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Imagine a class which is split completey regarding previous knowledge which is in some way needed for the class.

How can you deal with such a class? How can you - without giving too much workload to those without previous knowledge - bring those students to the level which is needed in order to understand new things - without bothering those who know this already?

(Background: In my university, the future high school teachers don't have to take a class in real analysis and measure theory. But they have to take a class in probability theory together with the bachelor students who did take the real analysis class. Of course you don't need to know everything about measures for the probability theory class, but some background is more than helpful. How much content and methods of measure theory would you repeat?

Another example is a course about partial differential equations where some students might have attended classes in real analysis and functional analysis; and some students haven't attended either of them. Here, you have usally very motivated students who want to take the course but they were not informed in the past.)

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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 many (or perhaps any) formal proof courses and quickly fell behind in the homework. I couldn't just fail half of the class, so I moved towards computational problems, which both groups excelled at.

It wasn't a difference of innate intelligence; they both did great at computations. They just hadn't been adequately prepared.

If the course is not essential, I believe you should set a certain level of requirements and make it very clear at the outset that you expect everyone to have met those requirements. An easy way to do this is to have a pre-test with a required minimum score.

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  • $\begingroup$ Do Math Ed students not typically go through any proof classes? At my school, one is required and I'm planning on taking another (two if I can fit it). $\endgroup$ – David G Mar 23 '14 at 23:31
  • $\begingroup$ @DavidG There are generally a couple of levels of proof classes. Usually there is an introductory course, and then abstract algebra or analysis begins real rigor, and then all classes after that have a similar rigor level. At the universities I've been to, math ed majors typically stop after the one real algebra or analysis course. So they aren't required to complete as many proof courses. $\endgroup$ – Brian Rushton Mar 23 '14 at 23:38
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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 someone familiar with the subject matter and related areas it might come very natural to suppose some knowledge as given, like real analysis and measure theory, and talk in terms of Stieltjes integrals simplifies probability enormously. But not everybody has this knowledge, and for the vast mayority of those who want to use this all those niceties are just superfluous baggage. Just using hand-crafted derivations (even if longer and more restrictive than needed) might well be enough for non-specialists. Besides, once you as much as mention non-Riemann integrals, you'll have lost them for good anyway.

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