Because words have multiple meanings, language users must often choose appropriate meanings according to the context of use. How this potential ambiguity affects first language learning, especially word learning, is unknown. Here, we present the first large-scale study of how children are exposed to, and themselves use, ambiguous words in their language learning environments. We tag 180,000 words in two longitudinal child language corpora with word senses from WordNet, focusing between 9 and 51 months and limiting to words from a popular parental vocabulary report. We then compare the diversity of sense usage in adult speech around children to that observed in a sample of adult language, as well as the diversity of sense usage in children's own productions. To accomplish this we use a Bayesian model-based estimate of sense entropy, a measure of diversity that takes into account uncertainty inherent in small sample sizes. This reveals that sense diversity in caregivers' speech to children is similar to that observed in a sample of adult-directed written material, and that children's use of nouns---but not verbs---is similarly diverse to that of adults. Finally, we show that sense entropy is a significant predictor of vocabulary development: children begin to produce words with a higher diversity of adult sense usage at later ages. We discuss the implications of our findings for theories of word learning.