# Making coefficient for centered binary variable meaningful?

I'm having a hard time making a centered binary variable in a multilevel model interpretable. Here is what I'm working with: at Level 1, I have a binary variable indicating whether a participant gets free/reduced lunch (0= No; 1= Yes). Participants are nested within schools (Level 2), so to disentangle person and school effects, I centered the free/reduced lunch variable within school (as recommended in various multilevel modeling texts).

The issue that I'm running into now is difficulty interpreting the coefficient for this variable. Each participant's score is now a 0 or 1 deviation from a school-level percentage of respondents that receive free/reduced lunch. For example, the range for my participant-centered lunch variable ranges from approximately .75 (a participant receiving free/reduced lunch in a school where 25% of participants get free/reduced lunch: 1.0 - 0.25 = 0.75) to -.34 (a participant that does not get free/reduced lunch in a school where 34% do: 0.0 - 0.34 = -0.34).

Now I'm not really sure what my fixed effect coefficients mean under this coding scheme. I initially thought that the raw coefficient from my multilevel model for the participant-centered binary variable (b = -1.6) indicates that the predicted change in the dependent variable decreases (on average) by 1.6 between participants that do not receive free/reduced lunch and those that do. That can't be right though because the centering.

Can anybody help me out?