INSTRUCTOR WITH SPI CROSSES THE ADIZ DURING TRAINING.

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I RECEIVED A MESSAGE AT XXXX LCL THAT A MR X OF ZMA WANTED TO SPEAK TO ME. HE ASKED ME WHY WAS I IN AN ADIZ WITHOUT FILING A DVFR FLT PLAN. MY RESPONSE WAS THAT I DID NOT KNOW WHAT HE WAS TALKING ABOUT. HE TOLD ME THAT OAKBROOK HAD ME IN THAT AREA AT ABOUT XX15/XX30. I HAVE NO IDEA WHO OAKBROOK IS. MY STUDENT AND I WALKED IN AND RECEIVED A WX BRIEFING FROM ST PETERSBURG FLT SVC. WE WERE TOLD THAT THERE WAS A STRONG EASTERLY FLOW COMING ACROSS THE CTR OF THE STATE DUE TO A LOW PRESSURE OFF THE TIP OF FLORIDA AND HIGH PRESSURE TO THE N THAT WAS LOCATED IN THE S EASTERN PART OF GA. PIE; TPA; LAL; SPG; WERE RPTING VISIBILITIES BTWN 7 AND 15 MI; SCATTERED LAYERS BTWN 700 AND 1200 FT; AND STRONG SURFACE WINDS OUT OF E. THE WINDS ALOFT WAS FORECASTED TO BE OUT OF THE E AT ABOUT 30 KTS AND NO NOTAMS THAT WOULD EFFECT US. WE DEPARTED SPG AT ABOUT XX45 IN A CESSNA 150 ON A HDG OF 240 DEGS AND CLBED UP TO 2500 FT THROUGH THE SCATTERED LAYERS. I MADE A PLT RPT TO SPG TWR THAT THE BASES WERE ABOUT 1100-1200 FT AND THE TOPS WERE 2000 FT. WE CONTINUED A CIRCLING CLB TO 3000 FT THROUGH A CLR AREA WHILE WE WERE OVER THE GULF OF MEXICO JUST OFF SHORE. WE LEVELED OFF AND BEGAN TO DO SOME BASIC FLT MANEUVERS OF TURNS; CLBS; DSCNTS; STRAIGHT AND LEVEL; FLT AT MINIMUM CTLABLE AIRSPDS; WITH AND WITHOUT FLAPS; STEEP TURNS AND APCH TO STALLS. I LOST SIGHT OF LAND DUE TO THE FACT THAT THE SCATTERED LAYERS AT THAT ALT LOOKED LIKE A SOLID OVCST. I THOUGHT I WAS ABOUT 5-7 MI INTO THE GULF WHERE OUR PRACTICE AREA IS. I WAS IN EXCELLENT CONDITIONS TO WHAT WE NEEDED TO DO. AT ABOUT XX30 I DECIDED THAT WE SHOULD HEAD BACK TO SPG SO AS TO ARRIVE BEFORE XX00 AM. WE TURNED TO A HDG OF 090 AND BEGAN A DSCNT FROM 3000 FT. AS WE WERE DSNDING I NOTICED THAT THE SPD OVER THE WATER WAS VERY SLOW; VISIBILITY WAS GOOD EXCEPT THAT FORWARD VISIBILITIES WERE RESTR DUE TO THE SCATTERED CLOUDS MAKING IT DIFFICULT TO SEE THE SHORE LINE. WHEN WE REACHED 1500 FT I WAS NOW UNDER THE SCATTERED LAYER AND WAS ABLE TO SEE THE SHORE LINE. I ESTIMATED MY POS TO BE ABOUT 5 MI FROM THE SHORE LINE AND 15 MI FROM SPG. AS WE REACHED THE SHORE LINE WE MADE OUR CALL TO SPG TWR. I WAS ADVISED TO RPT 2 MI W OF THE FIELD ON FINAL TO RWY 6. WE WERE GIVEN TFC INFO AND ASKED IF WE WOULD 'GIVE WAY' FOR THEM. I 'ROGERED' THAT REQUEST AND CONTINUED OUR APCH. WE LANDED; TAXIED TO THE TIE DOWN AREA; SECURED THE ACFT AND LEFT THE ARPT. DURING THE ENTIRE FLT I FELT I WAS OPERATING LEGALLY AND SAFELY.