Clever Hans, The Horse Who Could Do Math

One reads in elementary science textbooks about “The Scientific Method“, as if there were only one. But even if there are as many methods as scientists, it is nevertheless possible to single out certain common good habits of scientific thought. The story of Clever Hans, the horse reputed for his seeming ability to solve mathematics problems, illustrates one of these important good thinking habits.

Clever Hans (perhaps named after the Grimm fairy tale of the same name?) was an Arabian stallion who lived about 100 years ago under the care of a Berlin mathematics teacher named Wilhelm Von Osten. Von Osten trained the horse to respond to numbers written on a blackboard by tapping the corresponding number of times with one foot. He then trained Clever Hans to similarly tap the answers to basic mathematics problems.

Clever Hans became a cause celebre, thanks to Von Osten exhibiting him throughout Germany. Von Osten claimed that animals have far more intelligence than we give them credit for, and Hans was his star pupil.

Clever Hans’s fame resulted in the forming of the Hans Commission to investigate whether Von Osten’s claims were valid. In 1904, the commission reported that no trickery was involved.

But, human nature being what it is, we humans are prone to wishful thinking, and therefore have an enormous capacity for fooling ourselves. This was indeed the case with Von Osten, as was shown by the careful study of psychologist Oskar Pfungst, who reported his results in 1907.

It turns out that Hans was not solving mathematics problems at all, but he was indeed clever. Hans was an acute observer, and learned to stop tapping his foot upon being cued by Von Osten. Now Von Osten was not consciously signalling Hans, but Hans was such a tremendous observer that he was able to detect the minute, unconscious cues that Von Osten was giving.

Pfungst, who did not believe that Hans was actually doing mathematics, was able to determine that Von Osten was signalling Hans by “blinding” the experiment. That is, he would arrange for Hans to be asked a question, and then he would either put blinders on Hans or otherwise place the questioner outside Hans’s field of vision, so that Hans could not see the questioner.

It was important to make sure that nobody who knew the answer to the posed question would be in Hans’s field of vision, as the cueing process was unconscious and very subtle. Pfungst observed that when Hans got close to the correct answer, the questioner would begin to subtly tense his body, and the tension would be released when Hans would reach the correct answer. Sensitive Hans could detect this subtle tensing and releasing.

I first learned about Clever Hans from John L. Casti‘s book, Paradigms Lost (see pages 210–211). Casti mentions:

In fact, Pfungst demonstrated that Hans had the uncanny ability to detect head movements of a s little as one fifth of a millimeter, thereby being able to “read” the slight, but unconscious, movements in his trainer’s head when he came to the correct number of taps for a given computation.

Scientific thinking recognizes this aspect of human nature (our tremendous capacity for unwittingly fooling ourselves), and supplies an antidote: Doubt. That’s right, doubt is one of the most important tools of science.

Rather than believe all things, a scientific habit of mind is to doubt new hypotheses, which leads one to question them, and then to search for evidence that either supports them or not. This is not a simple process, and requires considerable imagination, as we can see by reconsidering Pfungst’s study. Could it really be that the horse is doing mathematics? If not, then what could explain his tapping? What else could be at work? This is the hard part … trying to imagine what else might be at work, and then designing experiments to test the hypotheses that something else is operative.

Nowadays, “blind” and “double-blind” experiments are used to avoid the erroneous conclusions initially made about Clever Hans. For example, if one is testing a new drug, then in a blind experiment one group of recipients receive the drug while a second group receives a placebo, but individuals do not know which group they are in. But maybe this will bias the results, because perhaps the doctor administering the drug will give some subtle, unconscious cues about which drug the patient is getting. To avoid this, a double-blind experiment also keeps the doctor in the dark. Each sample of drug/placebo is given a code number, so that neither doctor nor patient knows what the patient is getting. Then, at the end of the experiment, the codes (which have been prepared and kept secret by a third party) are revealed, and the efficacy of the drug can be analyzed.

Doubt is an extremely valuable and fruitful thinking habit, and essential to progress in science. But where do we teach students the systematic use of doubt as a thinking tool? It is undoubtedly taught in experimental design courses, but it ought to be front and centre in basic science education.

I’ll close with a couple of quotes from page 179 of Dictionary of Quotations, edited by Bergen Evans:

“If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts, he shall end in certainties.” — Francis Bacon, Advancement of Learning I

“Doubt is not a pleasant condition, but certainty is an absurd one.” — Voltaire, Letter to Frederick the Great, April 6, 1767


About Santo D'Agostino

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. Further resources, and online tutoring, can be found at my other site
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One Response to Clever Hans, The Horse Who Could Do Math

  1. John Somerville says:

    Dr. D’Agostino, What a great site. I was pulled in by your discussion of Smullyan’s puzzle(s), and then saw the ‘Clever Hans’ article. Unfortunately for many people the take-away is that anthropomorphizing is bad; my veiw is that it can be bad. What I got from the ‘Hans’ story when I first read it is: horses can read subtle gestures and at least some of them can be very obediant in performing (for the horse) meaningless tasks.

    I shall be back often.

    Yrs John Somerville (

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