Nature of Mathematics MCQ Quiz - Objective Question with Answer for Nature of Mathematics - Download Free PDF

Van Hiele’s theory of geometric thinking outlines levels through which students progress in understanding geometry. These levels describe how learners perceive and reason about shapes, gradually moving from visual recognition to logical deduction based on properties and relationships.

Key Points

• Rina can identify a triangle and describe its sides and angles, which indicates she recognizes shapes based on their attributes rather than just appearance.
• However, she struggles to understand the relationships between different triangles (like how an equilateral triangle is a special kind of isosceles triangle).
• This indicates she has not yet reached the level of informal reasoning about relationships.
• Her understanding aligns with the Analysis level (Level 2) in Van Hiele’s theory, where learners can describe parts and properties of shapes but do not yet relate or organize these properties across categories.

Hint

• Visualization (Level 1): Children recognize shapes based on appearance, not properties. Rina is beyond this.
• Informal Deduction (Level 3): Learners understand relationships and can form logical arguments beyond Rina's current level.
• Formal Deduction (Level 4): Involves rigorous proofs and axioms; typically not expected at primary level.

Hence, the correct answer is Analysis.

In mathematics, properties or laws such as the associative, distributive, and commutative laws help students simplify and manipulate expressions efficiently. When learners apply these properties, they demonstrate a deeper understanding of number operations and flexibility in problem-solving.

Key Points

•  In the given example: 6 × 15 = (2 × 3) × 15 = 2 × (3 × 15) = 2 × 45 = 90, the student breaks 6 into 2 × 3, then groups the numbers differently as 2 × (3 × 15), and finally multiplies to get 90.
• This method clearly uses the associative law of multiplication, which allows the grouping of numbers in a multiplication problem to change without affecting the result: (2 × 3) × 15 = 2 × (3 × 15).
• The commutative law, which involves changing the order of numbers being multiplied (a × b = b × a), is not applied here, as the order remains consistent throughout.
• The distributive law, which involves distributing multiplication over addition or subtraction (like a × (b + c) = a × b + a × c), is also not used in this problem.

Therefore, the student has applied only the associative law.

Hence, the correct answer is Only (a).

Van Hiele’s theory of geometric thinking outlines how learners progress through different levels of understanding geometry. These levels are not based on age but on the quality and type of instruction provided. The theory emphasizes a sequential and developmental process in learning geometric concepts.

Key Points

•  Progression through Van Hiele levels depends on instruction, not biological age. That is, well-structured teaching and experiences are essential for a learner to move from one level to the next.
• Additionally, learners must pass through the levels sequentially, they cannot skip a level, as each stage builds the foundation for the next. These characteristics are central to Van Hiele’s framework and help teachers structure geometry instruction appropriately.

Hint

•  Visualization is not the highest level; it is actually the first level of geometric thinking in the Van Hiele model.
• The highest level is rigor, which involves formal deduction and proof. Therefore, statement (c) is incorrect.

Hence, the correct answer is (a) and (b).

A closed-ended problem typically has a specific, finite solution that can be answered with one or more concrete answers. It does not leave room for interpretation or extended exploration, making it distinct from open-ended problems.

Key Points

• Identify all prime numbers between 10 and 20: This problem has a definitive, specific answer. The prime numbers between 10 and 20 are 11, 13, 17, and 19.
• Provide two fractions equivalent to 3/8: This problem can have multiple solutions (like 6/16, 9/24, etc.), making it an open-ended problem since there are many possible answers.
• List all integers between -5 and 5: Though it has a finite list, this is an open-ended problem as it invites listing multiple possible answers (such as -4, -3, -2, -1, 0, 1, 2, 3, 4), giving room for more exploration.
• Solve for x in the equation 2x + 5 = 15: This is a closed-ended problem with a unique solution, which is x = 5.

Hence, the closed-ended problem is Solve for x in the equation 2x + 5 = 15.

Mathematics problems can be categorized as open-ended or closed-ended based on the number of possible correct responses.

