A CFI INSTRUCTED HIS STUDENT TO DEPART ON RWY 34L AT HEF; EVEN THOUGH THE STUDENT HEARD NO SUCH INSTRUCTION FROM TWR.

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A PRIMARY STUDENT AND I HAD JUST FINISHED OUR RUN-UP AND PRE-TKOF CHKS; AND HAD TAXIED UP TO AND WERE HOLDING SHORT OF RWY 34L AT MANASSAS. WE HAD JUST SWITCHED FROM THE GND FREQ TO THE TWR FREQ; AND HEARD 'CHEROKEE (OUR CALL SIGN); NO RPTED TFC INBOUND; WIND CALM; CLRED FOR TKOF; RWY 34L.' IT IS SLIGHTLY UNUSUAL FOR THE TWR TO CLR AIRPLANES FOR TKOF THAT HAVE NOT MADE A RADIO CALL INDICATING THAT THEY ARE READY FOR TKOF; BUT THE TWR HAS BEEN KNOWN TO GIVE THOSE CLRNCS; ANTICIPATING THE RADIO CALL FROM THE AIRPLANE. THE STUDENT LOOKED AT ME AND SAID HE HADN'T HEARD THE RADIO CALL FROM TWR. I HAD HIM ACKNOWLEDGE THAT WE WERE CLRED FOR TKOF ON RWY 34L AND MADE SURE HE USED OUR CALL SIGN. AS WE BEGAN OUR TKOF ROLL; PROBABLY 3-5 SECONDS AFTER ADDING PWR; TWR SAID 'CHEROKEE (OUR CALL SIGN); IS THAT YOU ROLLING ON THE RWY?' I REPLIED 'AFFIRMATIVE; TWR;' TWR THEN SAID 'CHEROKEE (OUR CALL SIGN); I HADN'T CLRED YOU FOR TKOF YET.' I IMMEDIATELY HAD THE STUDENT REDUCE PWR TO IDLE AND WE ABORTED THE TKOF. WE EXITED RWY 34L AT THE NEXT TXWY; AND I SAID 'MANASSAS TWR; CHEROKEE (OUR CALL SIGN); WE UNDERSTOOD THAT WE HAD BEEN CLRED FOR TKOF; WE APOLOGIZE IF WE UNDERSTOOD INCORRECTLY.' I HAD THE STUDENT ASK TWR FOR A CLRNC TO TAXI BACK TO RWY 34L; AND TWR TOLD US TO TAXI RWY 34L. LOOKING BACK; I SHOULD HAVE HAD THE STUDENT ASK THE TWR TO REPEAT OUR TKOF CLRNC; SO THAT HE COULD HEAR IT FOR HIMSELF. THAT ALSO WOULD HAVE REINFORCED IT FOR ME. I WAS SURE I HEARD OUR CALL SIGN IN CONJUNCTION WITH CLRED FOR TKOF; RWY 34L; BUT BECAUSE WE WERE THE FIRST AIRPLANE IN POS AND THE ONLY ONE CAPABLE OF TAKING OFF ON RWY 34L AT THAT MOMENT; I MIGHT HAVE BEEN EXPECTING TO HEAR OUR TKOF CLRNC AND MISHEARD THE CALL SIGN. IT IS ALSO POSSIBLE THAT THE TWR CTLR MEANT TO CLR AN AIRPLANE BEHIND US FOR TKOF; AND DID NOT REALIZE THAT WE WERE THE FIRST ACFT IN POS. FURTHER; WHEN WE READ BACK OUR TKOF CLRNC; THE TWR DID NOT CATCH OUR ERROR; IF WE IN FACT MADE AN ERROR.