> Synthesis problems and time pressure don't feel fair together.
If you really want to grade the students its fair to put very hard problems on the exam to separate out mastery from merely very good understanding or recital.
Hard problems introduce noise to the grades. A student who gets a hard problem right could be good or lucky. In order to grade the students properly, you need multiple problems of similar difficulty. If you have both ordinary and difficult problems in the exam, it's probably long enough that it should take the entire day.
Undergraduate exams tend to be short. Which means that a perfect grade should be interpreted as "meets the expectations".
Which means that a perfect grade should be interpreted as "meets the expectations".
None of the mathematics exams I wrote during undergrad at Waterloo (as a math major) would fit that description. Nearly every single one of them had midterm grades with unimodal distributions centred below 70%, tending toward 60%. Typically, only 1-5 people in the class (of 100-200) would score a perfect grade. In upper year pure mathematics courses it was common to not have any perfect grades (in a class of about 20-30).
Mathematics is notorious for exams like that. But if you look at the reasons why people fail to get a perfect grade at undergraduate level, it's almost always due to honest mistakes or because they didn't learn what they were supposed to learn.
In my experience, studying mathematics is a bit weird. If you are ready to learn a topic, it's probably the field where you can get top grades with the least effort. But if you can't learn something with reasonable effort, hard work is unlikely to help. Doing something else and trying again after a few months might help.
If you are ready to learn a topic, it's probably the field where you can get top grades with the least effort.
Depends on your school. At my school, someone who was not super-talented in math but who works hard and is actually smart about studying is a 70s-80s student. The students who got 100s were basically IMO-level elite mathematics kids who were heavily recruited by the school and given full ride scholarships.
The course exams were essentially designed so that a "meets expectations" was a final grade of 65%. A grade of 100% is someone they were looking to recruit into research internships, grad school, and potentially a tenure-track position.
What skill are you grading? Can you expect some students to be able to churn out the problems? Are they doing design work? Is it more inventive or more creative? Is there more than one solution? How rigorous is the solution?
My point is that you can design an exam to test for speed and ability to do the day to day well. You prepare for this exam with lots of practice. There are a few problem types but you might not have seen every one. There's a basic level of extrapolation you're expecting.
You can design an exam for abstracting theory, and you prep for this exam by doing the hardest problems you can find, reading more theory, etc. You might not have seen anything like this but the test is to use the tools you're given in a class well. The test is the extrapolation.
I had theory courses where I expected the latter and practical courses where I expected the former.
There's more than one type of mastery; I bet your math professors would get destroyed by gifted 14 year olds in algebra competitions that involved arithmetic and timing. Do those 14 year olds not have more mastery?
Undergraduate exams tend to be short. Which means that a perfect grade should be interpreted as "meets the expectations".