So here goes.
As regular readers of DovBear and Yonatan Schreiber know, the Tiferes Yisroel has a view of this subject, and it is one we did not learn in Cheder. Even the more modern hebrew acadamy I attended for high school kept it a secret. Our rabonim insisted the world was 5740+ years old, and openly mocked paleontology and geology ("It is not true, by our definition.") The science faculty was too timid to protest.
I have a few theories as to the reason the Tiferes Yisroel's view of the subject is not taught in our schools, and now that I know Dave is awarding a prize for best series I may spread it out over a few posts, so let's start with the Tiferes Yisroel himself.
In short, he suggests that the world was "created" many times before Adam. This isn't his own idea. The essay in which he puts it forward, the Derush Or ha-Hayyim, is based on important sources including Bahya, Ramban and the Ibn Ezra, rishonim who agree that the universe is very old.
Money quotes from the Derush Or ha-Hayyim:
Age of the Universe
In the year 1807... they found in Siberia... a great elephant... whose skelteon now stands in the Zoological Museum in Petersburg... We already know of a giant creature found in... the city of Baltimore... bones of this creature have been found in Europe, too. This creature had been named mammoth... they have found... iguanodon... whose height was 15 feet, and whose length was as much as 90 feet...there is yet another creature called megalosaurus... from all this it is clear... [citing kabbbalists, Gemarahs, RAbaynu B'chaya, the Ramban, and Ibn Ezra's] that the world has been destroyed and renewed over and over again as many as four times...One of the kabbalists he cites has a formula, using this idea of cycles, that suggests the universe is several billion years old. His own writing suggest he thought each of those 4 cycles was at least 7000 years, and perhaps as many as 49,000 years. More on that later.
Early Man / Cave Men
Those humans who lived in the primordial world, known as "Pre-Adamites" in the vernacular, that is the humans who lived in the world before Adam was created are identical to the 974 generations mentioned in Shabbat [88b] and Hagigah [14A]Many Orthodox Jews feel they must challenge every scientific statment regarding paleontology or geology. However, the Tiferes Yisroel, a prominent achron, saw such discoveries as supporting longstanding Torah ideas. He didn't run from what his own eyes told him; rather he saw that new discoveries were not hostile to the traditions he received from those who came before him.
[Related: Taking it to Simcha]