How to Learn Mathematics and Science: Beyond Rote Memorisation
Introduction
This article is inspired by Barbara Oakley’s A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra), published by Tarcher/Penguin in 2014 which its central message is that success in mathematics and science is a matter of being gifted with numbers but involves understanding concepts, practicing problem solving, developing useful mental patterns, knowing when to concentrate intensely and dealing with difficult problems. A Mind for Numbers — Penguin Random House
For students studying mathematics, physics, biology, chemistry, engineering, medicine, computer science, or related subjects, rote memorization alone is therefore not enough, although memorizing formulas, definitions and procedures has a legitimate role, but students must understand why a method works, recognize when it applies, test alternatives when a familiar method fails. National Academies described problem solving as a fundamental part of mathematics. National Academies — Data and Computing in K–12 Education: Foundational Competencies
This is also reflected in professional education. ABET criteria for engineering programmes require graduates to be able to identify, formulate and solve problems using mathematics, science and engineering principles, as well as design solutions while considering social, environmental, economic and ethical factors. ABET — Criteria for Accrediting Engineering Programs, 2026–2027 The Association of American Medical Colleges identifies critical thinking, quantitative reasoning and scientific inquiry among the competencies expected of students entering medical education. AAMC — Premed Competencies for Entering Medical Students
Why rote learning alone is insufficient
There is an important distinction between memorisation and learning. Memorisation gives you facts and procedures that can be retrieved quickly, which is useful in mathematics and science but genuine proficiency also requires being able to use that knowledge in unfamiliar situations.
McDermott showed that actively recalling information can strengthen long-term retention and make knowledge more readily retrievable than simply rereading material and also support more flexible use of knowledge in different contexts and retrieval practice finds strong benefits for long-term retention, and retrieval can help learners use knowledge flexibly rather than merely recognizing information by the book. McDermott, “Practicing Retrieval Facilitates Learning,” Annual Review of Psychology Therefore, you should memorise the essentials, understand the underlying principles, and apply them.
Mathematics and science require flexible problem solving
Mathematics is abstract in all of its representations, while the natural sciences constantly move between observations of the physical world and abstract models, mathematical descriptions and theoretical explanations. National Academies emphasizes abstraction and algorithmic thinking because STEM disciplines use abstraction to create generalizable concepts for solving problems across different contexts. National Academies — Data and Computing in K–12 Education
The reason why students should resist the assumption that every problem must be solved in exactly the same way is because of the Einstellung effect, sometimes called a mental-set effect. Previous experience can make a familiar solution come to mind so strongly that a person fails to notice an appropriate alternative. Abraham Luchins demonstrated this phenomenon with water-jug problems, and Merim Bilalić, Peter McLeod, and Fernand Gobet has continued to study how previously learned procedures can interfere with finding better solutions. ScienceDirect — “Why good thoughts block better ones: The mechanism of the pernicious Einstellung (set) effect”
A method can valuable, but knowing when not to use it is equally important.
Focused mode and diffuse mode
Oakley described two complementary approaches: focused mode and diffuse mode. In her framework, focused mode involves concentrating attention on a specific task, such as working through a mathematics problem. Diffuse mode refers to a more relaxed state in which the mind can make broader and less direct connections between ideas. She argues that effective learning often requires volleying back and forth between the two. ( Barbara Oakley — A Mind for Numbers excerpt )
Focused thinking is particularly useful when you are learning a procedure, following a proof, calculating a result or carefully analyzing evidence. It helps you maintain attention on the immediate task, but intense concentration is not necessarily the best state for every stage of problem solving. Oakley argues that diffuse mode can allow a learner to approach it from a different perspective which is consistent with Menon who shows that mind wandering and internally directed thought can be associated with processes such as memory, imagination and creativity. Menon, “20 years of the default mode network: A review and synthesis” — PubMed
A paper named “Default mode network spatio-temporal electrophysiological signature and causal role in creativity” also provides evidence that the brain's default-mode network participates in creative cognition and internally generated thought, although this should not be interpreted as proving that the brain literally switches between two simple “modes” like a machine. Bartoli et al., Brain — Default-mode network and creativity
Allowing your attention to relax can be part of a productive learning process, but unstructured avoidance of work is procrastination. This is known as flexibility in thinking. There is an illusion where you understand something, which can be avoided by active retrieval and problem solving.
Chunk your knowledge
Once you begin understanding a concept, the next challenge is organizing what you know.
Chunking means grouping smaller pieces of information into meaningful units. . Thalmann, Souza & Oberauer has found that chunking can reduce the load placed on working memory because a familiar chunk can be retrieved as a more compact representation rather than treating every individual element as completely separate. Thalmann, Souza & Oberauer — “How does chunking help working memory?” These larger patterns are built from many smaller pieces of knowledge. Chunking is therefore not a replacement for learning details but it allows these details to become organised knowledge.
Mathematics and science can teach habits of mind
The value of mathematics and science therefore extends beyond the formulas contained in a textbook.
Mathematical reasoning teaches people to work with abstraction, structure, quantitative relationships and logical constraints. Scientific inquiry teaches people to ask questions, develop explanations, examine evidence and revise conclusions. These intellectual practices have applications well beyond STEM itself and shows why quantitative reasoning, evidence evaluation and problem solving are useful intellectual tools across modern society.
Conclusion
The strongest students in mathematics and science are not necessarily those who can memorise the greatest information but who can understand, retrieve, connect, question, adapt and solve.
The broader research base shows that effective learning is not one technique, but involves retrieval, organised knowledge, practice, flexible problem solving and the ability to recognize when a familiar strategy is no longer appropriate. McDermott — Retrieval Practice Review Thalmann, Souza & Oberauer — Chunking and Working Memory ScienceDirect — Einstellung Effect
Reference
Oakley, B. (2014). A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra). New York: Jeremy P. Tarcher/Penguin. Publisher record — Penguin Random House
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