INSTRUCTOR WITH STUDENT PILOT MIS-HEARS GND CTL. CROSSES RWY WITH ACFT ON TKOF ROLL.

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I WAS INSTRUCTED TO HOLD SHORT OF RWY 26L AT INTXN 'E' AT BRACKETT ARPT LA VERNE. AFTER WAITING ABOUT 1 MIN; MY STUDENT AND I WERE DISCUSSING OUR LESSON WHICH HAD JUST OCCURRED WHEN WE HEARD A CLRNC TO CROSS A RWY AND TO CONTACT GND ON THE SOUTH SIDE. THINKING THIS CLRNC WAS FOR US; MY STUDENT APPLIED PWR AND BEGAN XING THE RWY. AT THIS TIME; THE LCL CTLR TOLD US TO STOP; HOWEVER WE WERE ALREADY HALFWAY ACROSS THE RWY. LOOKING TO MY L AND OBSERVED A SMA APCHING US. AT THE TIME; HE WAS ABOUT 800 FT TO THE E OF US. SEEING THE APCHING SMA I APPLIED FULL PWR AND EXITED THE RWY BEFORE THE SMA REACHED MY POS. THE SMA WAS ON A TKOF ROLL AND WAS ALREADY AT VR WHEN I CROSSED 26L. HIS LAST RADIO STATEMENT WAS 'NO PROBLEM; WE'LL JUST FLY OVER HIM.' AFTER THIS; I CALLED THE TWR. IN MY OPINION; THIS WAS A SIMPLE MISINTERP OF ATC COM. WHEN INSTRUCTOR/STUDENT ARE TALKING IN THE COCKPIT; IT IS EASY TO INTERPRET COM INTENDED FOR ANOTHER ACFT TO BE SIMILAR TO WHAT YOU SHOULD RECEIVE. ALSO; THERE WAS SOME NOISE ON THE FREQ; FURTHER ADDING TO THE CONFUSION.