This article about a mathematically gifted Labrador retriever is a perfect example of the kind of undiscovered animal abilities Temple Grandin writes about in Animals in Translation (see my post from a few days ago). Grandin would not likely credit Beau with being able to calculate square roots or do algebra, but she would agree that he has an amazing ability.
Grandin told about a horse, Clever Hans, who could tap a hoof the right number of times to answer math questions. His owner was convinced his horse could count, since he wasn’t signalling Hans when he had reached the right number. A psychologist finally was able to show that the owner (or anyone else asking Hans questions) really was signalling Hans, they just were doing it without being aware of doing so. Whatever it was they were doing, another person could not detect, but Hans could. When the questioner was put out of Hans’ sight, or the questioner didn’t know the answer himself, Hans could not give the correct answer.
Beau’s owner and others who are convinced the dog is a math genius evidently haven’t read Grandin’s book. The article offers, as evidence that Beau is really doing math and not just watching for signals, that he can answer questions even when his owner is out of sight. But there is no indication that they have tested Beau with questions when the questioner is out of sight, or the questioner does not know the answer. It does say that Madsen (his owner) will ask him a question so complicated that you (the observer) are still trying to figure it out when Beau gives the answer. But Madsen presumably does have the answer figured out, and Beau is watching him very intently.
Grandin’s conclusion is that animals such as Hans and Beau are very intelligent – but it’s not the kind of intelligence that does math calculations. As she points out, no one knows how to train an animal to do what Hans did, and Beau is doing (though Madsen believes he taught math to Beau). These animals taught themselves to observe something so hard to detect that people have no idea how they do it. Other animals have taught themselves to predict when their owners are going to have a stroke, and no one knows how they do that, either. (They were trained to respond to seizures, but on their own they went beyond that to react before the seizure starts, something that humans do not know how to do.)