WRT the second design.  With today's software why not build classroom into the analysis. Is it really practical to split a class into two methods of teaching?

Art Kendall, Social Research Consultants

I agree with Art. The second option (randomize within classrooms) doesn't per  se get rid of the classroom effects anyway. (Classroom effects could be from  teachers, or certain people self-selecting into classrooms, or contamination between students among others I'm sure.) So I'm not sure what you gain from it besides head-aches in implementation. (On-line classes it seems doable, in person off-hand it seems very difficult.)

Also two presents a confound with the status quo of teaching that could have undesirable effects. Say for instance both experiment and control do worse than prior data would suggest. It could be from splitting the class in half it causes adverse effects.
I'd suggest picking up any book on multi-level modelling. Experiments of  this kind were one of the main motivations for development of the field. Controlling for classroom isn't difficult with current software (as Art mentioned), and it can potentially add useful insight into the experimental  analysis. (It certainly is more true to how it would be implemented in  practice anyway.)

Andrew W

I also do not see how students in the same physical class could be taught different methods.  
A variation I would consider would be to have the same teachers administer both methods to different classes. Such a design would allow you to evaluate the teacher-BY-method interaction effect. You would then likely have a greater risk for diffusion of methods.

I agree that you ought to learn about and/or consult with an expert in research designs for hierarchical designs

Ryan Black

This makes much more sense in terms of eliminating the bias due to a specific teacher. Although it does seem that some measure of implementation would be useful since the same teacher may excel at one method and not do well at the other (e.g. s/he prefers/understands/resonates with one method over the other.)

Melissa Ives