KINDERGARTEN — numbers, shapes, and play.
The very first steps: counting real things, naming shapes, and spotting patterns, written for a 4–6 year old (or a grown-up reading aloud).
In a nutshell
After Kindergarten, you can count real objects up to 20 without mixing them up, name the four basic shapes on sight, and figure out what comes next in a simple pattern.
PRIMARY / ELEMENTARY — everyday number skills.
Ages roughly 7–10: adding, multiplying, fractions, money, time, and measuring the world.
In a nutshell
After Primary/Elementary, you can add and subtract multi-digit numbers with carrying and borrowing, multiply and divide using times tables, and handle everyday fractions, money, time and measurements.
MIDDLE SCHOOL — building the toolkit.
Ages roughly 11–13: the number systems, algebra's language, and the first equations and functions.
In a nutshell
After Middle School, you can work confidently with all the number systems, simplify and factor algebraic expressions, solve linear equations and inequalities, and read and transform the graph of a function.
HIGH SCHOOL — deeper structure.
Ages roughly 14–17: curved growth, geometric proof, trigonometry, and exponential change.
In a nutshell
After High School, you can solve any quadratic, prove geometric facts with Pythagoras and similarity, use trigonometry to find unknown lengths and angles, and work with exponential growth and logarithms.
COLLEGE / UNIVERSITY — rigor and abstraction.
18+: sequences, the calculus of change and accumulation, probability, several variables, linear algebra, and the proofs that hold it all together.
In a nutshell
After College, you can differentiate and integrate functions, reason rigorously about limits and infinity, quantify uncertainty with probability and statistics, work in several variables and dimensions, and construct an airtight mathematical proof.