Asking questions is an important element of real-life collaboration on reasoning tasks like question answering. For example, a legal assistant chatbot may be unable to make accurate recommendations without specific information on the user's circumstances. However, large language models are usually deployed to solve reasoning tasks directly without asking follow-up questions to the user or third parties. We term this problem task-oriented asking (TOA). Zero-shot chat models can perform TOA, but their training is primarily based on next-token prediction rather than whether questions contribute to successful collaboration. To enable the training and evaluation of TOA models, we present a definition and framework for natural language task-oriented asking, the problem of generating questions that result in answers useful for a reasoning task. We also present fact-level masking (FLM), a procedure for converting natural language datasets into self-supervised TOA datasets by omitting particular critical facts. Finally, we generate a TOA dataset from the HotpotQA dataset using FLM and evaluate several zero-shot language models on it. Our experiments show that current zero-shot models struggle to ask questions that retrieve useful information, as compared to human annotators. These results demonstrate an opportunity to use FLM datasets and the TOA framework to train and evaluate better TOA models.