So I know that when hooking two batteries in series, the voltage increases and the current stays the same. Well, I am doing a project where there are two power sources (a supercapacitor and a battery) in series (there are two pathways for the current). There is a micro-controller, which always keeps the circuit at 12 V (Battery Voltage = 12 - Supercapacitor Voltage). In actual testing, the peak current flow of the battery decreases as the battery voltage decreases. I understand arbitrarily why, but I would like math equations that show this case.
That description doesn't match the circuit. There is a DC/DC converter.
Do you mean the microcontroller controls the DC/DC converter output voltage to keep the motor voltage
constant?
In that case as the converter output current equals the motor current, but the battery current doesn't as
the converter converters power.
I am doing a project where there are two power sources (a supercapacitor and a battery) in series (there are two pathways for the current).
This doesn't make sense, if they are in series there is only 1 path for the current. Please supply a schematic.
There is a micro-controller, which always keeps the circuit at 12 V (Battery Voltage = 12 - Supercapacitor Voltage).
How does it do that?
In actual testing, the peak current flow of the battery decreases as the battery voltage decreases. I understand arbitrarily why, but I would like math equations that show this case.
I think you lack the knowledge of the basic maths involved in electrical circuits. You need to understand Ohm's Law and Kirchhoff's circuit laws
These laws are fundamental to all electrical circuits and if you learn them a lot of the mysteries of electricity will become clear. In many cases applying both laws to a circuit will provide whatever answer you are looking for.
So I know that when hooking two batteries in series, the voltage increases and the current stays the same.
This is not correct. For any component, the current will increase with an increase in voltage.
This does not include certain special active circuits.
JohnRob:
This is not correct.
I think they meant that the current available from two cells in series is the same as from one.
hannah_mackinlay:
I think they meant that the current available from two cells in series is the same as from one.
He might but he is still wrong. With two batteries in series the source impedance doubles, thus limiting the maximum current available from the battery to half what it was with only one battery.
However, what he probably means is that the mAh or capacity of the battery stays the same. He is still not 100% right but is close enough.
This circuit is nonsense!
Grumpy_Mike:
With two batteries in series the source impedance doubles, thus limiting the maximum current available from the battery to half what it was with only one battery.
That is also - sadly - totally confused. What were you thinking? 
Double the source impedance at double the voltage (presuming both batteries the same) means the same current. If the load is simply resistive at least.