[TravisH]

Toss the frame. Theres probably a lot of hairline cracks that you can't see underneath rust and grime. With that much damage I would be surprised if the frame wasn't tweaked as well.

As for your welding instructor, hes mostly correct. When you weld a piece of metal the rapid heating and cooling changes the properties of it (eg. brittle vs ductile). To get different properties involves preheating, postheating and submersion cooling, all of which is out of the scope of the course you're taking right now. You'll probably touch on it but won't do any work with it. Now its possible to weld that frame and have it returned to the original state, but you would need to find out the exact process and temperatures you are dealing with. Its not something I'd recommend to a journeyman welder.
That being said you might wonder why a lot of people (including myself) are welding on these frames. You've got basically two types of frame, light passenger and heavy truck. Light passenger vehicles such as our trucks don't see a whole lot of stress in comparison to heavy trucks and therefore any change of temper is pretty much negligable on a solid frame, if the weld is allowed to cool naturally or with post heat. Different story on a heavy truck and thats where the properties from heat treating really come into play, so much so that on semi-tractor you are not allowed to have a weld within the axles. The frames go through enough repeated flex under heavy loads that it will crack period. DOTs see that and your off the road, big fine, and I wouldn't be surprised with criminal charges. However other heavy trucks that don't carry trailers can be welded on.

Sorry if I got a little off topic on your question but hopefully now you understand what your teacher was talking about. The reinforcement methods talked about above all work but in your case scrap the frame. It shouldn't be much more than a couple hundred bucks for a roller and I guarantee theres more wrong with your frame than you think