1. Learning mathematics is a skill. If you teach/learn this skill, the effect of other barriers will be reduced. 2. Learning to communicate is a skill. Instead

1. Learning mathematics is a skill. If you teach/learn this skill, the effect of other barriers will be reduced. 2. Learning to communicate is a skill. Instead of saying "the fraction of people who have both skills" you could say "the fraction of people willing to learn both skills"... then it becomes more clear that if someone is teaching poorly it is because they are not putting in the time to acquire the skill of communicating/teaching. I don't think people are talking about problems with great communicators in this thread. The complaints are usually about below average communication. Anyway, I am grateful to have been taught by both great communicators and great mathematicians. The great communicators have a huge impact, don't underestimate it. The issue is that you might not even understand the impact of what the great mathematician is telling you, unless it is communicated in a clear way. All I remember from one geometry class is the pictures... and they make the fundamental idea crystal clear.