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My friend, who is currently teaching an abstract algebra course, recently provided me with an example of an incorrect proof that I think showcases a few types of faulty reasoning that are common among students. I'm sharing it here with his permission. Proposition: Suppose $R$ is a ring and $I$ is a prime ideal in $R$. For all ideals $J$ and $K$ in $R$, if $... 1 This is simple, but effective. Consider doing a weekly, period-long test, every Friday. I remember having that in HS for pre-calc and calc. At first was surprised by the frequency. But ended up admiring the pedagogic effect. Tests are some of the most effective drill (because of the drive for prep and the higher stakes...and the learning from the ... 2 The concepts behind limits are actually very important to engineering (in the form of error/precision analysis), but are rarely phrased that way. Given a function$f$, we can imagine an engineering situation where there is some desired range of outputs from the function, but the engineer has control over the value of the inputs of the function. If$f(c) = L$,... 4 It is probably not the place of mathematics educators to decide what mathematics courses engineering majors should take. But a good reference point is ABET accreditation. Over 600 universities in the US have ABET accredited engineering programs. We should defer to the professionals who set these standard and assess outcomes. Here is a description of their ... 1 No, it's definitely not "necessary". I'm not an engineering major, but roomed with one, did a general engineering minor, and worked in/around mechanical, nuclear, mining and chemical engineering (had electrical on staff too). Passed my EIT and was at one time, about to take the PE (mechanical) exam. Most engineers in the workplace don't even use ... 4 Well it doesn't really feel right to get degrees in engineering and gain years of engineering experience without even knowing what a limit actually is. And even though many engineers will do just fine without having been exposed to the rigorous definition of a limit, some engineers will need to be familiar with rigorous definitions/proofs if they ever pursue ... 2 Expanding a bit on my comment, there is (1) a new textbook available for project-based intro stats, (2) an online syllabus describing a course based on community projects, and (3) an academic paper concluding that "the project-based course ... provides a promising model for getting students hooked on the power and excitement of applied statistics." ... 0 There's a recent book by Nicholas A. Scoville: Discrete Morse Theory, AMS Student Mathematical Library, 2019. Publisher's page: https://bookstore.ams.org/stml-90 MAA review: https://www.maa.org/press/maa-reviews/discrete-morse-theory It looks very accessible to the undergraduate - some background in proof writing is recommended as is some linear algebra, but ... 1 That's pretty normal to have a lot of ground to cover. Even with the more friendly books, it still ends up being a lot of concepts and formulas. Given this, I think you sort of have to make your peace with the idea that kids will not master everything, especially in the long term. I probably wouldn't try some fundamental change to improve things since it ... 1 That book gets ripped pretty hard on Amazon. The Dover texts by Trudeau and Chartrand are supposedly easier and friendlier, per reviews. And will be cheap, since Dover. If you want to develop familiarity and speed, I would certainly not eschew (i.e. I would do) problems that are repetitive. You'll get more practiced at the concept. Also more practiced at ... 3 I'd say yes, and I'd go with binary if you had to do any one alternative base simply because it's so relevant to computing and technology, and in my experience teaching discrete math, once you understand binary, related bases like octal and hex are pretty simple to pick up. But I don't think the converse is necessarily true. Ideally I'd like math and CS ... 4 I can't answer the OP's questions, but I'll just mention that a local 6th-grade teacher (in the U.S.) has a successful unit on base-$5\$. It is mentioned in the recent article below. Sometimes he called it "star-fish math." James Henle. "Math for Grades 1 to 5 Should Be Art." Mathematical Intelligencer. 42, pages 64–69, Dec. 2020. ...