In the development of a particular branch of mathematics, the mathematician is chiefly concerned with a logical rigorous treatment of the subject matter, whereas the teacher is usually concerned with its psychological organization and presentation. It is the curriculum organizer who is called upon to integrate the two approaches.

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 Geometry teaches Logical thinking and natural design.