Mathematics is like a language, although technically it is not a natural or informal human language, but a formal, that is, artificially constructed language. Importantly, educators use their natural everyday language to teach the formal language of mathematics. At times, however, instructors encounter problems when the technical words they use, as formal parts of mathematics, conflict with an everyday understanding or use of the same word, or related words. This article discusses this problem, including some examples, and offers some suggestions for handling the difficulties. The author offers four cases to discuss conceptual complexity and apparent contradictions in mathematics language: (1) graphed functions; (2) temperature variations; (3) southbound sailing ships; and (4) flowing rivers. The author also offers three recommendations for handling difficulties: (1) Be alert for possible confusion in word meanings and usage; (2) Use student talking to negotiate and construct correct understanding; and (3) Examine new terms, symbols, techniques, diagrams, and technical "apparatus". (Contains 4 figures.)