12

I think it helps to make it abundantly clear whether or not you would expect an average student in your course to come up with such a "stroke of genius". If you're presenting something that might induce such a "WTF?!" moment, be very up front about it. I even like to put myself on the students' level if it's a particularly ingenious stroke, and will say ...


8

As per the comments above, I'm going to tackle this explicitly for discrete math and set theory being a shared course. What follows is some strategies I've tried and how well they worked (or didn't work!). Group Work. Asking the students to do group work (either in class or on the homework), and assigning groups can really help this issue. Imagine that ...


8

Solutions that could not have been concieved by the average student It is important to realise that many more solutions can't be concieved by the average student than we usually realise. Even solutions we think are quite routine can seem like strokes of genius to students new to the area. Students often say to me, "I can follow the lecturer's proof/...


7

Yes, I think most arguments are "routine" in the sense of being very similar to others in the relevant context, while now-and-then there is something "out of the blue". The latter can be deconstructed a bit, by admitting that all the goofy attempts that failed were not shown, but only the one that succeeded. Thus, possibly, if all the other attempts were ...


6

A strange question. Almost every time you show something new to the students, it will be something that would never have occurred to them. Try to remember the first time you saw the derivation of the formula for the quadratic proved, or the $\epsilon - \delta$ definition of limit, or Cantor's diagonal argument. So this isn't the exception, it is the norm in ...


5

If you are not going to formally split the classes (with different exams, class names, etc.), I would advise just teaching to the basic level. [But I like Adam's idea to add ungraded extra hard problems for the students who just elect to do that for edification and enjoyment.] There is also nothing wrong with just letting the advanced kids know that this ...


4

If the course is required for graduation, and if the curriculum does not adequately prepare the students, then I strongly feel you should teach to the less prepared students. I am teaching a capstone course based on knot theory. It is the last course before graduation for math and mathed majors, and it is the only section. The mathed majors have not had ...


4

In the case of Desargues' theorem, and many others, the solution is not so unexpected when you know the history. Desargues was one of the inventors of projective geometry, whose original goal was to develop a mathematical theory of perspective drawing. He also worked as an architect, where he prepared accurate perspective drawings from blueprints. If you've ...


3

Get them to work on problems in small groups (4 to 6 students) and walk around watching/listening and giving advice. This works best if they are working at a white/black board. If you don't have enough board spaces, dark whiteboard markers work well on windows, and butcher's paper is ok at a pinch. You need to make your expectations clear on what is ...


3

Perhaps the real solution to the conundrum is to take out one of the restrictions. Limit the material (it is just impossible to cover much material in any depth in 50 minutes), stretch time (perhaps by asking the students to prepare beforehand, or leave homework(ish) questions for later), or match abilities (ask the more quick students to help out others, ...


1

If the knowledge really is required for the class (and can't be supplemented with a short "fill in" for those who lack it), and there is no way to redefine the evaluation as @BrianRushton says, the course is just badly defined (or has the wrong requisites). Even in the mentioned cases I believe you should think of defining the course anew. Note that for ...


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