\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}
