FIELD GUIDES
Make a plan for learning math.
Find your algebra or calculus gaps with original diagnostics, work through examples, and check your own solutions. Other roadmaps help you plan the next step.
A Calculus Roadmap: From Functions to Applications
An original four-chapter unit with a readiness check, worked examples, practice, and explained answers on limits, derivatives, and integrals.
Start the full unit: diagnostics, examples and practice
Start with the language of functions
Calculus does not start with derivative rules. It starts with functions, graphs, rates of change, and limits.
Before learning derivatives, make sure you can read linear, quadratic, exponential, logarithmic, and trigonometric graphs.
Build limits, derivatives, and integrals first
Limits describe approaching behavior, derivatives describe instantaneous change, and integrals describe accumulation. Study series later, after these foundations are secure.
Pair every rule with a graph and an application such as velocity, area, marginal cost, or probability density.
Practice deliberately
For each topic, solve concept questions, standard computations, and applied modeling problems.
When you miss a problem, record whether the error came from algebra, concept confusion, or notation.
Resources for this path
- OpenStax Precalculus 2eFollowing a full precalculus sequence with written examples and exercises.
- 3Blue1BrownUnderstanding why an idea works before formal practice.
- Paul's Online Math NotesChecking a specific method or working through a solution step by step.
- OpenStax Calculus Volume 1Reading a complete explanation after an introductory video or lecture.
A Linear Algebra Roadmap: Structure Behind Matrix Computation
A guide to connect vectors, matrices, systems, determinants, and eigenvectors with geometric meaning.
See matrices as transformations
The central idea is not memorizing determinant formulas, but seeing how matrices move, stretch, rotate, or compress space.
Matrix multiplication is best understood as applying one linear transformation after another.
Recommended concept order
Study vectors and linear combinations, matrix multiplication, linear systems, bases and dimension, determinants, then eigenvalues and eigenvectors.
For every concept, keep three views connected: algebraic computation, geometric interpretation, and application.
Common traps
A determinant is not only a formula; it measures area or volume scaling.
An eigenvector is not only a solution artifact; it is a direction that stays on the same line after transformation.
How to Choose Math Learning Resources
A framework for combining courses, videos, textbooks, exercises, and tools without constantly switching resources.
Compare four specific courses and textbooks
Define the goal first
The same topic can serve exams, research, engineering, or curiosity. The goal changes the best resource choice.
Exam goals need structured courses and feedback; exploration benefits from visual videos; applied goals need tools and projects sooner.
Choose a main path and supporting tools
Choose one textbook or course as your main path. Try problems after each small section, then use a video, diagram, or tool when you need another explanation.
Videos explain motivation, textbooks build structure, exercises expose gaps, and tools help test conjectures.
Evaluate quality
Strong resources explain why ideas work, not only what steps to follow.
If a resource lacks examples, exercises, prerequisites, or error-correction paths, treat it as a supplement rather than the main path.
Resources for this path
- Khan AcademyRebuilding fundamentals and practicing one skill at a time.
- OpenStax College Algebra 2eLearners who want a stable reference alongside a class or self-study plan.
- MERLOTDiscovering specialized activities or teaching material beyond a standard course.
Rebuild Algebra: From Equations to Functions
An original four-chapter unit that diagnoses gaps, explains equations and functions, and checks understanding with practice and worked answers.
Start the full unit: diagnostics, examples and practice
Find the actual gap first
Use a few equation and graph questions to check signed numbers, fractions, ratios, and linear equations. If one step is shaky, practice that step before moving on.
Record whether each mistake came from arithmetic, notation, translating words into an equation, or reading a graph. A targeted review is usually more useful than replaying an entire course.
Move from equations to graphs to functions
Practice linear equations and inequalities, then connect solutions to number lines and coordinate graphs. Next, learn slope, intercepts, and function notation.
Once you can explain slope and intercepts in words, move to quadratics, polynomials, and exponentials. Sketch each new family and describe how its graph changes.
Check transfer each week
Use the same idea in a word problem, a graph question, and a symbolic calculation instead of repeating only one exercise type.
If you cannot move between those representations, strengthen the current topic before opening the next chapter.
Resources for this path
- Khan AcademyRebuilding fundamentals and practicing one skill at a time.
- CK-12 FlexMathReviewing one school-level concept before moving to harder exercises.
- Math Is Fun: AlgebraA quick plain-language explanation before trying a practice set.
- OpenStax College Algebra 2eLearners who want a stable reference alongside a class or self-study plan.
Start Statistics: From Data to Inference
Learn to read data before moving into probability, sampling, and statistical claims.
Describe data before testing claims
Begin with data sources, variable types, graphs, and distributions. Explain what a mean, median, and outlier each tell you.
For any chart, ask where the sample came from, how the axes were drawn, and whether important groups were omitted.
Connect probability to sampling
Use simple experiments to understand conditional probability and independence, then distinguish a population from a sample and sampling error.
Before computing a confidence interval or p-value, state the question, the assumptions, and how the data were collected in plain language.
Use a real question to check the conclusion
With each new method, use a small data set and write down the result, what it supports, and what it cannot establish.
Keep correlation separate from causation, and distinguish statistical significance from practical importance.
Resources for this path
- OpenStax Introductory Statistics 2eA course-like reference when learning statistics for the first time.
- OpenIntro StatisticsLearning statistical reasoning with a full text and companion material.
- StatQuest Video IndexGetting intuition before returning to textbook examples and data exercises.
- Mathigon PolypadTesting a geometric or algebraic idea by moving objects yourself.