The Nationals are batting .240 as a team and rank 14th in the National League in runs scored and on-base percentage.
Last season, they were 5th in the NL in runs scored.
Eckstein, 40, was hired as the Nationals' hitting coach in October 2008.
Schu, who had a nine-year playing career in the majors, has spent the last four seasons as the Nationals' minor-league hitting coordinator.
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Schu also has experience working with infielder Chad Tracy and recently acquired outfielder Scott Hairston during his time in the D-backs' organization, where he started his coaching career in 1998. Schu later spent parts of four seasons as the D-backs' hitting coach (2004, '07-09) before joining the Nats' organization. Schu's 16 years of coaching experience followed a nine-year playing career, during which he spent time with five clubs, including one season with the Expos in 1996.