C172 flight instructor reported the need to take evasive action to avoid a collision with opposite direction traffic.

Narrative

My student and I were climbing up to an appropriate altitude to conduct CFI demonstration stalls after getting through most of the required maneuvers for his proficiency (clearing turns had been completed for each maneuver with consistent scans to the Multi-function Flight Display (MFD) for traffic as well as outside). We were on flight following and in constant communication with ZZZ Approach who was giving us active traffic advisories. During the climb; we witnessed a tail dragger (tailwheel) aircraft overtake us from the opposite direction and flew directly over us within approximately 250'-300' vertically (judging with visual distance). Evasive action was taken immediately on our end to avoid the aircraft; and because of that I could not catch the tail number. This aircraft did not have a transponder/ADS-B out and was not visible on the traffic map (which was configured appropriately per the checklist). Immediately following the incident; I verbally verified with ZZZ Approach that we had not lost radar contact; and they confirmed that they could still see us. The tail dragger continued to not show up on the traffic map and proceeded southbound. I thoroughly debriefed with the student the importance of outside vision; proper visual scanning techniques; and how we need to account for what others do in the practice area. We also talked about the consequences of a lack of transponder/ADS-B use. Practice area congestion is growing significantly in this area and this incident illustrates the importance of collision avoidance in all phases of flight; even with appropriate precautionary measures in place. It also illustrates the importance of the usage of ADS-B out and appropriate transponder usage and how the presence of this type of equipment on the other aircraft; as well as radio communication from the other aircraft; could have helped prevent this incident. Reach out to me with any additional questions.