Elementary mathematics: The students we are facing are primary schools and middle schools, which are relatively simple.

Advanced Mathematics: Students are undergraduates and undergraduates, which is relatively difficult.

2, the basic content is different

Elementary mathematics:

(1) Primary school: Elementary school's elementary school's elementary school's elementary school's elementary school's elementary school's elementary study of integer, fraction and four operations, number and algebra, space and graph, simple statistics and possibility, one-dimensional linear equation, circle, positive and negative numbers and solid geometry.

(2) Junior high school: rational numbers (positive numbers and negative numbers and their operations), real numbers (operations of roots), plane rectangular coordinate system, basic functions, simple statistics, acute trigonometric functions, equations, (one-dimensional linear equations, two-dimensional linear equations, one-dimensional quadratic equations and three-dimensional linear equations), factorization, algebra, fractions and one-dimensional linear inequalities.

(3) High school: set, basic elementary function (exponential function, logarithmic function, power function, higher order function), quadratic function root distribution and inequality, permutation inequality, elementary determinant, trigonometric function, analytic geometry and conic curve, complex number, sequence, higher statistics and probability, permutation and combination, plane vector, space vector, space rectangular coordinate system, derivative and relatively simple definite integral.

Advanced mathematics: series, limit, calculus, spatial analytic geometry and linear algebra, series, ordinary differential equations.

3. Different contacts

(1) Advanced mathematics can provide a theory for mathematical methods commonly used in elementary mathematics.

In the current middle school textbooks, we only talk about how to use the commonly used mathematical method-mathematical induction instead of proof of principle. Middle school textbooks consider the knowledge level, age characteristics and the teaching purpose of middle school mathematics. But for a future middle school teacher, it is more important to know why.

The rationality of mathematical induction is guaranteed by the inductive axiom of natural numbers, which is also provided by the inductive axiom. Starting from this axiom, various inductive proof methods can be evolved: first mathematical induction, second mathematical induction, backward induction, infinite descent induction and so on.

(2) Advanced mathematics plays a guiding role in the learning and teaching of elementary mathematics.

It is very limited to study the increase and decrease, concave and convex, extreme value and maximum value of functions by means of elementary mathematics. In advanced mathematics, we can use the knowledge of limit, derivative and series to study the characteristics of functions completely.