The three lies (by omission) are:
1. This solution is how I solved it. If you look at problem solving as A* search then the solution is not what was done to solve it. Learning mathematicians are supposed to learn to search trough the space of mathematics by examples of solutions. Each step presented on the blackboard is a step in the right direction, omitting steps in the wrong direction, and backtracking in the initial efforts to solve (or even define) a problem.
2. This idea is abstract and has nothing to do with reality. Mathematics is a distillation process of abstraction. In this purification process all preconceptions of reality are filtered out, so the result is rigorous and universal. This often leads to a denial of any connection to reality. (A well known counterexamples is Conway's Analysis of go games, leading to the surreal numbers.)
3. This is the only and right way to do it. You are presented with a single solutions, definitions or axioms, omitting alternative definitions. This get better in higher education (think of how many ways there are to define numbers).
This is not what it looks like doing math. It leads to frustration in students who learn (wrongly) that math is about doing everything right the first time, intuitive thinking is wrong, and basically only about what you are not allowed to do (as the feedback they get are highlights of mistakes in exams).
Our task should be cleaning that up and reteach exploration, reteach that math - like any universal language - can describe everything however you want, finding solutions is trial and error, intuition is good and fast but fuzzy and can mislead you, to not stop with a solution but explore around it more for better understanding, and while we have good cause to define things as we do, it's not necessarily the only way to do things (looking at you, axiom of choice).
To use an analogy: Otherwise we stick to teaching orientation to scouts by fastest satnav routes.