Ace
Short answer: study maths by doing problems, not rereading. Start with worked examples, then gradually solve more steps yourself; mix different problem types in one session so you learn to recognise which method fits; explain each step's reason; and revisit topics over several weeks. Keep a list of the mistakes you make and redo those problems later.
Why does maths need a different study approach?
In many subjects you can get by remembering facts. In maths, knowing a formula is only the start: you need to recognise when to use it and carry out the steps reliably. That's a skill, and skills come from practice. Rereading solutions creates a dangerous illusion: every step makes sense when someone else has already done it.
How should you study maths?
1. Start with worked examples
When a topic is new, studying fully worked examples is more effective than struggling with problems straight away. Students who studied worked examples learned algebra faster and made fewer errors than those who only solved problems (Sweller & Cooper, 1985).
2. Fade the examples gradually
Don't stay on examples forever. Move from complete examples to examples with the last step missing, then two steps missing, then whole problems on your own. This gradual "fading" helps learners move smoothly from studying to solving (Renkl et al., 2002).
3. Mix problem types
Textbook exercises usually group problems by method, so you always know which one to use. Exams don't. Mixing different types of problems in one practice session feels harder, but it teaches you to choose the right method, and it led to much better test results in one study (Rohrer & Taylor, 2007).
4. Explain why each step works
After each step, ask yourself why it's allowed or why it was chosen. Students who explain worked examples to themselves understand them more deeply and solve new problems better (Chi et al., 1989).
5. Test yourself on methods and definitions
Before a problem set, try writing out a theorem, definition or method from memory. Retrieval practice strengthens memory far more than rereading (Roediger & Karpicke, 2006), and knowing the tools by heart frees your attention for the problem itself.
6. Keep a mistake list
Every time you get a problem wrong, note why: a concept gap, a careless sign error, choosing the wrong method. Redo those problems a few days later. Your mistakes are the most useful part of your practice.
7. Space your practice out
Return to old topics every week or two rather than finishing a chapter and leaving it. Spaced practice leads to much better retention than massing it all at once (Cepeda et al., 2006). Our spaced repetition schedule shows how.
For formula-heavy topics, Ace can turn your own lecture slides into practice questions, including interactive simulations where you change the values in a formula to reach a target, with every answer showing the line from your notes it came from.
What does a good maths study session look like?
- 5 minutes: write the key formulas and methods for the topic from memory, then check.
- 15 minutes: study two worked examples, explaining each step to yourself.
- 30 minutes: solve problems on your own, mixing in a few from earlier topics.
- 10 minutes: check your answers, add mistakes to your list, and redo one.
See also how to study physics.
Frequently asked questions
How do I get better at maths fast?
Do more problems, and spend more of your time on the ones you get wrong. Mixed practice and a mistake list improve skill faster than redoing problems you can already solve.
Should I memorise formulas?
Yes, the core ones, but also understand where they come from and when they apply. Knowing when to use a formula is what exams test.
How many practice problems should I do?
Enough that you can solve a new problem of each type without looking at an example. Quality and variety matter more than a fixed number.
Sources
- Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89.
- Renkl, A., Atkinson, R. K., Maier, U. H., & Staley, R. (2002). From example study to problem solving: Smooth transitions help learning. Journal of Experimental Education, 70(4), 293–315.
- Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498.
- Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182.
- Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249–255.
- Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354–380.
