Although recessions in the United States are officially determined by a committee of the independent and nonpartisan National Bureau of Economic Research (NBER)—usually long after the fact—a rough-and-ready definition of a recession is the period of time when real GDP (the actual amount of goods and services produced) is falling, if it falls for a period of at least six months. This period of time when real GDP is falling is very different from the period when real GDP is “in the hole” compared to its peak, let alone the period when real GDP per person is in the hole.
For the same level of GDP,
GDP/Population goes down when Population goes up,
and Population in the United States is growing.
Americans are used to real GDP not only growing, but keeping up with the growth of population, plus a couple of percent more each year. This link shows a graph of real GDP per capita since the beginning of the Great Recession. Since at least the beginning of 2008, real GDP has been doing quite a bit worse than Americans are used to.
Catherine Mulbrandon has made a great set of graphs on her Visualizing Economics website.
Here is a graph showing the history of the logarithm of real per capita GDP in the U.S. since 1871. With the logarithm on the vertical axis, the slope of the curve shows the percentage growth rate.
Here is a graph showing the history of real per capita GDP in the U.S. since 1871. You can see what the miracle of compound growth does.