Is consistent dual n-back training likely to improve the performance of analysis (mathematics) students? Suppose you isolate a class of undergraduate mathematics majors who are about to enroll in a semester of real analysis (rigorous, proof-based calculus) and split them into two groups. Then, you schedule the first group to study n hours of real analysis every week and the other group to study (n-m) hours of real analysis every week and perform m hours of n-back training (assume that n and m are reasonably well tuned; m is probably much smaller than n). 

Also assume that the second group does not take shortcuts in doing the training (where certain patterns are exploited).


Which group do you conjecture to perform better in the analysis
course and why?
To what degree will one group perform better and why?
Does having a wider short-term memory horizon of numbers (which I
assume n-back produces) result in being able to keep track of more
variables and rigorous proof steps in a sharper way (working memory)
and in faster comprehension of mathematical concepts (fluid
intelligence)?
Does the training result in a larger attention span and therefore to
better grades?
I assume that the group that  spends 100% of their time studying real analysis and 0% of their time doing n-back training will do best in any subsequent real analysis course.

Cognitive skill acquisition does not generalise all that much (for a review see VanLehn, 1996). Transfer is often limited. I'm sceptical of any claims that short term training can lead to far reaching cognitive change as implied by n-back training. 

The demands on working memory of problems in a domain get less as the knowledge and rules implied by a domain get better acquired. Over time students will start to recognise patterns, and will learn to apply rules specific to the domain. Domain specific practice is probably the best way to acquire that knowledge. And in general, the more time that is spent on learning that knowledge, the more knowledge that will be acquired. 

Of course, much could be said about creating an optimal learning environment for real analysis. For example, consolidation of prerequisite knowledge, providing practice at the right level of difficulty, providing feedback to the learner, providing a good mix of presentation of content with problem exercises, distributing practice over time, and so on are all important components of an effective learning environment. However, these have nothing to do with n-back, and everything to do with good pedagogy. In particular, if you want to teach something other than real analysis, but your aim is effective performance in real analysis, then you should examine what are the relevant skills and prerequisites for effective performance in real analysis. And combine this with an examination of student difficulties and knowledge gaps.

Empirical Evaluations

@Ofri Raviv pointed out two references (Shipstead et al, 2012; Melby-Lervag et al, 2013) that provide reviews of working memory training and are generally critical of its utility.

Melby-Lervag and Hulme (2013) did a meta-analysis of working memory training studies:


  To be included in the review, studies had to be randomized controlled
  trials or quasi-experiments without randomization, have a treatment,
  and have either a treated group or an untreated control group.
  Twenty-three studies with 30 group comparisons met the criteria for
  inclusion. The studies included involved clinical samples and samples
  of typically developing children and adults. Meta-analyses indicated
  that the programs produced reliable short-term improvements in working
  memory skills. For verbal working memory, these near-transfer effects
  were not sustained at follow-up, whereas for visuospatial working
  memory, limited evidence suggested that such effects might be
  maintained. More importantly, there was no convincing evidence of the
  generalization of working memory training to other skills (nonverbal
  and verbal ability, inhibitory processes in attention, word decoding,
  and arithmetic). The authors conclude that memory training programs
  appear to produce short-term, specific training effects that do not
  generalize.


References


VanLehn, K. (1996). Cognitive skill acquisition. Annual review of psychology, 47(1), 513-539.
Shipstead, Zach; Redick, Thomas S.; Engle, Randall W. Psychological Bulletin, Vol 138(4), Jul 2012, 628-654. doi: 10.1037/a0027473 
2   
Melby-Lervag, M., &amp; Hulme, C. (2013). Is working memory training in children effective? A meta-analytic review. Developmental Psychology, 49, 270-291. PDF