Optimal daily allocation between vocabulary and grammar in second- language study: a normative control model
This paper employs a normative optimal-control model to demonstrate that the widely recommended 98% vocabulary coverage target is optimal in only a small fraction of parameter scenarios, with the true optimal allocation driven primarily by the free rehearsal provided by reading, vocabulary retention rates, and the study horizon rather than grammar-specific parameters.
Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
Every day, millions of people sit down with a foreign language book or a language app, trying to decide how to spend their limited time. They face a persistent dilemma: should they spend their hour memorizing new words, or should they focus on understanding the rules of grammar? For decades, a widely accepted piece of advice has guided them. It suggests that to read a text comfortably, a learner needs to know about 98 percent of the words in it. This number has become a standard target, a finish line that promises fluency. But this rule of thumb treats language learning as a simple matter of counting words, ignoring the complex reality of how the brain actually holds onto information. It assumes that words and grammar are just two buckets of knowledge that can be filled at will, without considering that the brain forgets things at different speeds, or that reading a book itself might help you remember words without you even trying.
A researcher named Tian-Yu Wang set out to test whether this 98 percent rule is actually the best strategy. Instead of asking learners what they do, Wang built a mathematical model that simulates a single learner trying to maximize their reading comprehension over a set period. This model treats language learning as a balancing act. It accounts for the fact that when you read, you naturally encounter words you already know, which helps keep those words fresh in your memory without any extra effort. In contrast, the model assumes that reading does not automatically help you remember grammar rules; to keep grammar sharp, you must actively study it. The model also includes the reality that we forget things over time, and that words and grammar work together to create understanding, but they cannot simply replace one another. By running thousands of simulations with different assumptions about how fast people forget, how long they have to study, and how effective their reading is, Wang explored what the truly optimal daily schedule looks like.
The results of these simulations challenge the long-held belief that aiming for 98 percent word coverage is the universal goal. In the vast majority of the scenarios tested, aiming for that high number was not the best strategy. The model found that the 98 percent target was only the optimal choice in a tiny fraction of cases—less than 5 percent of the time. This high target only made sense when three very specific conditions were met simultaneously: the learner had to be reading a great deal (which provided free memory maintenance for words), they had to have a very long time horizon to study, and they had to have a natural ability to hold onto words for a long time. For most learners, who do not meet all these strict conditions, the model suggests that the best strategy is to stop adding new words much earlier and shift their focus to grammar.
Perhaps the most surprising discovery was that the specific number of words a learner should aim for is not a fixed constant. The study showed that the ideal target depends entirely on how one measures word frequency and coverage. Because different researchers use different methods to count words, the "perfect" number of words to learn changes depending on which counting method you use. In some simulations, the optimal coverage was around 47 percent, while in others, it was closer to 80 percent. This means there is no single magic number that applies to everyone. Instead, the study identified what actually drives the decision. The most important factor is the "free rehearsal" that comes from reading; the more a learner reads, the less they need to spend time deliberately memorizing new words. The second most important factor is how long a learner can remember a word before forgetting it, followed by the total amount of time they have to study. Surprisingly, the specific rules of grammar and how easily words can be swapped for grammar rules had almost no effect on the decision of when to stop learning words.
The study also looked at the rhythm of learning. It found that the best approach is not a constant, back-and-forth switching between words and grammar. Instead, the optimal pattern is usually a single, distinct phase. A learner should build up their vocabulary first, then switch their focus to grammar for the remainder of their study time. Occasionally, if they start to forget too many words, they might briefly return to vocabulary for a short period to top up their memory, but this does not turn into a repeating cycle. This "build, switch, and replenish" pattern appeared in about 10 to 14 percent of the scenarios, but never as a continuous loop.
Ultimately, this research suggests that the one-size-fits-all advice to memorize thousands of words to reach 98 percent coverage is likely inefficient for most people. The model indicates that learners should tailor their goals to their own situation. If a student reads a lot and has a long time to learn, they can afford to aim higher. But if their reading is limited or they forget words quickly, they should stop adding new words sooner and focus more on grammar to get the best reading comprehension. The study does not offer a new number to memorize, but rather a new way of thinking: the best strategy is not a fixed target, but a flexible plan that responds to how much free practice reading provides and how well the learner retains what they have already learned.
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