Your teacher is mistaken on two counts.
First of all, he is simply ignorant of the overwhelming amount of historical evidence, archeological and otherwise, that there is for the Exodus. I'll mention some of this evidence later.
Secondly, even if there were no archeological evidence to prove the Exodus, that would not necessarily disprove it. The only way to disprove something is either to find evidence against it, or to find a lack of evidence that should be present and for which there is no plausible explanation for its absence.
But simply to say that something was not proved is a meaningless statement. For instance, the fact that they didn't find a harp with David's name on it or a pair of shoes with the name "Moses" written in them does not disprove these people's existence.
Now, not finding trash could be "evidence against" the Exodus if the archeologists knew exactly where such trash should be. But how would they know that? Do they know the exact route that we traveled in the desert? How extensive was their digging?
And exactly what kind of trash were they looking for? Candy wrappers? The manna that fell from Heaven wasn't wrapped in candy wrappers. Were they looking for human waste? The Talmud says that the manna was miraculous food that was totaly absorbed by the body; a person didn't have to go the bathroom after eating the manna. Perhaps, then, they expected to find worn out clothing? The verse says, "Your clothes did not wear out from upon you these forty years (Deuteronomy 8:4)."
Now, an archeologist may say, "But I don't believe in these miracles; I'm looking for evidence of a purely natural Exodus."
In other words, the Jewish version of the Exodus is rejected at the outset; instead, evidence is sought for a different event, an event which we never said happened.
This is like saying: "If the Jews crossed the Red Sea, they must have had boats. If we don't find these boats, it disproves their story." But this won't disprove our story; we never claimed we crossed the Red Sea in boats! So, too, our version of the Exodus does not imply necessarily that we would have littered the dessert with all sorts of artifacts.
As I said in the beginning, there is overwhelming historical evidence, archeological and otherwise, for the Exodus. For one, we have an unbroken historical record of these events. Our record is both a written record, recorded in our Torah, and an oral record passed on by word of mouth from parents to their children (like we do the night of Passover). See our "Historical Verification of the Torah"
Finally, let me suggest a few books that bring extensive archaeological evidence for the Exodus: The best book on the subject is Israel in Egypt by James Hoffmeier. See also Biblical Personalities and Archaeology by Leah Bronner and Permission to Receive by Leib Keleman.