AN SMA INSTRUCTOR PLT CUT OUT ANOTHER ACFT IN THE TFC PATTERN. NMAC.

Narrative

MY STUDENT AND I ENTERED THE PATTERN ON A L DOWNWIND TO RWY 32 AS INSTRUCTED. WE WERE VERY BUSY IN THE PLANE AND WE WERE HAVING A HARD TIME GETTING OUR RADIO CALLS IN BECAUSE THERE WERE SO MANY PEOPLE ON THE FREQ AND IN THE PATTERN. ATC SAID WE WOULD BE FOLLOWING SMT TFC ON A 2 MI R BASE. APPARENTLY I DIDN'T HEAR ALL OF THE CALL; I HAD TFC ON THE L BASE AND I THOUGHT THAT WAS WHAT I WAS FOLLOWING. WE TURNED TO THE BASE LEG AND WERE ABOUT TO TURN FINAL WHEN ATC ASKED IF WE HAD THE TFC WE WERE FOLLOWING IN SIGHT. I REPLIED THAT WE HAD TFC ON SHORT FINAL. ATC THEN TOLD US TO MAKE AN IMMEDIATE HARD L 360 DEG TURN. AS I STARTED THE TURN I SAW THE SMT BEHIND US; A COUPLE HUNDRED FT AWAY. THE SMT PASSED US AND LANDED; AND WE FOLLOWED AND LANDED. I BELIEVE FACTORS WHICH CONTRIBUTED TO THIS SIT INCLUDE THE FACT THAT MY ATTN WAS DIVIDED BTWN RECEIVING AND XMITTING RADIO CALLS; CORRECTING FOR A DIRECT 16 KT XWIND; INSTRUCTING MY STUDENT; AND WATCHING ALL THE TFC IN THE AREA; WHILE SETTING UP FOR LNDG. BECAUSE I WAS SO BUSY; I WAS NOT LISTENING AS CAREFULLY AS I COULD HAVE BEEN. FORTUNATELY ATC SAW THE SIT AND INFORMED ME; UPON WHICH I TOOK IMMEDIATE ACTION TO CORRECT THE SIT. TO PREVENT A RECURRENCE IN THE FUTURE; I WILL HAVE TO BE MORE CAREFUL WHEN LISTENING TO ATC AND LOOKING FOR TFC; ESPECIALLY WHEN I HAVE AN INCREASED WORKLOAD IN A BUSY ENVIRONMENT.