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I noticed recently that for Krias Shema, Reform Jews only say the first paragraph (V'ahavta) and the last couple verses of the third paragraph
(for an example: scroll down to page 18).

I know the Reform movement changed a lot of davening liturgy, but any idea why they specifically changed the text for Krias Shema?

Rabbi Richard Sarason, a faculty member at Hebrew Union College (the Reform seminary), talks about the history of these passages in an article about the newest siddur, Mishkan T'filah. He writes (emphasis mine):

The earliest Reform congregational prayer book (Hamburg, 1819) includes all three paragraphs of the Sh'ma. [...] The deletion of the second and third paragraphs of the Sh'ma in the more radical Reform prayer books (Berlin, followed in North American by Leo Merzbacher's Seder Tefillah at Temple Emanuel in New York, 1855; David Einhorn's Olat Tamid in Baltimore, 1856; and the Union Prayer Book , 1895, and its successors) was justified in terms of both length and theology. In order to shorten the worship service (on the theory that less is more), repetitions were omitted. Thus, the second paragraph, which contains much of the same language as the first, was deemed redundant. But there were also theological problems with the second paragraph: It affirms that God rewards the observance of the mitzvot through rainfall in its proper season, and punishes violations through drought. To the modern scientific mind, this seemed rather primitive and gross, both as an account of the weather and as an understanding of divine providence. [...] In more radical Reform circles, the third paragraph, (or minimally, its first part, dealing with the mitzvah of tzitzit ), was deemed expendable because the tallit, as distinctively Jewish, non-western prayer garb, was also deemed expendable. Also, the passage describes the function of the tzitzit as reminders to perform God's mitzvot---but many of these, too (particularly the ritual ones), were deemed archaic and dispensable in the modern world.

(Rabbi Eliot Stevens writes more about early Reform siddurim in the last article on this page.)

From the mid-19th century, the Reform movement largely rejected the obligations of "ritual" mitzvot (while adhering to ethical ones). If you reject mitzvot then you pretty much have to reject the torah's views of reward and punishment. And the Enlightenment movement drove a desire to "pass", to blend in and not be distinctive, including through clothing.

Leaders of the movement now seem to understand that they went too far. Early drafts of Mishkan T'filah included both the second and third paragraphs (my minyan was part of a pilot study). There was fierce debate among both lay people and rabbis at the time. In the end, the third paragraph was kept but the second was removed. R' Sarason writes:

Nonetheless, the verses describing an angry God "sealing up the heavens" were printed in smaller type [in the pilot version], indicating the theological difficulties with this image. The paragraph ultimately remained sufficiently problematic to require both a recommendation from the Siddur Editorial Committee and a vote by the CCAR Executive Committee. Both decided to uphold the earlier Reform deletion of the paragraph in its entirety: even though subject to mitigating non-literalist interpretations, the text itself remains difficult for a modern Jew to recite in the liturgy. The third paragraph, on the other hand, has been restored as an option, since many Reform Jews have embraced the tallit as distinctively Jewish prayer garb; viewing the tzitzit while praying can usefully remind us of our religious obligations.