of development in which the child is made or led to discover truth for himself. 
• Some merits of inductive method is given below 
  • It is a scientific method because knowledge attained by this method is based on real facts.
  • The knowledge gained by the use of this method is more durable becuase in this method child himself attains the knowledge by examples, observation and testing. 
  • The critical observation and logical power of children are developed by inductive method. 
  • In this method children themselves attain the knowledge with the help of examples so they don’t feel bored (fatigue). 
• They remain active to attain new knowledge. This is certainly true that inductive method is a very important and useful method of teaching Mathematics. 
• Inspite of having so many merits. It has some demerits also which are as follow
  • This is a very slow process, so gaining knowledge by this method costs more time and labour.
  • It needs sharp mind, proper planning and enough labour. So it is not easy to attain knowledge by this method for students of all levels.
  • Only an experienced and able teacher can use this method successfully.
  • The ability and capacity of problem solving cannot be developed by the use of this method.

Deductive Method

• Deductive method is exactly opposite to inductive method. 
• Deductive logic is used in this method. Deductive method is mainly used in algebra, geometry and trigonometry because different relations, laws and formulae are used in these sub-branches of Mathematics. 
• It is impossible to verify each law and formula practically. 
• In this method help is taken from assumptions, postulates and axioms of Mathematics. 
• This method is used for teaching Mathematics in higher classes. 
• This is based on deduction. 
• Some merits of deductive method is given below 
  • By using this method the speed of gaining knowledge increases because students directly use the formula for solving the problem. 
  • This method should be used when there is shortage of time. 
  • This method is used for teaching theorems and axioms of geometry, tables in arithmetics etc. 
  • Both the teacher and pupil do not find any difficulty in using this method. 
  • More knowledge can be attained in less time by the use of this method. 
  • This method is short as well as practical. 
• Like Inductive method, Deductive method is also an important method of teaching Mathematics. 
• It has its own qualities but still it has some demerits and limitations. Following are the main demerits of this method 
  • This method is not in accordance with psychological principles. 
  • In this method more emphasis is given on cramming than understanding or discovering.
  • In this method there is no scope of developing powers like logical thinking and investigation. 
  • Children don’t get opportunities to gain new knowledge by using this method.

Analytic Method

• The original meaning of the world Analysis is to unloose or to seperate things that are together. 
• Analysis starts with ‘‘what we have to find out and traces the connection between it and the data. 
• With the help of this method, the difficult parts of any problem can be analysed to find out the solution of the given problem. Thus, separation of different part of a problem is known as analysis. 
• Some merits of analytical method are given below 
  • This method is based on psychological principles. 
  • The analysis is an explanatory procedure. 
  • (iii) It creates creativity and originality in the child and develops analytic and reasoning power.
  • This develops self-confidence and logical abilities in the child.
  • It is a formative method and based on inductive reasoning. 
• Inspite of being so important and useful method of teaching Mathematics, it has some limitations which are as follows 
  • Every teacher cannot use this method successfully. 
  • The whole syllabus cannot be completed with in the certain period. 
  • The use of analytical method is possible only when we have the knowledge of known facts and unknown conclusions.

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Synthesis Method

• It is reverse of the analytical method. Synthesis means to place together things that are apart or ‘‘to join seperate parts’’. 
• In this method we proceed ‘‘from known to unknown’’ or we start with hypothesis and end with conclusions. Thus synthesis begins from the data and connects them with the conclusion. 
• Some merits of synthesis method are given below 
  • It is a short and quick method. 
  • It glorifies the memory of the child. 
  • It formulates, records and presents concisely the discovered facts. 
  • If omits the trials and errors like in analysis. 
  • This is the method of setting out the solution in a concise form. 
• Some demerits of synthesis method are given below 
  • It creates many doubts in the mind of the child. 
  • There is no opportunity for developing thinking, reasoning, and other mental abilities. 
  • The recall of each step cannot be possible for every child.

Laboratory Method

• To make mathematics more interesting and meaningful, laboratory method is used in teaching of Mathematics. 
• In this method students get the opportunity to acquaint themselves with the facts through direct experiences individually. In this method students themselves verify the facts and laws of Mathematics with the help of experiments. 
• This method needs a laboratory in which equipments and other useful teaching aids related to Mathematics are available. 
• Some merits of laboratory method are given below 
  • This method is based on psychological principle, ‘‘learning by doing’’ so that students take interest in their work. 
  • The knowledge acquired by this method is more solid and durable. 
  • It helps in developing the habit of discovery and self-study. 
  • It develops scientific attitude in the students. 
  • This method helps in developing problem solving ability. 
  • The children get the opportunity of creative and practical work. 
  •  The children enjoy as they remain active in the laboratory. 
• Some demerits of laboratory method are given below 
  • It is an expensive method so due to financial constraints, all schools are not able to adopt this method. 
  • It takes more time to teach so vast syllabus cannot be completed with in the time. 
  • This method requires laboratory equipped with different apparatus. 
  • Individual attention cannot be paid to classes with large number of students.

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Heuristic Method

• Like other method, heuristic method also has a special place in Mathematics teaching. 
• The world heuristic is believed to be originated from Greek word ‘‘Heurisco’’ which means ‘‘I find out’’. The profounder of this method was Prof. Henery Edward Armstrong. 
• By the use of this method scientific and mathematical attitude can be developed in students. Herbert Spencer has thrown light on this method and stated that—‘‘Students should be told minimum and as much as possible they should be encouraged to discover.’’
• Some merits of heuristic method are given below 
  • This is a psychological method. 
  • It develops self-confidence, self-reliance and scientific attitude. 
  • It develops ability of observation and spirit of enquiry to solve the problems. 
  • Contemplation and awakening increases in the children. 
  • Pupil get the opportunity to develop the mental and thinking powers. 
  • This method is based on the principle of activity. 
  • Individual attention of the teacher is possible and the relation between the teacher and learner becomes more intimate. 
• Some demerits of heuristic method are given below 
  • It is not suitable for lower classes. 
  • It is a very slow method. 
  • There is lack of text books written on heuristic approach. 
  • This is formational method rather than informational