*Some broad answers to the question posed.*

For a college student in a math class (I’m thinking of a first year course, but this could apply to students in more advanced courses) what are the distinct challenges that they face (that they might not expect)?

- Contextual Knowledge – Most students will have seen lots of mathematics by the time they reach college and will have developed lots of ideas and practices related to doing math. However, they usually only know this in a specific context and often haven’t been really challenged to apply these ideas in new scenarios (i.e. non-textbook) and in places where they might not work the same way. Thus they don’t always have the ability to recognize what ideas are relevant in a situation or the ability to check if the applied method actually works. The most common instance of this is ‘forgetting’ basic algebra when it is encountered in a calculus class.
- Fragile Knowledge – Related to Contextual Knowledge, but involves applying advanced mathematical processes in a slightly different way, that might test a different aspect of the mathematics. In this case, although they may ‘know’ the process, they don’t know it well enough to actually use it widely. The result is often panic and reverting to some primitive (and often wrong) mathematics. An example of this would be working with rational expressions, and asking questions that are related to questions about whole number fractions, and the student ends up doing things like (12)/(16) = (
~~1~~2)/(~~1~~6) = 1/3. - Variable Overload – Students can handle a certain number of variables in an expression, but when that number is exceeded, they can’t make sense of it anymore. This also extends to the use of nonstandard variable names and Greek (and other non-Latin) symbols. This can be made worse in contexts where the number of variables used is almost always 2 (e.g. x,y), and something with an extra variable comes along. The weirdest instance of this is confusion when facing standard formulas, like V = π r
^{2}h for the volume of a cylinder in a class like Calculus where you’ve worked primarily with (x,y,t). The students may not even know what to do with π, like is it a variable or what? - Abstraction – This is related to variable overload as college math instructors, being math people, are often quick to go to the abstract case; you’d think that we count: 1, 2, 3, n. Students will have been exposed to abstraction and will have some specific relation to it (usually hate), but it will be fragile and in very specific contexts (sometimes limited to abstract versions of well-understood concrete versions) , and usually won’t be something they do on their own or are comfortable with. In most college math classes, abstraction comes early (and often) and it expected that students can connect the specific details of examples, etc. with the general theory.

The big issue is whether the responsibility to deal with these challenges belongs to the instructor or to the student. As with most classroom issues, it probably belongs to both with the instructor responsible for course structure that helps students become aware of these challenges and provide opportunities and feedback to help them make adjustments according to these challenges. The students would have the responsibility to self-assess and make adjustments and accommodations based on their own level of difficulty with the challenges. Both would have the responsibility of not ignoring these challenges, but to face them head on throughout the course.