This is simply how i teach it. I've done a TON of big horse showmanship successfully, and have recently trained my first mini. Someone will probably find an issue with something, and you don't need to follow it, but, it might be useful!
As per pivot - I use the blunt handle of a crop/whip, (a jumping bat works well). By that, i mean the little rubber knobby thing at the end. Take a step towards your horse's throatlatch, like you are asking for a pivot, then touch him with the handle (i find the middle of the next works well). Add more and more pressure until one front foot crosses over the other, then pull back off the pressure, give him a break, then ask for another crossover. If he moves his back end, back him up a few steps quickly as a "punishment,"or jerk (not YANK) the chain - but don't get mad, he is learning. Always use a chain halter like you would use in the ring when practicing, and especially while training. No need for them to learn it twice! It may take a few minutes for him to move his front end, but he cant stand like that forever - at some point he needs to do something you can respond to! I start verrrryyyyy slow. 3 consecutive "cross overs" are good for me for each training session in the first week. Keep building and building until he can go a 1/4 turn, a 1/2 turn, then a full turn! Eventually, he will learn that as you set up to pivot, and walk towards him, to do it. Personally, I find a pivot one of the top two most frustrating things to teach- mostly because it goes sooo slow. After he crosses over well at a slow pace, THEN speed it up.
As for sidepassing, the first thing i work on is yielding to pressure. Apply pressure on the middle of his belly, right below the second or third to last rib. Much like the pivot, apply dull, firm, increasing pressure, until he moves sideways, then immediately pull back. I continually say the word "over" like a mad woman, until they get it as a vocal cue too. Once again, or or two sideways steps are good for the first 2 or 3 sessions!!!!
After they "get" going sideways, then practice going sideways over stuff! (the bane of my stallions existance LOL). I start with a cotton lead line. Lay it out straight on the ground. Put his front feet on one side and his hind on the other. Sidepass until you are at the other end, and keep the feets on their respective side! We then move up to crazy colored PVC pipes, and ground poles, cause if he'll sidepass over that, he'll sidepass over anything!
Hope this helps! Let me know if you need clarification.