STUDENT PLT WITH INSTRUCTOR TOOK OFF AFTER TKOF CLRNC HAD BEEN CANCELED; RESULTING IN A NMAC WITH ANOTHER ACFT MAKING A GAR.

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OUR SCHOOL'S INSURANCE POLICY REQUIRES THAT PRIOR TO STUDENT SOLO; ANOTHER INSTRUCTOR (OTHER THAN THE ONE WHO TRAINED THE STUDENT); FLY WITH HIM/HER TO EVALUATE IF THEY ARE READY TO SOLO. I WAS FLYING WITH THIS STUDENT FOR THE PRE-SOLO CHKOUT EVALUATION. THIS STUDENT WAS NOT ONE OF MINE; AND THEREFORE; I HAD NOT FLOWN WITH THE STUDENT PREVIOUSLY AND WAS UNAWARE OF THE STUDENT'S CAPABILITIES. THE STUDENT COMPLETED PREFLT AND RUN-UP DUTIES; AND WE WERE SUBSEQUENTLY CLRED FOR TKOF ON RWY 15. AS WE WERE ON THE TKOF ROLL; TWR CALLED OUR NUMBER AND ALL I HEARD WAS 'DEPARTING ACFT RWY 10.' I ASSUMED WE WERE TO FOLLOW THE ACFT SINCE WE WERE ALREADY CLRED FOR TKOF. (WE DID NOT USE HEADSETS; SINCE THIS PARTICULAR STUDENT DIDN'T HAVE ANY.) TWR THEN CALLED 2 MORE TIMES AND SAID 'DO YOU UNDERSTAND.' THE STUDENT ACKNOWLEDGED BOTH TIMES. WHEN WE LIFTED OFF; WE SAW ANOTHER ACFT AHEAD OF US THAT HAD JUST TAKEN OFF RWY 10. I INSTRUCTED THE STUDENT TO WIDEN OUT AND FOLLOW BEHIND ACFT A WHO DEPARTED RWY 10. (THERE WAS PLENTY OF DISTANCE SEPARATION THROUGHOUT.) AS WE MADE A L TURN AND OVERFLEW THE FIELD TO THE NW; I HEARD TWR AGAIN CALL OUR NUMBER AND SAY; 'DID YOU HEAR YOUR TKOF CLRNC WAS CANCELED?' THIS WAS THE FIRST TIME I HEARD THAT IT HAD BEEN CANCELED; AND AS I LOOKED OVER AT THE STUDENT; HE PROCEEDED TO CALL THE TWR AND SAID 'YES; I DID'. THEN TWR SAID; 'SO WHAT HAPPENED?' AT THIS POINT MY MOUTH WAS OPEN IN SHOCK! I TOOK THE MIKE AWAY FROM THE STUDENT AND I TOLD TWR I WAS THE INSTRUCTOR ON BOARD; AND THAT I HAD NOT HEARD A CANCELLATION OF OUR TKOF; I THOUGHT WE WERE ONLY BEING INSTRUCTED TO PASS BEHIND THE DEPARTING ACFT OFF OF RWY 10. WE CAME BACK IN AND LANDED; AND AFTER I INFORMED THE STUDENT HE WOULD NOT BE AUTH TO SOLO AND SENT THE STUDENT HOME; I WENT AND TALKED TO THE CTLRS. WHAT HAD HAPPENED WAS AN ACFT THAT WAS CLRED TO LAND ON RWY 10 DID A GAR. SINCE WE WERE STILL ON THE GND ON RWY 15; THE CTLR CANCELED OUR CLRNC WHEN THEY SAW THE OTHER ACFT GOING AROUND. HOWEVER; ALL I HEARD OF THIS XMISSION WAS THE LAST HALF; 'FOR DEPARTING ACFT OFF RWY 10;' AND SINCE THE STUDENT ACKNOWLEDGED TWR; I ASSUMED THAT WE WERE ONLY TOLD TO PASS BEHIND THE OTHER ACFT. (NATURALLY; IF STUDENT HEARD A TKOF CANCELLATION; AND ACKNOWLEDGED; HE WOULD ABORT AND STOP?) WHEN I ASKED THE STUDENT WHY HE ACKNOWLEDGED THAT HE DID HEAR THE TKOF CANCELLATION; HE TOLD ME HE THOUGHT THE CTLR WAS ASKING IF HE HAD RECEIVED THE TKOF CLRNC. WAYS TO PREVENT THIS IN THE FUTURE INCLUDE: ALWAYS WEARING HEADSETS. NEVER ASSUME THE STUDENT KNOWS ANYTHING; EVEN IF THEY ACKNOWLEDGE A RADIO XMISSION; DON'T ASSUME THEY KNOW WHAT THEY'RE ACKNOWLEDGING! WHEN FLYING WITH ANOTHER INSTRUCTOR'S STUDENT THAT I DIDN'T PERSONALLY TRAIN; NEVER EXPECT THAT THEY ARE AT THE SAME LEVEL OF CAPABILITY THAT I REQUIRE MY OWN PRE-SOLO STUDENTS TO DEMONSTRATE. AND LASTLY; LISTEN MORE CLOSELY TO XMISSIONS WHEN INSTRUCTING; AND IF UNSURE OF SOMETHING ASK TWR TO REPEAT RATHER THAN ASSUME THE STUDENT IS ACKNOWLEDGING THE PROPER XMISSION. IN OTHER WORDS; RECHK AND VERIFY. NEVER ASSUME THAT ONCE WE GET THE MAGIC WORDS 'CLRED FOR TKOF' THAT IT'S ETCHED IN STONE. I FIND MYSELF SOMETIMES TUNING OUT MY HEARING AT THIS POINT AND BEGINNING TO INSTRUCT. TKOF CLRNCS CAN AND WILL BE AMENDED OR CANCELED WHEN YOU LEAST EXPECT IT!