In examples like these you are assuming that the insulation is such that there is no significant transfer of heat to the surroundings on the experimental timescale. Your teacher is wrong. The system itself is increasing or decreasing in temperature.
If the reaction is exothermic, heat is given out. In an isothermal set-up, this heat would escape to the surroundings. In an adiabatic (no heat exchange) set-up, this heat can't escape, so it heats up the system. In effect, the heat content of the system remains the same, but some is converted from chemical energy to thermal energy. Or to put it another way,
Q = ΔHrxn + mCpΔT = 0
Therefore ΔHrxn = -mCpΔT
Thus a rise in temperature means an exothermic reaction and a fall in temperature means an endothermic reaction. This has nothing to do with how you measure the temperature.
As to work, chemists and physicists have different conventions as to whether positive work is work done by the system or work done on the system. Make sure you know which convention you are working with.