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Peppered Moths

Summary of problems with claim:

Textbooks do not use peppered moths as an example of something new being created, they use it to demonstrate what natural selection can do in mere decades.

Full discussion:

Explore Evolution claims:
Critics question whether the peppered moth story shows that microevolution can eventually produce large-scale change. They point out that nothing new emerged.
Explore Evolution, p. 93, emphasis original

Textbooks which present this example typically use it to illustrate the process by which biologists investigate natural selection, not to demonstrate the origins of biological novelty. In Raven and Johnson's Biology (5th edition), the discussion of peppered moths and industrial melanism is in a section titled "Natural selection explains adaptive microevolution," and never claims that the example illustrates anything other than the process by which scientists have investigated the effects of natural selection. Ridley's Evolution (2nd ed.) discusses peppered moths first in a section explaining how "Natural selection operates if some conditions are met," and later in a chapter entitled "The Theory of Natural Selection," in a section discussing how "the model of selection can be applied to the peppered moth." Ridley first demonstrates the reasons why natural selection was invoked by observers of a pattern, and then proceeds to describe the particular ways in which researchers investigated the hypothesis: determining the heritability of coloration, experimenting to determine the fitnesses of various genotypes under different conditions, and concluding with a discussion of ways in which "the details of the story are now known to be more complex."

As Ridley explains:

In conclusion, the industrial melanism of the peppered moth is a classic example of natural selection, and illustrates the one-locus, two-allele model of selection. The model can be used to make a rough estimate of the difference in fitness between the two forms of moth using their frequencies at different times; the fitnesses can also be estimated from mark-recapture experiments. However, the one-locus, two-allele model is only an approximation to reality. In fact, several alleles are present (and their dominance relations are not simple); selection is not simply a matter of bird predation in relation to camouflage; and it seems that migration, as well as selection, is needed to explain the geographic pattern of gene frequencies.
Mark Ridley (1996), Evolution, 2nd ed., p. 109

Explore Evolution claims that "the experiment [does] not show what the story says it's supposed to," but misrepresents what scientists claim it illustrates. It is not an experiment meant to illustrate speciation, and Explore Evolution does not discuss those experiments, such as the examples in Drosophila discussed by Ridley in his chapter on speciation.

In fact, Explore Evolution does not even discuss the process by which melanism would have originated in peppered moths. The genetics of melanism have been well understood since the 1960s, when researchers showed how several different mutations to the same genes could produce similar sorts of melanism (Lees, David R., 1968, "Genetic Control of the Melanic Form Insularia of the Peppered Moth Biston betularia (L.)," Nature 220(5173):1249-1250). Natural selection is the process by which those mutations increased in frequency over several generations, exactly what scientists and textbook authors claim this example demonstrates.