Date: 202012
Local time of day: 1801-2400
Location: ZZZ.TRACON, US
Aircraft: Small Aircraft; High Wing; 1 Eng; Fixed Gear
Operator: Personal
Operating under: Part 91
Flight plan: IFR
Flight phase: Cruise
ATC / advisory: TRACON ZZZ
Reporter organization: Government
Reporter function: Approach; Instructor
Human factors: Distraction; Confusion; Training / Qualification
Second reporter function: Trainee; Approach
Anomaly: ATC Issue All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT
Detector: Person Air Traffic Control
When detected: Pre-flight
Result: Air Traffic Control Issued New Clearance
Contributing factors: Human Factors
Primary problem: Human Factors

Narrative: Training on combined sectors; moderate traffic; two practice approach aircraft in satellite airspace all other traffic in departure sectors.Aircraft X came off ZZZ on the published missed approach for the RNAV RWY XX approach; he checked in and was radar identified while in the published left turn towards the missed approach fix; ZZZZZ.  The trainee subsequently verified the pilot's previous request to return to ZZZ1 on an IFR clearance.  The trainee cleared Aircraft X direct ZZZ and instructed the aircraft to maintain 2;000'.  At the time; the aircraft was within a 2;000' MVA [Minimum Vectoring Altitude].  I believe both the trainee and I anticipated that the pilot would make a right turn direct to ZZZ1; not recognizing that a continued left turn may be more appropriate because the aircraft was already in the published left turn on the missed approach.  With the anticipation of a right turn in mind and the aircraft in a 2;000' MVA my attention as the OJTI turned to the other departure traffic; which was going to require the trainee to merge and weave multiple departures off multiple airports (ZZZ2/ZZZ3/ZZZ4) to get them on course and through our airspace.  This was an area the trainee had shown struggles in in previous sessions as well as earlier in this training session; so I was attempting to stay ahead and have a plan of my own if needed.  Upon reviewing the incident; I became aware that in the moments following Aircraft X's clearance I was suggesting an alternative course of action for one of the trainee's departures in an effort to illustrate a way to expedite getting a departure on course which also got them out of the way of succeeding departures quicker.  Unfortunately; this conversation kept my focus on the departure traffic as opposed to the continued turn that Aircraft X made into the higher MVA.  Aircraft X subsequently continued a left turn towards ZZZ1 and entered a 2;500' MVA at his assigned altitude of 2;000'.  The trainee was aware of the incident almost immediately.  The trainee issued a climb to 2;500' to Aircraft X when the aircraft was only a third or a half a mile inside the 2;000' MVA.  I was not aware of the situation until I heard the trainee issue the climb to Aircraft X.  I subsequently requested that we be relieved.  The trainee and I debriefed; after which I informed management of the incident when I requested time for myself and the trainee to review the incident and fill out our report.  To my knowledge; Aircraft X climbed to the MVA and subsequently landed at ZZZ1 without further incident.Numerous things could have prevented this occurrence.  Had the trainee climbed Aircraft X to 3;000' instead of descending them to 2;000' they would have been above the MVA.  Had the trainee left the aircraft on the published missed approach for even a few miles they would have climbed above the MVA and a subsequent clearance may have included a more appropriate altitude.  Had the aircraft made the anticipated right turn instead of continuing in a left turn they would have remained in the 2;000' MVA.  Had the trainee issued the clearance to ZZZ1 via radar vectors; I believe they would have assigned a vector that would have remained within the 2;000' MVA and avoided the situation.


Second narrative: Recertification time on [positions] (departures) and Article 67 training on[positions](south satellite). Aircraft X departed ZZZ on the published missed from the RNV runway XX  approach. Requested clearance to ZZZ1 airport. I observed Aircraft X in a 2;000ft MVA [Minimum Vectoring Altitude] and told pilot to maintain 2;000' and proceed direct ZZZ1; which is north east of ZZZ. A direct flight path would have kept the aircraft away from the 2500ft MVA east of their position. Pilot was in a left hand turn though; and continued a left hand turn direct ZZZ1. I was expecting a right hand turn direct. The left hand turn put the aircraft 2-3 miles further east than planned and inside the 2;500ft MVA. I climbed the aircraft to 2;500ft and let them continue on course. I; as the controller should have been more aware of the chance the aircraft could have entered higher MVA. I should have climbed Aircraft X to 2;500ft before sending direct ZZZ1 to account for a left or right turn.


Synopsis: Instructor and Re-certification Controller reported a MVA [Minimum Vectoring Altitude] violation when an aircraft under their control turned the wrong anticipated direction leading to the violation.