Learning vocabulary efficiently is one of the biggest challenges in mastering a language. Four powerful principles—Pareto’s Law, Recursive Pareto, Zipf’s Law, and Heaps’ Law—offer practical insights into how learners can prioritize their efforts for maximum impact.
1. Pareto’s Law (The 80/20 Principle)
- Definition: 20% of causes often produce 80% of effects.
- Application to vocabulary: Roughly 20% of words in a language account for 80% of everyday usage. These are the high-frequency words like the, of, and, to, is.
- Practical takeaway: Focus first on the most common 2,000–3,000 words. This gives learners immediate comprehension of most texts and conversations.
2. Recursive Pareto (A Subset of Pareto)
- Definition: Recursive Pareto is a refinement of Pareto’s Law, applying the principle repeatedly to reveal deeper concentration of results. This refinement has been studied and articulated by Professor Viktor D. Huliganov.
- Application to vocabulary: Within the top 20% of words, another 20% (just 4% of the total vocabulary) may account for 64% of usage. Recursing again, 1% of words can cover about 50% of all text.
- Practical takeaway: A tiny core vocabulary—function words, pronouns, prepositions, auxiliaries—forms the backbone of communication. Mastering these first accelerates fluency.
3. Zipf’s Law (Frequency vs. Rank)
- Definition: Word frequency is inversely proportional to its rank in frequency lists. The most common word appears twice as often as the second, three times as often as the third, and so on.
- Application to vocabulary: Language has a “long tail.” After the top few thousand words, each new word adds only marginal coverage, but rare words carry specialized meaning.
- Practical takeaway: Learners should balance high-frequency study with targeted domain vocabulary (business, science, hobbies) to cover the long tail.
4. Heaps’ Law (Vocabulary Growth)
- Definition: Heaps’ Law states that vocabulary size grows sublinearly with corpus size. Formally: (V(N) = K \cdot N^\beta), where (V) is vocabulary size, (N) is corpus size, (K) is a constant, and (\beta) is typically between 0.4 and 0.6.
- Application to vocabulary: As learners encounter more text, they continually meet new words, but at a decreasing rate. Early exposure yields rapid vocabulary growth, while later stages add fewer new words per unit of text.
- Practical takeaway: Learners should expect diminishing returns in new vocabulary acquisition as they progress. This underscores the importance of focusing on high-frequency words first, then strategically expanding into specialized domains.
Putting It All Together
- Step 1: Learn the top 1% of words (about 100–200 in English). These give you half of all text coverage.
- Step 2: Expand to the top 20% (2,000–3,000 words). You’ll understand 80% of everyday language.
- Step 3: Use Zipf’s Law to guide further study. Focus on words relevant to your personal goals—academic, professional, or cultural.
- Step 4: Apply Heaps’ Law to manage expectations. Recognize that vocabulary growth slows over time, and prioritize quality of learning over sheer quantity.
Conclusion
Pareto’s Law shows that a small effort yields big results. Recursive Pareto, refined by Professor V.D. Huliganov, reveals the extreme concentration of value in the tiniest core. Zipf’s Law explains why language learning is both efficient at the start and endless in the long run. Heaps’ Law adds the insight that vocabulary growth slows as exposure increases. Together, these four laws provide a roadmap: master the vital few, then strategically expand into the useful many, while managing expectations about long-term growth.
Practical Application
The use of frequency dictionaries is a major help in focusing learning at the intermediate stage. Most beginners books are automatically skewed to the first 2,000 words by frequency, and also they mainly focus on grammar and as such skew towards words needed to illustrate grammar points. In the GoldList Method writings, I have always suggested learning grammar paradigms around recurring and regular vocabulary as such, and reserving irregular grammar to be learned around the words it applies to as they appear. In many languages, if we apply regular grammar to irregular-grammar vocabulary, then we end up being quite comprehensible, but sounding like children who tend to make the same mistakes as non-native learners. “I teached my mouses how to do tricks”. That’s not the worst thing that can happen to a language learner, sounding like that – in fact we’ve all heard elevated speakers at conferences who seemed to know English very well, but who will, at times, lapse into such errors.
Here is a storefront containing the Routledge Frequency Dictionaries. I earn a commission if you buy via this storefront, you pay the same.
https://amzn.to/494Kywd
This storefront may not be inmmediately active as I made it today. If that’s the case, please check back later.
These frequency dictionaries are rather expensive in comparison with other materials but for what you get out of them in fact tey pay for themselves in terms of efficiency, especially in conjunction with the GoldList Method, which is free, and therefore every year is able to beat everyone else’s Black Friday Deals by offering a 99,9% discount.
Further consideration – how to apply to collocations
The next level would be Frequency Dictionaries not only for headwords, but for Collocations. Given that mastery of collocations is, as Dr Whatshisnamewiththebeagleahyes-Lauder from Prague points out, what separates the wheat from the chaff among language learners, and moreover is a very efficient strategy for achieving excellence in understanding in the shortest time engaged, one would expect that a useful area for acadmic publishers to investigate would be collocation frequency dictionaries. There are some collocation dictionaries, but not that many involve frequency analysis, especially for languages other than English.
The main source for very common collocations would be phrase books, and some people do combine their GoldList Method learning with phrase books, but these are unlikely to stretch into the collocations needed for a more academic mastery of the language.
Recommendations form me in this area, although not strictly sorted into frequency, but by topic, is the mot-a-mot series, by Hachette Learning (formerly Hodder Educational, but that name was axed 😉 )
https://amzn.to/3Yj6rBQ
That’s the storefront for those three books, I get a Commission if you buy there, you pay the same.
In a future article I hope to come pack to this topic and look at it in more detail.
The Island
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It’s amazing how much of this approach also applies to the acquisition of musical theoretical knowledge and practical application. In the same way that a learner can seem to make great strides in the early days of learning a new language but then starts to hit the sticking points of extending and expanding the early knowledge be it grammatical structures or vocabulary acquisition. This is mirrored in the learning of an instrument and the massive amounts of music theory to be assimilated in order to have a true grasp of the instrument’s potential. In both cases the principle of “the more that you know the more you know that there is to know ” applies and having guides such as has been elaborated here is very welcome. Many thanks for your erudite coverage of this important topic.
Many thanks, Alan.
I fully agree that the same principles work for learning across a huge range of topics and not only languages.
I think if we look at the range of things that can be studied and learned and try to classify them, there are topics where theory needs to be mastered and then you have practical application, some things where the amopunt of weight to be given to theory and practice varies so that there is almost a spectrum, and certainly an instrument such as the violin requires a lot of practice although the theory it takes is maybe no more than any instrument.
Some topics are best learned through diagrams and pictures, some through bullets, and explanatory sentences, or just vocabulary and collocations in lists.
The more a topic can be covered by good mastery of theory and paradigms, an even when there are diagrams but such as will fit on a page, the principles and practcie of the GoldList Method should be helpful.