Maths Pedagogy Paper 1 (Nature of Mathematics) - CTET & State TET MCQ

Questions in mathematics can be classified on the basis of the nature of the questions and developed to collect qualitative and quantitative data from the students. It could be close-ended, open-ended, contextual, or essay-type questions.

Key Points

• Close-ended questions
  • These are also known as convergent questions, as the children can only react in a restricted manner of ways such as "yes" or "no", and they have fixed answers.
  • They offer limited insight but are simple to utilize when analyzing quantitative data. 

​​Hence, it is clear from the above points that this question is an example of a close-ended question. As there is a fixed answer to this question, i.e. 9875.

Additional Information

• Open-ended questions- ​These are the type of questions that allow students to express themselves freely. They give tremendous freedom to the children to answer according to their interpretations.
• Contextual question- It usually contain reference expressions to refer to previous questions and their answer.
• Essay type question- These questions are the one that demands lengthy answers.

Different methods of teaching mathematics have been proposed by different educators. Knowledge of these methods may help in working out a teaching-learning strategy. . A teacher should adopt an approach considering the nature of the children, their interests and maturity, and the resources available. There are different methods like lecture method, play-way method, heuristic method, inductive deductive method.

Key Points“Heuristic methods of teaching are methods which involve our placing students as far as possible in the altitude of the discoverer - methods which involve their finding out instead of being merely told about things”.

• In Heuristic method, teacher should present every lesson to learners in form of an inquiry. Learners are asked to identify the problem and work as independent enquirers. They can discuss with their peers, teachers, and others before starting their investigations. The teacher may provide them with some written instructions like what should be followed and what should be avoided. Learners are asked to keep the record of each and every step and show it to teachers after finishing their investigation.
• This method can also be named the 'discovery method'. It is in contrast to the lecture method. Instead of merely the teacher telling everything the student finds out everything for himself. It demands complete self-activity of self-learning on the part of the student.
• Through this method, the student learns to the reason for himself. The teacher is not even required to guide, help or encourage the student. This method helps in the development of a scientific attitude in the learner. It develops self-confidence, originality, independence of judgment, and thinking power in the learner to make him an ever successful student.

Thus, it can be concluded that, in this method, The student is put in the place of independent discovery.

Mathematics is the study of geometrical figures, their co-relation, and their dependency on each other. It deals with quantity, measurement, and spatial relationships.

• Anxiety refers to a chronic condition caused by the feeling of stress, worry, uneasiness, nervousness, etc. It includes some tensed and worried thoughts coming into a person's mind which is visible through his physical behavior.

• Failure is the state or condition of not meeting a desirable aim and may be viewed as the opposite of success. There are many cognitive, physical, intellectual, scholastical, emotional, and cultural causes for the failure of students.

Key Points

Major reasons of anxiety and failure in mathematics classroom includes:

• Emphasis on testing of procedural knowledge as assessment process emphasizes on testing of procedural knowledge than mathematization of abilities.
• Learning gap between word problems and numeral problems as most of the times numeral problems are not taught in context or by relating them to real-life.
• The strict nature of mathematics teacher enables anxiety as children nor feel comfortable with such teacher neither share their thought, ideas and problems.

Hint

Teaching fundamental mathematical skills by connecting concepts to learners' daily life is not a major reasons of anxiety and failure in mathematics classroom as it will help learners in:

• retaining information and concepts for a longer period.
• enhancing skills and a better understanding of the concept.
• nurturing their curiosity and interest in the learning process.
• gaining concrete experience by actively engaging with content.
• assimilating practical knowledge by applying theoretical knowledge.

Hence, it could be concluded that major reasons of anxiety and failure in mathematics classroom does not includes connecting concepts to learners' daily life.

The mathematical manipulative tools are also a type of teaching-learning materials that are specifically used by the students to internalize the abstract mathematical concepts.

• The students used the manipulative tools by themselves to explore, investigate, and internalize the concepts to enhance their mathematical competency.
• These are the teaching tools or materials used to engage students in learning mathematics to provide direct and hands-on experience.

Key Points

Howard Gardener believed that there are multiple intelligences with autonomous "intelligence capacities" and he divided the intelligence into different types which are known as Gardner's theory of multiple intelligence. Spatial intelligence is one of them.

• Spatial intelligence includes the capacity to perceive, understand and use spatial and visual information effectively.
• The word "spatial" comes from the Latin word "spatium" that deals with space and its perception of objects in space.
• The person who possesses this type of intelligence can easily visualize the world with the mind’s eye and modify the surrounding based upon their perception.
• It helps in recreating the aspects of an individual's visual experiences.
• Tangram can be used to check the visual-spatial intelligence of an individual.
  • It includes the different pieces of geometrical shapes and each part is known as "tans" that are put together to form the outline of a shape or any design.
  • The only condition in using tangram is there should not be any overlapping.
• Tessellation is another name for tiling that refers to the placement of shapes alongside each other to fill space completely.
  • The shapes should be placed in a way that there should not be gaps between them just like the tiles covers the floor.
  • It is more used by the artists than the mathematicians that use either a single shape which may or may not be regular or at most a few shapes, to cover the plane.

