Note, looking into this is officially preparation for one of my fall classes. Work work work.
Matthew Seymour has done more incredible work in service of the connection game Hex, this time in the form of a beautifully-implemented collection of 500 "white/black to play and win" puzzles of the sort that chess and go enthusiasts take for granted as study tools:
The most recent Mathematical Gazette has some really lovely articles. If you have access to the Gazette at all, I can recommend these two:
Demystifying Beethoven's Große Fuge
(Personally I don't think it needs "demystifying," exactly, but anyway – if you want to spend 50 minutes working through the score, there you go)
A nice page of recent writings about abstract strategy games (mostly connection games:
(Clicking the tags at the bottom of the page leads to many more good, unindexed articles)
For some reason, I feel that when the whole and only point of the question is "think about this reasoning," students pause and take some time to do that. If the question asks for a number, then it's like they have free license to skip the time-consuming "reasoning" step, multiply some things, get a number, and move briskly along to multiplying some different things in the next problem.
Francis Su: "7 Exam Questions for a Pandemic (or any other time)"
My students are taking a quiz with a counting question about hands of cards of a particular type. They have multiple attempts at the question with hints in between, and I can see the responses they've submitted. This gives me a fascinating problem of reverse-engineering the attempted strategies from the numbers that were submitted. It's instructive.
Elemenatry combinatorics lectures or textbooks tend to present mostly, or only, correct solutions to counting problems, which are easily seen to be correct. But when you're learning to count you'll spend most of your time looking at (your own) incorrect solutions and trying to figure out what's wrong with them. Or looking at your own solutions and trying to figure out whether they're correct or not. I dunno, maybe there's a moral there.
When I'm grading proofs by hand I usually run through the stack once marking any zeros or full credit ones and pulling them out, and keeping the ones that will get partial credit insertion-sorted as I go. Once I've seen the full spectrum of mistakes and misconceptions, I go back and assign the partial credit points.
This is tougher to do with electronic submissions.
Herzlich Willkommen in Minnesota!
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