In fact, p-values are now finally 'out of fashion' as well: http://www.nature.com/news/psychology-journal-bans-p-values-1.17001. Null hypothesis significance testing (NHST) produces little more than a description of your sample size.(*) Any experimental intervention will have some effect, which is to say that the simple null hypothesis of 'no effect' is always false in a strict sense. Therefore, a 'non-significant' test simply means that your sample size wasn't big enough; a 'significant' test means you collected enough data to 'find' something.
The 'effect size' represents an attempt to remedy this, by introducing a measure on the natural scale of the problem. In medicine, where treatments always have some effect (even if it's a placebo effect), the notion of a 'clinically meaningful effect' is introduced to guard against the 50% prior probability that a 'treatment' will be found to have 'a (statistically) significant positive effect' (however minuscule) in an arbitrarily large study.
If I understand the nature of your work, Clarinetist, then at the end of the day, its legitimate aim is to inform actions/interventions that improve education in the schools under your purview. Thus, your setting is a decision-theoretic one, and Bayesian methods are the most appropriate (and uniquely coherent) approach.
Indeed, the best way to understand frequentist methods is as approximations to Bayesian methods. The estimated effect size can be understood as aiming at a measure of centrality for the Bayesian posterior distribution, while the p-value can be understood as aiming to measure one tail of that posterior. Thus, together these two quantities contain some rough gist of the Bayesian posterior that constitutes the natural input to a decision-theoretic outlook on your problem. (Alternatively, a frequentist confidence interval on the effect size can be understood likewise as a wannabe credible interval.)
In the fields of psychology and education, Bayesian methods are actually quite popular. One reason for this is that it is easy to install 'constructs' into Bayesian models, as latent variables. You might like to check out 'the puppy book' by John K. Kruschke, a psychologist. In education (where you have students nested in classrooms, nested in schools, nested in districts, ...), hierarchical modeling is unavoidable. And Bayesian models are great for hierarchical modeling, too. On this account, you might like to check out Gelman & Hill .
: Robert, Christian P. The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. 2nd ed. Springer Texts in Statistics. New York: Springer, 2007.
: Gelman, Andrew, and Jennifer Hill. Data Analysis Using Regression and Multilevel/hierarchical Models. Analytical Methods for Social Research. Cambridge ; New York: Cambridge University Press, 2007.
For more on 'coherence' from a not-necessarily-beating-you-on-the-head-with-a-Bayesian-brick perspective, see .
: Robins, James, and Larry Wasserman. “Conditioning, Likelihood, and Coherence: A Review of Some Foundational Concepts.” Journal of the American Statistical Association 95, no. 452 (December 1, 2000): 1340–46. doi:10.1080/01621459.2000.10474344.
(*) In , Meehl scourges NHST far more elegantly, but no less abrasively, than I do:
Since the null hypothesis is quasi-always false, tables summarizing research in terms of patterns of “significant differences” are little more than complex, causally uninterpretable outcomes of statistical power functions.
: Meehl, Paul E. “Theoretical Risks and Tabular Asterisks: Sir Karl, Sir Ronald, and the Slow Progress of Soft Psychology.” Journal of Consulting and Clinical Psychiatry 46 (1978): 806–34. http://www3.nd.edu/~ghaeffel/Meehl(1978).pdf
And here's a related quote from Tukey: https://stats.stackexchange.com/a/728/41404