What makes a glowing splinter glow and a burning splinter burn

This question bugs me for years since I started learning the oxygen, carbon dioxide and hydrogen tests. I know you can make a glowing splinter burn when doing the oxygen test due to enough oxygen. But why won't it burn in ordinary air?

So, my question is what factor differentiate a glowing splinter and a burning splinter

• I'm guessing you are defining a burning splinter as having visible smoke, while a glowing splinter does not. Is this correct? If so then both splinters are burning. The burning in the glowing splinter is complete. – Agriculturist Jan 24 '16 at 20:22

Ordinary air is made up of only approximately $21~\%~\ce{O2}$, the primary component being nitrogen. According to the partial pressure rule, at standard laboratory temperature and pressure, this is a partial pressure $p(\ce{O2}) = 21~\mathrm{kPa}$. In a test tube filled with pure oxygen, e.g. from water electrolysis, the partial pressure of oxygen is $p(\ce{O2}) = 100~\mathrm{kPa}$ — five times as much.
According to the rate law, the combustion of wood ($\ce{C_{$n$}H_{$m$}O_{$l$}}$) with oxygen will be dependent of oxygen’s partial pressure somewhat. Thus, we can assume at least a five-fold increase in reaction rate. A five-fold rate increase also means a five-fold increase of enthalpy per time-frame which means five times the heat is liberated which means that the glowing splinter will get hotter much more quickly and finally ignite.