\documentclass{article}
% Did you know?
% Mathmode and things like \hbox actually corresponds to groups!
\begin{document}
\def\foo{Original}
Expect Original : \foo
Expect a + Original: $ a + \mbox{\foo}$
Expect Original : \foo
Expect InBox + Original:  \hbox{\def\foo{InBox}\foo} + \foo
Expect Original : \foo
Expect a + InlineMath: $ \def\foo{InlineMath}  a + \hbox{\foo}$
Expect Original : \foo
Expect a + DisplayMath : \[ \def\foo{DisplayMath}  a + \hbox{\foo}\]
Expect Original : \foo
% Note that text mode inside mathmode groups a 2nd level!
Expect a + InText + InlineMath: $ \def\foo{InlineMath}  a + \hbox{\def\foo{InText}\foo} + \hbox {\foo}$
Expect Original : \foo
\end{document}