So, it is concluded that tangram and tessellation are best that can be used to check the spatial intelligence of a child.

Hint

Additional Information

Howard Gardner nine types of intelligence are:

The process of teaching includes both strategies and techniques of teaching and involves the choice of what is to be taught that helps in interpreting the world of knowledge to a pupil's mind. It helps a teacher to understand "what to teach", "how to teach", "how to approach it".

• The teacher must be flexible while using teaching methods to encourage the participation of students in the teaching-learning process.

Key Points

Questions can be classified on the basis of the nature of the questions and are constructed to gather the data from the students. It may be close or open-ended questions.

Close-ended questions:

• These questions are also known as the convergent questions where the respondents answer in limited ways, like responding in ‘yes’ or ‘no’, putting the sign ‘correct’ or ‘incorrect’, etc.
• They provide limited insight but can be easily used to analyze quantitative data.
• For example, what is the difference between 23 and 17 or 23 - 17 = 6 because it has limited ways of responding.

Open-ended questions:

• These questions are also known as the divergent questions where the respondents are free to share, clarify and put their views. 
• These questions are generally framed in a single statement that requires a longer response to compel the students to think beyond the textbooks. So, it helps in developing logical thinking and creative expressions among students.
• For example, which two numbers can be added to get 100. There can be so many numbers that can be added to get 100 such as 70 + 30, 50 + 50, 25 + 75, 45 + 55, and so on.

Hint

"The average mass of 15 fishes caught in a pond is 2.5 kg. The value of the mode is 3 kg. What are the possible masses of the 15 fishes? Explain your thinking". 

• The question asked by a mathematics teacher in class is an open-ended question as the possible masses of 15 fishes will not be the same. Each fish will have a distinct value for its mass. 
• The average mass of fish is the mean value of the mass of fishes whereas the mode is the value that is used frequently in a set of numbers.

So, it is concluded that the above-asked question is an open-ended question.

Learning disabilities are diagnosed particularly when children start going to schools and are engaged in academic activities with other children in the school. It may be caused due to premature birth, low birth weight, and malnutrition.

• The differently-abled children feel difficulty in performing the basic processes required in understanding or using language, spoken or written. It may be seen as the imperfect ability to listen, think, speak, read, write, spell, or do mathematical calculations.

Key Points

Diagnostic test:

• It helps to find out the weakness or deficiency of a child in learning.
• In this, both the performance and background of the children are needed.
• No scores are made for a right answer, only wrong answers are taken into view in the sequence of contents.
• The teacher diagnoses the learning difficulties or errors of children so that remedial teaching improves the performance of children.
• It also helps a teacher to identify the children with learning disabilities i.e., those who are unable to perform basic mathematical computations, difficulty in reading, writing, listening or speaking, etc.
• It paves the way for further improvement and measures to be taken to help the child in his learning and development.

Hint

Hence, it is concluded that diagnostic test is used to assess children with learning disabilities in mathematics.

Mathematics pervades all aspects of our lives. Any person, he/she may be a farmer, daily labourer, artisan, teacher or scientist uses the principles of mathematics in his/ her day-to-day activities in different situations. Thus mathematics holds a key position in our life. Therefore, mathematics has enjoyed a privileged or sheltered position in the school curriculum.

• The concept of the school curriculum can be best described as the set of all activities designed within the school to promote the intellectual, emotional, social, and physical development of the pupils.

Key Points

• One of the major objectives of teaching primary mathematics is to enable children to solve speedily and accurately the numerical and spatial problems which they encounter at home, in school and in the community.
• It should help children develop an understanding of key mathematical concepts through appropriate experiences with the physical world and the immediate environment. For maintaining the relevance of the mathematics curriculum to the societal and personal needs of the learners, it should continuously undergo change in light of changing national goals and priorities. 
• Mathematics curriculum in order to be realistic, relevant and meaningful has to be in tune with the pedagogical goals and the nature of the mathematical content. It should be based on systematic research about the nature of the learner, the learning process and the presentation of the mathematics curriculum to the learners. 

Important Points

• ​The primary school stage is the basic platform on which the later stages need to be built. Two things have to be considered here while choosing the content.
• Firstly, this is the stage at which we need to create among children a positive attitude towards the subject of Mathematics. Naturally, young children like to play games and indulge in activities. Hence, ample scope must be given to including more Mathematical games, puzzles and other recreational activities at this stage.

