I'd be furious to get some lofty theoretical synthesis problem on an engineering exam where the other problems are grindy analysis of "looks like concept you know but expand into a form you haven't seen and then will take 1/n time on your n question exam". I suppose these are already 1 extension in, so 1 more logical extension isn't that far off
Expecting a final exam consisting only of calculation and memorization type questions is wholly inappropriate in a mathematics course full of proofs.
Off-topic, I know, but I always disliked the time pressure of exams. All the bad scores I got were down to not feeling quite right at the time of the exam: maybe I didn't sleep well, the food I ate did not quite sit with me, had a fever, etc. None of these things actually impair too badly but if the time is short, there is an impact.
I suppose it's similar to sports where you have to, at least somewhat, adapt on the go. Many teams botch a game here and there. But... your final exam? Botch it and you're screwed. It just does not feel right.
maybe this is good for weeding out students who would not be a good fit for high stakes speed math.
It's really no different from a software dev interviewee who can't solve basic programming questions without being handheld and coached through the entire process.
I didn't get those exams in math courses. Engineering courses I did get those where some of the point of the test is the speed of your pattern matching ability (did you use design skills to reason about it or did you brute force a solution).
Math you're already extrapolating theory and you need ample time for that.
That’s what the exam is testing for. If you need unlimited time to write the proof, then you haven’t studied the material yet, and should be expected to perform poorly on the exam.
It’s like the difference between someone who has never touched React before the interview, and someone who has been programming with React for several months. You expect the person with several months of experience to know how to solve problems with React several times faster than the person who doesn’t know it at all. If you’re hiring and looking for someone with a minimum of 1 year of React experience then you shouldn’t expect the person to take a full day to do basic tasks with React (especially if you the interviewer or anyone else on your team can do those tasks in just a few minutes).
Maybe it has been a while since you have worked on something difficult, then? Difficult for you, I mean. Maybe the difficult things are now easy.
> If you need unlimited time to write the proof, then you haven’t studied the material yet, and should be expected to perform poorly on the exam.
I mean, maybe. I remember some classes and exams where I had simply never seen the topic before, and the problem relied on having picked up on metapatterns that weren't explicitly taught during the class and applying them in novel ways on the spot. Though my guess was that at least some of those problems were markers for "if you can one-shot this, we want to know".
> If you’re hiring and looking for someone with a minimum of 1 year of React experience then you shouldn’t expect the person to take a full day to do basic tasks with React.
If you're talking about the basics, then yeah I would agree. I guess I don't really know what counts as a synthesis problem in this context, and if we were looking at specific examples it would be more obvious why one might be fair and another might not be. I've heard that line about TAs being touted as being able to solve the tests in just a few minutes, yet somehow unsolvable problems fell through the cracks to the students in the same class more than once in mine.
That's the crux of it, really. For a lot of students, the kind of proofs we do in undergrad pure math courses are very difficult. For a math professor? They're extremely basic.
There are levels to this. If a typical high school math student is at level 2 or level 3 (of math ability), a 3rd year pure math course may be at level 10, but the professor teaching the course is at level 100. Okay, so these are made up numbers, but I hope they help to illustrate.
For someone who has never tried to write a computer program in their life, writing Fizzbuzz may be quite difficult. For a working lead developer with 10 years of experience? Quite basic.
I quite fondly remember a situation with a two question three hour open everything exam where I stared at the 1st problem and relevant equation for 20 minutes because I had no idea what I was looking at and was wondering if I should just leave and drop out.
Then I realized that a bunch of terms could cancel out and the problem became 3 lines trivial. The professor later admitted he wanted to give a "easy" question that you couldn't just copy out of your notes and had to think and to spend the rest of the exam on the actually difficult one.
Being able to do deep thought quickly isn't a valuable skill, and you don't pick it up on the way ti mastery with lots of practice like you do the speed for rote problems
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.
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".
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).
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.
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.