New adjunct math professors should prioritize mastering their subject matter and preparing thoroughly for classes to foster student comprehension. Clear communication of expectations on the first day is crucial, including grading policies and available resources. Engaging students through examples, a conversational tone, and interactive techniques can enhance learning and comfort in the classroom. It's important to adapt teaching methods based on student feedback and to be prepared for varying levels of understanding among students. Overall, quality teaching involves patience, preparation, and a willingness to learn from both successes and mistakes.