I have instructed for community colleges and adjuncted for major universities for just over fifteen years now.
In all my time doing this job, I've found that the worst manifestation of so-called "mathphobia" inevitably appears in my Statistics courses when we move to the material dealing with inferential methodology.
I have used many different source materials (different textbooks, websites, Youtube videos, example problem writeups posted in addition to lecture, etc.) but it invariably follows the same pattern each semester: one or two of the more vocal students will eventually interrupt me mid-lecture and will claim that they do not understand anything that's going on. For reasons that are probably psychological and similar to "giving up," they almost always claim they haven't understood anything in the class at all, including very elementary concepts like measures of center and introductory empirical probability (e.g. from small tables that give counts for just two binomial variables). This will invariably snowball into the whole class claiming the same lack of understanding, even including those who are earning the highest grades in the section. Among all the math classes I've taught, this part of this course presents the toughest psychological barriers I have to help students overcome, and, ironically, since spreadsheet formulas usually crunch the numbers for us on almost all the problems we encounter, this part also represents the least computation-focused part of any of my courses.
After facing this scenario again yesterday, I have reached two possible contributors to this phenomenon:
The processes of confidence intervals and hypothesis testing is difficult and counterintuitive for students learning it for the first time. Even though I've attempted to use analogies (e.g. the "innocent until proven guilty" analogy for hypothesis testing), students have a hard time understanding the concept of using a sample to draw inferences about parameters, and do not understand why error is needed, even if I show them that they already implicitly understand that even the most representative sample still leads to error.
Students are looking at the increasingly involved equations needed to compute standard error, z or t-scores for levels, and for test statistics, and math panic sets in even if I tell them that they will far more often lean on technology for these computations and instead should focus more on actual interpretation.
These issues inevitably pop up to form mass confusion and delayed lecturing schedules, even if I convey multiple times that patience and disciplined practice is the true teacher in learning inferential statistics.
A great example of this phenomenon is the example I used yesterday comparing two population proportions: one student just would not stop arguing with me that .51 and .5 (results for two sample proportions I compared in an example) were different numbers and therefore had to mean that the two proportions were different; the student simply would not accept my explanation that the margin of error for the difference implied that a true parameter difference could in fact be zero with the .01 difference appearing out of acceptable error due to the fact that we are sampling. The idea wouldn't sink in even after I slowly reviewed the idea of inference and error with her to her agreement; when we revisited the problem, she just slammed her fist on the table, declared a .01 difference meant two different numbers again in anger, and (as is usually the case) demanded the age-old question of "where she would ever have to use this stuff."
I apologize for venting a bit, but I guess my central point is to ask if any other math educator finds inferential statistics to be one of the top points of difficulty with students, especially math-phobic ones, and what you do to help students get past the frustration and the tendency to discount the presence of error and the fact that we are estimating (confidence interval) or testing a population claim (hypothesis testing) by use of a representative but much smaller sample. Either I have been consistently teaching the subject wrong for years, or else this should be an ordeal many other teachers face when teaching these concepts to students for their first time.