To my view, the most effective strategies deal with anxiety as a collective - and thus political and cultural - issue. The answers provided by adamblan and Mandy Jansen are quite to the point, in that regard, and what I'll add here is just a complement.
Anxiety can be defined as overreaction to falsely perceived risk. Risk taking is socially divided, just like labor is. Risk avoidance behavior is commonly present in children. It can obviously be reinforced positively or negatively, depending on what kind of parental guidance is given, and these decisions are heavily informed by the cultural background of the parents and/or caregivers. It has been argued that biological characteristics are relevant to determining what degree of exposure to danger can a person tolerate (so there would be inclinations based on gender), but anthropological research tells us that human behavior is extremely plastic, so universal truths are scarce here.
That said, what kind of risk is presented to a child in learning mathematics? There is, of course, the social risk of being outcast by failing to fit properly into the assigned stereotype. This fear has been brilliantly addressed by previous answers, so I'll skip it here.
There is possibly, though, a different kind of risk perception, more subtle, and with deeper roots than social prejudice. To put it in simple terms: the human brain consumes a lot of energy, and streams of thought that are (considered) energy efficient tend to be strongly preferred. This also has a social signature: anti-intellectualism is the way societies have to indicate to the individual that to be "lost in thought" is dangerous to herself and to the larger group. Being considered the way of thinking that is more liberated from experiential constraints, mathematics may have been historically associated with the biggest of threats.
In other words, if thinking "complicated" thoughts has been considered, in itself, wasteful and possibly dangerous to the cohesion of the group - and thus risky - then thinking mathematically has to be extremely problematic. This kind of prejudice affects more directly, obviously, those that are not in a position of power. In this case, though, it won't be enough to address cultural stereotypes, because the anti-mathematical prejudice is largely unconscious, and even more pervasive. I'd say that it lies outside of what can be dealt with in a pedagogical environment.
This predisposition can cause enormous suffering, inside and outside of the classroom. It is a clinical problem, and a social one. The issue is particularly urgent on our present time, when the ability to think mathematically is, maybe for the first time in human history, not an accidental feature (virtue or menace), but the most valuable resource that we have, for our continued survival on this planet.