SMA X NON ADHERENCE TO ATC INSTRUCTIONS; CUT Y OUT ON FINAL APCH AND Y MADE GAR.

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X HAD BEEN CONDUCTING TKOFS AND LNDGS AND WERE CLBING UPWIND TO TFC PATTERN ALT. TWR WAS GETTING OVERLOADED AND REQUESTED WE CHANGE TO OVERFLOW FREQ (128.75). WE CHANGED OVER AND ASKED FOR FULL STOP ON RWY 21L. TWR WANTED R TFC FOR US SO WE MADE R X-WIND. SEVERAL OTHER AIRPLANES WERE IN THE PATTERN (APPROX 6) AND A COUPLE MORE WERE ON THE R 45 DEG ENTRY FOR RWY 21R. ON DOWNWIND; WE AGAIN ASKED RWY 21L FULL STOP; BUT THE CTLR WAS UNABLE TO PROVIDE RESPONSE BECAUSE OF THE DIFFICULTY SEQUENCING TFC. THE CTLR COULD NOT VERIFY POS OF AT LEAST 2 ACFT IN THE PATTERN. WE WERE TOLD TO LAND RWY 21R #4. WE ACKNOWLEDGED WITH NEGATIVE CONTACT RWY 21R. I HAD 2 ACFT IN SIGHT ON FINAL BUT COULD NOT FIND THE THIRD. WE CONTINUED DOWNWIND BUT STILL HAD NEGATIVE CONTACT. MY STUDENT FINALLY MADE CONTACT BUT IT WAS WITH THE SECOND ACFT ON FINAL. THE SECOND ACFT WAS ABEAM OUR POS SO MY STUDENT MADE THE TURN TO BASE. JUST AS WE MADE THE TURN; I SAW THE THIRD ACFT Y ON LONG FINAL (3-4 MI FINAL). THE SPACING WASN'T THAT FAR OFF AND IF HE HAD SLOWED UP WE WOULD HAVE HAD SIGNIFICANT SPACING BTWN US; BUT HE DECIDED THE BEST COURSE OF ACTION WAS TO GAR. WE CONTINUED OUR APCH AND LANDED. HOWEVER; DURING OUR TURN TO BASE THE CTLR ASKED US FOR OUR POS AND TO ROCK OUR WINGS. I BELIEVE THE CTLR LOST THE ACFT SEQUENCED THIRD TO LAND AND THOUGHT WE WERE THE THIRD ACFT. THE ACFT MAKING THE GAR WAS THEN ASKED FOR HIS CALL SIGN BECAUSE THE CTLR THOUGHT THAT HE WAS US. THE CHANGE TO OVERFLOW FREQ MESSED EVERYBODY UP INCLUDING THE CTLR. WE WERE FLYING DOWNWIND; APPROX HDG 030 DEGS; FACING THE SUN AND COULD NOT MAKE OUT OTHER TFC COMING OPPOSITE DIRECTION. THIS WAS MY SECOND INSTRUCTIONAL ACTIVITY AT AN ARPT THAT GETS EXTREMELY BUSY; AND I DID NOT EXPECT THE DOWNWIND TO GET SO EXTENDED OUT (4 MI). I SHOULD HAVE ASKED THE CTLR TO CALL OUR BASE SINCE I DIDN'T HAVE TFC. I SHOULD HAVE NOT LET THE STUDENT MAKE THAT TURN UNTIL I HAD THE TFC. I WISH THE CONGESTION WASN'T SO BAD BECAUSE IT REALLY MAKES IT DIFFICULT TO TEACH AND ACKNOWLEDGE EVERYTHING. I WILL WORK HARDER TO BE MORE AWARE. I LEARNED MORE THAN MY STUDENT AGAIN!!