That is a [North American BC-1][1], a North American NA-16 derivative, the predecessor of the  [North American T-6 Texan][2]

[![North American BC-1 with marking 31][3]][3]

<sub>Source: [nationalmuseum.af.mil][4] </sub>

Some observations that helped to determine the precise model:

 - The fuselage is covered with fabric. Later models used metal
 - The trailing edge of the rudder is rounded, later models had a straight rudder 
 - The bottom corner of the rudder is squared, some other models are rounded (e.g BT-9) 

 - The fuselage shows the crest of the [USAAF's Technical Training Command][5]: 

[![The crest of the USAAF's Technical Training Command, similar to the one shown on the fuselage of the aircraft][6]][6]

<sub>Source: [Picturing Meteorology][7] </sub>

The aircraft belonged likely to Chanute Field Technical Training Command (Illinois), located some 200 miles south of Madison. 


  [1]: https://en.wikipedia.org/wiki/North_American_NA-16
  [2]: https://en.wikipedia.org/wiki/North_American_T-6_Texan
  [3]: https://i.sstatic.net/wMERk.jpg
  [4]: https://www.nationalmuseum.af.mil/Upcoming/Photos/igphoto/2000538187/
  [5]: https://en.wikipedia.org/wiki/Technical_Division,_Air_Training_Command
  [6]: https://i.sstatic.net/c6SLC.jpg
  [7]: https://picturingmeteorology.com/home/2017/12/22/gi-dinner-christmas-1943