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.
@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
- 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
- Melby-Lervag, M., & Hulme, C. (2013). Is working memory training in children effective? A meta-analytic review. Developmental Psychology, 49, 270-291. PDF