Flight Instructor reported an NMAC event in the traffic pattern requiring evasive action to avoid a collision.

Narrative

After completion of a touch and go on XXR we turned crosswind at the AIM (Aeronautical Information Manual) recommendation of 300 before pattern altitude. Landing after us was Aircraft Y; a Cessna 172. Aircraft Y was issued a call out prior to turning crosswind advising them us our position in the down; clearance to land number 3; and instructions to follow us. The Cessna 172 acknowledged the transmission and included they were searching for traffic. immediately after this exchange the Cessna appeared to turn crosswind nearly directly into us. Tower then issued a traffic alert to the Cessna which stated traffic was less than a mile. The Cessna Acknowledged the transmission with 'searching' and did not seem to have a sense of urgency or take action. At this point we decided we needed to increase our spacing by completing a 360 turn to reposition us on downwind and advised tower of our turn. After resolving the conflict we re-entered downwind and reported the aircraft got in very close proximity to us. At ZZZ the published pattern altitude on XXR is 1300 feet. I typically fly the pattern at 1500 for extra margin and standardization particularly during instruction. Had we been at 1300 a collision could have occurred.