 Key Points

• A closed-ended problem has a limited number of correct answers, usually just one or a few clearly defined ones. These types of problems are useful when specific knowledge or accuracy is being tested.
• Writing four equivalent fractions for 5/7 is a closed-ended problem because there are specific, correct answers formed by multiplying both the numerator and denominator of 5/7 by the same number.
• For example, 10/14 and 15/21 are valid equivalent fractions. These answers are definite and follow a fixed rule, making the task structured and precise.

Hint

•  Writing integers less than 5, or rational numbers between two given values, or natural numbers in a range allows for many correct answers and encourages different responses from students. These are open-ended in nature as they permit flexibility and multiple correct outputs.

Hence, the correct answer is write four equivalent fractions for 5/7.

Basic mathematics was created to meet the basic demands of individuals who want to learn the fundamentals of mathematics and how to use them in their daily lives. Basic mathematical concepts such as addition, subtraction, division multiplication, percentage, profit, and loss among others are essential for everyone in daily life.

Key Points

Hence, we conclude that the correct statement is only D.

Mathematics is the study of numbers, shape, quantity, and patterns. Mathematics is the ‘queen of all sciences’ and its presence is there in all the subjects. 

• Mathematics relies on logic and connects learning with children's day to day life. It acts as the basis and structure of other subjects. 
• It visualized as the vehicle to train a child to think, reason, analyze, and articulate logically.

Key Points

The Nature of Mathematics is Logical as it relies on: 

• evaluation of truth or likelihood of statements.
• development of skills like speed, accuracy, estimation.
• improvement of reasoning power, analytical and, critical thinking.

• enhancement of scientific attitude like estimating, finding and verifying results.

Hence, it becomes clear that the nature of Mathematics is logical.

Pre-number concept: These are defined as math skills that are learned by pre-nursery or kindergarten kids to make them understand the different variations in shapes, sizes, colors, etc. These concepts can be developed in children during the preschool years. i.e. before attaining 7 years of age(before the concrete operation stage).

Important PointsStages of Pre-number concepts include:

• Classification: Children need to look at the characteristics of different items and find characteristics that are the same and classify them accordingly.
• One to one correspondence:- The ability to count one object while saying one number is known as one-to-one correspondence. If you are counting items, for example, you can point to the first one and say '1' then, ;to the second and say '2', and so on.
• Patterns:- It refers to the understanding of the repeated arrangement of numbers, shapes, and designs and making a generalisation based on some rules and structure.
• Matching:  Matching forms the basis for our number system.
• Comparing: Children look at items and compare by understanding differences like big/little, hot/cold, smooth/rough, tall/short, and heavy/light. 

Thus, it is concluded that Classification, patterning and one-to-one correspondence are part of pre-number concepts in young children.

Base 2 system provided in the Binary number system. In Binary number system (0,1) only two no. are taken.

Hindu Arabic number system base 10 is provided.

• Hindu Arabic number are as follows (0,1,2,3,4,5,6,7,8,9)
• They are additive in nature.
  • Additive: a + b = b + a
• The position of a digit in a number dictates its value.
• It is multiplicative in nature.
• This is a base-ten (decimal) system since place values increase by powers of ten. 
• Furthermore, this system is positional, which means that the position of a symbol has to bear on the value of that symbol within the number. 

Mathematics- Mathematics is concerned with the logic of shape, quantity, and arrangement. It aids in our understanding of the world and is a good tool for developing mental discipline. Logical reasoning, critical thinking, creative thinking, abstract or spatial thinking, and problem-solving abilities are encouraged by math.

Key Points

•   Nature of mathematics-
  • ​Mathematics has its language which consists of mathematical terms, concepts, formulae, theories, principles, signs, etc.
  •  It involves the conversion of abstract concepts into concrete ones.
  • It helps the students to build the habit of self-confidence by eliminating any uncertainty in their minds regarding theories, ideas, and concepts.
  • Mathematics information is precise, systematic, logical, and clear and it cannot be forgotten once acquired.
  • Mathematical rules, laws, and equations are universal and may be checked at any time anywhere.
  • A problem can have multiple ways of solution in mathematics as it helps children to develop flexibility and support the understanding of concepts.

Hence we can conclude that concerning the nature of mathematics, it has its language and the mathematical concepts are abstract.

Mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.

It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. 

