Writing a mathematical proof is an essential skill for qualified science, technology, engineering, and mathematics (STEM) students, especially those majoring in mathematics, data science, artificial intelligence (AI), and computer science. However, many first-year students complain that it is tremendously difficult for them to learn mathematics well, particularly mathematics proofs. New teaching staff are also often frustrated during their first couple of years of teaching undergraduates mathematics, wondering why something so seemingly simple is so difficult to teach. Although most universities have experienced this problem, there has been little deep reflection and investigation on why learning of mathematical proofs is so difficult for first-year undergraduates and how to mitigate such difficulties. The aim of this paper is to fill such a gap by reflections and investigation which focus on the recognition and learning of the mathematical proof process.