I like your chemistry teacher -- you asked a GREAT question, and she asked you to find out. Hope you did, but if not, here're some thoughts... (and tell her she's a GREAT teacher... ;-)
Start with PROPORTIONS. Early chemists figured out that you'd mix 1 part of X with 2 parts of Y. Let's say we're talking about water (H2O) -- and let's just pretend that we live on a world where we can weigh some hydrogen or oxygen gas as easily as we weigh water or flour, mmm-kay?
Now - we know that water is 1 part oxygen, and 2 parts hydrogen. How? We split water with electricity, collect the gas, and measure the volume. According to Boyle's law (or the ideal gas law, etc. -- hey -- I'm not going to feed you EVERYTHING!), if you've got the same number of molecules of gas -- ANY gas! -- at the same temp & pressure -- you've got the same VOLUME.
So - you electrolyze some water, you get two gases, one volume of which is EXACTLY (as near as you can measure) double the volume of the other. So - you hypothesize that the first gas (hydrogen) exists in water in exactly 2x proportion to the other (oxygen).
Now you weigh them. Hydrogen is the lightest thing you've ever found, so let's just assign it a weight of "1". (Doesn't matter what the units are -- carats per bushel is fine -- the only thing that matters is the relative weights -- which actually should be masses, and in this case are, but never mind. >;-) You weigh the amount of Hydrogen, then the equivalent amount of Oxygen, and find that, after adjusting for the 2x number of "units" (in this case, either atoms or molecules, as both H and O form diatomic molecules) -- that the Oxygen is 8x more massive than the Hydrogen.
Bingo. Hydrogen: atomic weight (arbitrary): 1. Oxygen: 8 x H.
(We are NOT going to get into why these numbers are off by fractional percentages -- your question was about the EARLY chemists. Until we introduce isotopes, quantum mechanics, neutrons, etc. -- let's just use nice, whole round numbers... ;-)