In the first chapter of this thesis we have shown that the natural and mathematical language interpret truth, logical connectors, definitions, and even whole sentences differently (compare implicatures in speech with implications in mathematics). With this, we have shown that superficial understanding of certain mathematical terms may lead to reliance on their meanings in everyday speech, and with this, to the development of misconceptions (e.g. the phrase ”getting closer” in the explanation of limits). In the second chapter, we focused on mathematical thinking and proofs, and we gave a few examples of different types of proofs. Even though the examples are simple, they clearly illustrate the rigor of mathematical proof in comparison to naive argumentation. In the third chapter we took a closer look at specific concepts that are familiar to nonmathematicians (number, infinity, line, and probability), and ways in which mathematics generalised and formalised intuitive concepts. This departure from intuition towards the abstract may make understanding those concepts in the context of mathematics more difficult. Finally, in the fourth chapter we analysed several common misconceptions – the illusion of linearity, problems that arise from misunderstanding limits and probability, and misuse of statistics. Common to all of them is that they arise from relying on intuition or improper generalisation of previously learned mathematics.