B737-700
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 154; Flight Crew Type 7000
Human-Machine Interface; Situational Awareness; Training / Qualification
ATC Issue All Types; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued New Clearance; Flight Crew Became Reoriented; Flight Crew Returned To Clearance
Human Factors
While descending on the ZZZZZ Arrival into ZZZ Airport we were given a heading off of the arrival and a speed and altitude.  We were previously cleared to descend via the ZZZZZ [arrival].  Approach gave us a heading and a descent to 8;000 feet and slow to 220 knots.  The First Officer (FO) dialed in the heading; set the speed and engaged Level Change.  Shortly thereafter Approach re-cleared us direct to ZZZZZ [fix] and descend via.  The First Officer selected direct ZZZZZ set 4;000 feet (the bottom altitude on the arrival) and engaged LNAV.I did not notice that we were still in Level Change or if they selected VNAV.  I looked out the window and commented that the arrival really brought [us] close to the terrain on this arrival.  I crosschecked the VNAV PATH on my display and noticed we were well below the path. I told this to the Pilot Flying and they immediately went to Vertical Speed; set the min altitude for ZZZZZ1 and started a slow climb.  I called Approach to let them know and I believe at the same time they were calling us.A few seconds later we were all on the same page; so to speak; and we were given the clearance descend via and landed in ZZZ uneventfully. It wasn't immediately apparent to me that we were below the path as I have flown into ZZZ1 many times; and ATC step-descends us fairly close to the same hills.  I have only been into ZZZ once before over a year ago.  We never received any Terrain Alerts and there was no yellow or red on my ND.  The radar altimeter was not registering yet.  Approach did inform us that we had gone below their minimum vectoring altitude.  There was no unnecessary conversation going on that I can recall; I just missed the fact the we did not re-engage VNAV when re-cleared to descend via. More diligence in confirming MCP selections on the FMA.
On the ZZZZZ Arrival in to ZZZ we were descending via; then Approach changed the clearance to a heading and descent to 8;000 feet. I selected the heading; airspeed; and altitude assigned using Level Change. We were then cleared direct ZZZZZ [fix] and descend via again. I selected 4;000 feet as the bottom of the arrival and continued in Level Change instead of reselecting VNAV. Before reaching ZZZZZ1; we realized we were off the VNAV PATH and immediately stopped the descent and started a climb back to path. We told ATC what we were doing and they said we were below the minimum vectoring altitude and assigned 7;000 feet. Once we reached 7;000 feet; we were back on path and continued the arrival. There was a breakdown of [CRM]. I failed to recognize the improper descent when selecting a new altitude.
B737 flight crew reported they inadvertently descended below minimum vectoring altitude during arrival due to not recognizing that the MCP had been incorrectly set. The flight crew subsequently climbed back to the correct altitude.