Mathematics is accepted as a branch of logic. According to C. G. Hempel “It can be derived from logic in the following sense:

• All concepts of mathematics i.e. arithmetic, algebra and analysis can be defined in terms of concepts of logic.
• All the theorems of mathematics can be deduced from these definitions by means of the principles of logic.

​Main branches of Pure mathematics:

• Algebra: Algebra is first accepted as a branch of mathematics. It is a kind of arithmetic where we use unknown quantities along with numbers. These unknown quantities are represented by letters of the English alphabet such as X, Y, A, B, etc. or symbols. The use of letters helps us to generalize the formulas and rules that you write and also helps you to find the unknown missing values in the algebraic expressions and equations.
• Geometry: It is the most practical branch of mathematics that deals with shapes and sizes of figures and their properties. The basic elements of geometry are points, lines, angles, surfaces, and solids.
• Trigonometry: Derived from Greek trigōnon, "triangle" and metron, "measure", it is a branch of mathematics that studies relationships between side lengths and angles of triangles.
• Calculus: It is a branch of mathematics concerned with instantaneous rates of change and the summation of infinitely many small factors.
• Statistics and Probability: The branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation, and display of numerical data. 

Also Note:

• Arithmetic: It is the oldest and the most elementary among other branches of mathematics. It deals with numbers and the basic operations- addition, subtraction, multiplication, and division, between them.
• Analysis: The analysis is a branch of mathematics that studies continuous changes and includes the theories of integration, differentiation, measure, limits, analytic functions, and infinite series.

Hence, we conclude that Mathematics is accepted as a branch of ​Logic.

The three basic groups of mathematical concepts that are essential in all topics included in the mathematics curriculum at the elementary school level are number and operations on numbers, spatial thinking and measurement. Note that:

• The mathematical concepts and processes at different levels, especially at the primary level, are arranged from simpler to complex order. 
• Such arrangement is associated with the development of their thinking abilities and with the growth of the learner. 
• Research has established a close relationship between the growth of thinking and the development of mathematical concepts. 
• As a teacher, one should be aware of such relationship so that one can develop an understanding of the strength and difficulties of every child in one's class in their learning of mathematics concepts and can take appropriate facilitating steps in that direction.

Hence, we conclude that a close relationship between the growth of thinking and the development of mathematical concepts is established in ​Research.

Generally, the questions asked by teachers during the teaching-learning process can be classified into two types- open-ended and closed-ended questions.

•  Open-ended and closed-ended questions-
  • Open-ended questions can have more than one answer. They provide valuable and specific information about the student's understanding and application of learning. They create opportunities for real-world application of math. They are also known as divergent questions.
  • Closed-ended questions can have only one answer where the student can answer in limited ways like answering "yes" or "no" are telling "correct" or "incorrect". They are also known as convergent questions.

Key Points

Here, it is concluded that A & B are open-ended type questions whereas C & D are closed-ended type questions.

Mathematics: Mathematics is a systematized, organized, and exact branch of  Science. It plays an important role in accelerating the social, economical, and technological growth of a nation. It helps in solving problems of life that need enumeration and calculation.

Important Points

The nature of Mathematics can be made explicit by understanding the chief characteristics of Mathematics:

• Mathematics is a science of discovery.
• Mathematics is an intellectual game.
• It deals with the art of drawing conclusions.
• It is a tool subject.
•  It involves an intuitive method.
• It is the science of exactness, precision, and accuracy.
• It is the subject of a logical and specific sequence.
• It requires the application of rules and concepts to new situations.
• It is a logical study structure and patterns.

Thus, it is concluded that expanded expression is not related to the nature of Mathematics.

In our day-to-day life, we need to quantify several things that we use or come across like members of the family, and students in the class/school. For counting objects, expressing different quantities, measuring length, weight, volume, expressing time, etc numbers are essential. Numbers have been so intimately associated with our life that we cannot think of anything without them.  Different systems of numbers were developed in different ancient civilizations. Note that:

• A numeral is a representation while a Number is an idea.
• A numeral is a symbol that represents a number. For example, 'three' can be represented as '3' and 'III'.
• The same number can be represented by different numerals as mentioned above.

Hence, both statements are correct.