PA44 FLT CREW HAS RWY INCURSION AT CYUL.

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RECENTLY; I FLEW WITH 2 STUDENTS FOR TRAINING ON INTL OPS TO CYUL. AFTER CLRING CUSTOMS AND PURCHASING FUEL AT THE FBO; WE PREPARED TO DEPART ON THE SECOND LEG OF OUR FLT TO OTTAWA. AFTER PICKING UP ATIS; MY STUDENT CALLED UP AND RECEIVED OUR CLRNC; AND THEN CALLED GND AND ASKED FOR A 'PROGRESSIVE' TO RWY 6L; WHICH WAS THE DEPARTING RWY. THE GND CTLR WAS QUITE BUSY SPEAKING WITH OTHER ACFT IN FRENCH. HE RESPONDED TO OUR CALL IN ENGLISH; BUT WITH A THICK ACCENT WITH OUR TAIL NUMBER AND 'TAXI TXWY R AND TXWY L FOR NOW.' HE THEN WENT IMMEDIATELY BACK TO SPEAKING TO OTHER ACFT IN FRENCH. WE BEGAN OUR TAXI FROM THE FBO BY TAXIING ONTO TXWY R AND THEN ONTO TXWY L. TXWY L IS THE TXWY THAT LEADS OUT ONTO RWY 6R. RWY 6R WAS CLOSED BY NOTAM AND INDICATED ON THE ATIS. FOR OUR TAXI ON ARR; GND HAD US TAXI UP RWY 6R TO THE FBO; AND SEVERAL OTHER ACFT HAD TAXIED UP THERE AS WELL DURING OUR TIME THERE. WHEN WE REACHED RWY 6R ON TXWY L; MYSELF AND MY STUDENTS DISCUSSED AND AGREED THAT WE WERE SUPPOSED TO CONTINUE OUR TAXI DOWN RWY 6R. AFTER TURNING OUT ONTO RWY 6R; GND CALLED AND ASKED FOR THE TAIL NUMBER OF THE ACFT TAXIING ON RWY 6R. WE RESPONDED; AND HE THEN GAVE US TAXI INSTRUCTIONS TO CONTINUE DOWN THE RWY AND EXIT AT THE LAST TXWY; THEN TO CONTACT RAMP CTL THERE. MY STUDENT MISUNDERSTOOD THAT CALL; AND THE CTLR REPEATED HIMSELF. WHEN MY STUDENT WAS STILL UNCLR WITH THE INSTRUCTION; I TOOK OVER THE RADIO; AND RESPONDED TO THE CTLR. AT THAT POINT; THE GND CTLR INFORMED ME THAT OUR ORIGINAL TAXI INSTRUCTIONS WERE ONLY AS FAR AS TXWY R AND TXWY L; AND NOT ALL THE WAY TO RWY 6L. I RESPONDED TO HIM THAT I HAD MISUNDERSTOOD AND THOUGHT WE WERE TO TAXI VIA TXWY R AND TXWY L. I APOLOGIZED SEVERAL TIMES FOR THE MISUNDERSTANDING; AND THE CTLR RESPONDED WITH 'THAT'S ALRIGHT;' AND TO CONTINUE OUR TAXI AS INSTRUCTED. SINCE THE RWY WAS CLOSED AND THERE WAS NO OTHER TFC ON IT; I BELIEVE THAT THE WAY WHICH WE WOUND UP TAXIING WAS HOW THE CTLR WAS GOING TO SEND US. THAT BEING SAID; I CAN APPRECIATE THE HAZARDS THAT MAY HAVE ARISEN BY CONTINUING TAXI; PARTICULARLY DOWN A RWY; BEYOND THAT WHICH THE CTLR INTENDED. IN RETROSPECT; WE SHOULD HAVE NEVER CROSSED THE HOLD SHORT LINE; EVEN ON A CLOSED RWY; WITHOUT BEING 100 PERCENT SURE THAT WE WERE CLRED TO. WE WOULD HAVE BEEN BETTER OFF BY STOPPING AND CONFIRMING THE CTLR'S INTENTIONS. A CONTRIBUTING FACTOR TO THIS EVENT WAS THE LANGUAGE BARRIER THAT ARISES WHEN DEALING WITH A FOREIGN CTLR. ALTHOUGH THEY RESPOND TO YOUR CALLS IN ENGLISH; THEY MAY NOT SPEAK IT WITH ALL ACFT ON FREQ. THE DIFFERING LANGUAGE CAN MAKE IT QUITE DIFFICULT TO MAINTAIN YOUR SITUATIONAL AWARENESS AND KEEP TRACK OF WHAT OTHER ACFT ARE DOING; AND WHERE THEY ARE. BECAUSE OF THIS; IT CAN BE HARD TO UNDERSTAND EXACTLY WHAT IS EXPECTED. THIS IS ESPECIALLY TRUE IF THEY ARE SPEAKING WITH AN ACCENT. OUR DIFFICULTY IN UNDERSTANDING THE CTLR; COMBINED WITH HIS USE OF NON-STANDARD PHRASEOLOGY IN HIS RESPONSE TO OUR FIRST CALL; WAS A BIG FACTOR IN OUR MISTAKENLY THINKING WE WERE CLRED TO TAXI BEYOND TXWY R AND TXWY L. THE LESSON LEARNED IS THAT ANY TIME AN INSTRUCTION IS RECEIVED THAT I AM NOT COMPLETELY SURE OF; I WOULD BE MUCH BETTER OFF TO STOP AND ASK FOR CLARIFICATION RATHER THAN CONTINUE AND ASSUME WHAT I THINK WAS SAID.