VECTORED INTO PROHIBITED AREA.

Narrative

I WAS IFR INBOUND TO NATIONAL ARPT; WASHINGTON; DC; I HAD ONE STUDENT IN THE FRONT LEFT SEAT AND A STUDENT AND AN INSTRUCTOR IN THE BACK. WE WERE IN A TWIN SMA AND THE LESSON INVOLVED CREW COORDINATION INTO A HIGH-DENSITY ARPT. WE HAD PICKED UP ATIS INFORMATION AND WERE EXPECTING THE ROSALIND LDA 18. WE HAD NO FURTHER CLRNC PASSING NOTTINGHAM VOR AND HAD DIFFICULTY GETTING THROUGH TO ASK FOR A VECTOR. WE WERE CLRED DOWN TO 4000' AND GIVEN A NORTHWEST VECTOR. IT WAS RAINING AND THE VISIBILITY WAS ABOUT 5 MILES. ATC SUDDENLY CLRED US DOWN TO 1800' AND SAID WHAT SOUNDED LIKE; CLRED FOR THE RIVER VISUAL RWY 21. WE LOOKED AND SAW THE ARPT AT 11 O'CLOCK AND ABOUT 2 MILES. THE AIRWAVES WERE SO BUSY THAT IT WAS IMPOSSIBLE TO GET ON TO ASK FOR CLARIFICATION. MY STUDENT SAID THAT THEY HAD SAID; FOLLOW THE RIVER TO RWY 21 SO WE BEGAN TO DO THAT WHILE TRYING TO LOSE A LOT OF ALTITUDE IN A VERY SHORT TIME. ATC CAME BACK TO US AND SAID; LEVEL AT 2000'; FLY A HEADING OF 270; LAND RWY 15. A FEW MINUTES LATER HE ASKED WHAT OUR HEADING WAS AND WE CONFIRMED THAT IT WAS 270'. BOTH MY STUDENT AND I WERE LOOKING TO OUR LEFT TRYING TO SEE THE RWY AND SET UP FOR LANDING. I REALIZED TOO LATE THAT WE HAD BEEN VECTORED INTO P-56. ONCE ON THE GND; ATC CONTACTED ME BY PHONE AND WE DISCUSSED THE INCIDENT. I EXPLAINED THAT WE WERE EXPECTING AN LDA AND WE WERE NEVER ASKED IF WE HAD THE ARPT IN SIGHT BEFORE BEING CLRD FOR THE VISUAL. HE SAID HE HAD HANDED US OFF TO TWR AFTER CLEARING US FOR 21. NONE OF US HEARD THAT SO POSSIBLY HE WAS STEPPED ON. FINALLY; HE ADMITTED THAT IT WAS THEIR FAULT. HE TOLD ME THAT FSDO WOULD PROBABLY CONTACT ME AND TO TELL THEM WHAT I HAD TOLD HIM. THIS IS THE THIRD TIME THAT I'VE FLOWN INTO DCA WITH STUDENTS AND THERE IS ALWAYS SOME DIFFICULTY. GENERALLY; THEY KEEP US TOO HIGH TO SET UP FOR A REASONABLE APPROACH. TO PREVENT THIS TYPE OF INCIDENT; I THINK IT WOULD HELP IF ATC COULD GIVE MORE ADVANCE NOTICE ON WHICH RWY TO EXPECT AND CONFIRM THAT THE PLT HAS THAT RWY IN SIGHT.