AN ACR LTT TAXIED ONTO A RWY WITHOUT CLRNC.

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MY COMPANY'S POLICY IS TO DO A RADIO CLOSE OUT OF THE FLT AFTER THE TAXI HAS STARTED. I HAD CONTACTED GND CTL AND GOT INSTRUCTIONS TO TAXI TO RWY 2L; HOLD SHORT RWY 31. I ACKNOWLEDGED; THE CAPT STARTED TAXIING AND I STARTED MY PAPERWORK. I CALLED COMPANY ON #2 RADIO AND GOT MY CLOSE- OUT NUMBERS AND WAS WORKING ON CLOSE OUT WHEN CAPT CALLED 'CLR L.' I LOOKED UP AND NOTICED THAT WE WERE XING RWY 31 AND DID NOT REMEMBER US BEING CLRED TO CROSS. I SAID TO CAPT THAT WE WERE TO HOLD SHORT. ALSO; AT THE SAME TIME; OTHER ACFT WERE XING THE SAME RWY AT OTHER TXWYS. CAPT NOTIFIED GND THAT WE MAY HAVE MISUNDERSTOOD A CLRNC. GND CTL MADE NO REMARK OTHER THAN TELLING US TO FOLLOW ACFT IN FRONT OF US; THEN DIRECTED ANOTHER ACFT TO FOLLOW US. I THINK THE SOLUTION TO THIS EVENT IS OBVIOUS. WHEN WE ARE IN A CRITICAL STATE OF FLT; I.E.; TAXI; BOTH PLTS SHOULD BE DOING NOTHING BUT ACCOMPLISHING THAT TASK. HAVING 1 PLT WITH HIS HEAD DOWN DOING MATHEMATICAL PROBS AND TALKING ON ANOTHER RADIO DEFEATS THE PURPOSE OF HAVING 2 PLTS IN THE ACFT. ALL BOOK WORK SHOULD BE DONE BEFORE ACFT LEAVES GATE.