Maths from first counting to a PhD
All of school and university maths in order, from first counting to original research. There are 37 steps. Each one lists what to learn, where to learn it, and how to tell when to move on.
The whole plan
Beginnercounting, shapes, first sums
- 1
Maths starts as play: counting real things, noticing shapes, and comparing sizes.
- 2
Numbers to 100 and the first ideas of adding and taking away.
- 3
Addition and subtraction facts, and what the tens and ones in a number mean.
- 4
Place value to 1,000, and adding and subtracting bigger numbers.
Easymultiplication, fractions, decimals, ratios
- 5
Multiplication and first fractions
Multiplication, division, and the first fractions.
- 6
Long division and equivalent fractions
Multi-digit multiplication, long division, and equivalent fractions.
- 7
Fraction and decimal arithmetic
Fraction and decimal arithmetic, volume, and the coordinate plane.
- 8
Fluency with fractions, ratios, and negative numbers. Everything later assumes it.
- 9
Proportional reasoning and the first real equations.
Intermediatealgebra, geometry, functions
- 10
Lines, slope, and functions: the doorway into algebra.
- 11
The language of all later maths. Take it slowly and do many problems.
- 12
Shapes, yes, but mainly a first taste of logical proof.
- 13
Widens the function toolkit: polynomial, rational, exponential, logarithmic.
- 14
Trigonometry plus the loose ends calculus needs.
Advancedcalculus, statistics, university maths
- 15
Rates of change. The first big idea of higher maths.
- 16
Integration technique and infinite series.
- 17
Reasoning from data under uncertainty.
- 18
Calculus in two, three, and more dimensions.
- 19
Arguably the most useful course on this list. Start it right after calculus 1.
- 20
Equations that describe change, used throughout science and engineering.
- 21
The turning point: from computing answers to proving statements. Do not skip it.
- 22
Probability done with calculus, the base for statistics and stochastic processes.
Expertmaths major and graduate topics
- 23
Calculus rebuilt from the ground up with full proofs. The standard gatekeeper course.
- 24
The study of structure: groups, rings, and fields.
- 25
Calculus over the complex numbers, where things work out surprisingly well.
- 26
Continuity and shape in their most general form.
- 27
Choose by what you enjoyed so far. These hint at your eventual research area.
- 28
The modern theory of integration, required for analysis and probability.
- 29
Linear algebra in infinite dimensions.
- 30
Undergraduate algebra again, deeper and faster, plus modules and representations.
- 31
Uses algebra to tell spaces apart.
- 32
Calculus on curved spaces.
- 33
Written or oral exams, usually in analysis, algebra, and topology or geometry. Passing them marks the move from student to researcher.
- 34
You narrow to one area. Sample widely before you commit.
- 35
Getting from textbooks to the research frontier.
- 36
The most important decision of a PhD.
- 37
Original research and dissertation
Years on problems nobody has solved. This part cannot be self-studied in the usual sense; it needs an advisor and a research community.
Before you start
- How long it takes
- At a school pace the whole plan takes about 24 years: about 15 before university, 4 at university and 5 to 6 for a PhD. An adult can move through the first three bands far faster. From Proofs and discrete maths onward, the steps mostly don't compress.
- Who the early steps are for
- Steps 1 to 7 are written for the adult who is helping, so their checks say what the learner can do. From step 8 on, the steps speak to the learner.
- The real gate is the proofs course
- Everything before it is computing answers. Everything after it is proving statements. Most people who stall, stall there.
- The last stage isn't self-study
- You can learn through the graduate core from books, but original research needs an advisor and a programme.