I've been having a conversation regarding teaching world views in schools ( manily naturalistic and materialistic world views) and we have gotten to discussing the supernatural. 

I must first admit to not being a scholar or having any formal education on the matter.  All I know is based on looking up basic information on the ideas and trying to deduct my personal view on the matters, so I may be completely wrong and don't mind at all being corrected. 

Here is what has been said regarding the matter and I would appreciate it if someone could help me better understand this better.

This is what I wrote regarding the matter: You can't believe in the supernatural and have a naturalistic world view. Supernatural refers to phenomena not bound by natural laws or observability. Supernatural claims are untestable and can't be held up with empirical evidence.

I don't know if anything real that doesn't exist in the material sense?

This is the reply I received: well it depends on what you mean. i think you are confusing the possibility of the supernatural with the method of naturalistic learning. check out ericsteinhart.com/abstract s.html for a person who is an atheist and believes in things that are supposedly supernatural are possible. i would not go as far as saying supernatural claims are untestable or unobservable. you are just ruling out he supernatural a priori. that is different from being skeptical of the supernatural. if you dont test it how do you know it does not violate a natural law. and natural laws are inductive. we expect that gravity will keep working minute to minute but i dont think there is a way to prove like in a math equation that it will always work. its a problem that drove David Hume to distraction. its true that the supernatural has never be reliably observed. if we observed something that broke natural law i.e. the regularity we observe like say gravity stopped working for a single person then we would have a supernatural event but it would be observable and testable. it would need to happen again and again to figure out which natural law is being broken. you could argue that it is a special case and that our induction just did reach this special case of a person but it was a greater regularity in a seemingly irregularity. for instance you could look at gravity in the relativistic sense as being a special case of gravity in general that is not covered by Newtonian mechanics. there is nothing supernatural about relativity. as for something not existing in the material sense in what way does 2+2=4 exist materially? it does not seem reducible to correspondence with a material fact. its a concept a social construct. it does not exist entirely in my brain or yours or anybody's. it happens to be very hard to prove logically. which brings me to another thing that is not physical but real i.e. logic which does not even have a way to prove i think. how can you argue against logic. it makes it hard to see that it is a social construct but it does not correspond to a material fact either. i dont think it is an illusion that our brains evolved to grasp the patterns in the world.