Thus, it is concluded that the Curriculum of mathematics at the primary level in school education should be content-centred.

Mathematics is a man-made, well-disciplined subject, which deals with abstract concepts and things. It has its own language, its own tools, and mode of operation to help people in a proper understanding of nature’s work and complicated problems of life. The three basic groups of mathematical concepts that are essential in all topics included in the mathematics curriculum at the primary school level are numbers and operations on numbers, spatial thinking, and measurement.

Children’s Development of Mathematical Concepts:

• Measuring: In pre schoolers or Kindergartner, a student may be learning simple measurements using non-standard units, such as measuring how long something is using paper clips or blocks, instead of a ruler. They come to learn how to classify objects based on weight (heavy/light); capacity (holds more/less); and length (long/short).
• Time: They understand concepts such as morning, afternoon, and night and words that describe time such as before, after, or next, to name a few. In school, they may be learning the days of the week, and how to read a calendar. They learn that a year is longer than a month, and a month is longer than a week, etc.
• Predicting: Through experiences, children start to make predictions about what will happen next. For example, if they see it is raining out, they may predict that there will be indoor recess at school.
• Problem-solving: Children at this age can solve simple problems. 
• Cause and effect relationships: They understand if they go out in the rain, they will get wet. If they take their friend’s crackers, they can predict the effects of their actions: they will have more crackers, their friend will have fewer, and their friend may become angry.

Important Points

Most appropriate to assess children's understanding of Mathematical concepts in Class I is observation.

• Classroom observation needs to focus on what the students are doing and how teacher practice affects this.
• Observing Classroom Activity when given a task to solve (Related to number System)
• Introduction to an Observation Framework which helps in the teaching-learning process.
• The Context of a Lesson Observation

Hence, we can conclude that observation is most appropriate to assess children's understanding of Mathematical concepts in Class I.

There is no single and definite way of learning mathematics even at the earliest stage of learning. Some points regarding the characteristics of the nature of mathematics learning as summarised by White Bread (Anghileri, 1995):

• Mathematics starts from ‘home learning’ established in the child before he/she comes to school.
• Mathematics is based on understanding.
• Mathematics puts great emphasis on the child’s own methods of calculating and solving problems and rejects the previous practice of heavy emphasis on standard written algorithms.
• Mathematics is regarded as a powerful tool for interpreting the world and therefore should be rooted in real experience across the whole curriculum.
• Mathematics is brought out of the child’s everyday situations.
• Mathematics with reason is rooted in action – learning through doing.
• Mathematics with reason puts less emphasis on representing numbers on paper as ‘sums’ and more emphasis on developing mental images in the child.
• The main tool for child and teacher to employ in the mastery of mathematics concepts is language, not pencil and paper exercises from textbooks. The child is encouraged to talk about what he/she is doing.
• Errors are accepted as an essential part of the mathematics learning process. The child, freed from the fear of criticism, will more readily experiment.

Hence, we conclude that Mathematics with reason is rooted in action- learning through doing.

Mathematics is an important component of school education. Its influence has been so fundamental and widespread that being numerate is becoming more important than being literate.

• It is the responsibility of a teacher to make students understand and conceptualize the mathematical concepts so that they can use the acquired knowledge in all aspects of their life.
• The students require the ability to solve a problem and conclude it accurately. Certain decisions in daily life require sufficient skill and understanding of quantitative relations. 

Key PointsSocial Aspect of Mathematics: 

• The social aspect of mathematics consists of the routine activities of daily life that require a mastery of a number of facts and processes such as percent, discount, commission, dividend, invoice, profit, and loss, wholesale and retail, taxation, etc.
• Mathematics help in bringing together the countries of the world which are separated from each other physically and helps to discover the mysteries of nature and to overcome superstitions and ignorance.
• Mathematical operations like addition, subtraction, multiplication, division, and so on, are used in our daily activities. From poor to rich, all have to use Mathematics in their real lives in one or another way. 
• For example, Suresh uses his knowledge of mathematics while eating panipuri i.e., counting the number of eaten panipuris, the total amount that is needed to be paid at the shop, and calculating how much panipuris will cost of how much money, and so on.
• The child should gain an appreciation of the role played by mathematics in many fields of work. Since scientific knowledge and technology are linked with the progress and prosperity of a nation, we should be able to appreciate the role of mathematics in these fields. 

Hint

• Mathematical Aspects of Mathematics: Mathematics develops the ability to perform necessary computations with accuracy and reasonable speed. It also develops an understanding of the processes of measurement and of the skill needed in the use of instruments of precision. 

Hence, it could be concluded that Mathematics develops the ability to perform necessary computations with accuracy and reasonable speed is not associated with the Social Aspects of teaching Mathematics.