Required Reading for Math Teachers I

Last week I promised I’d write about something worthwhile this time, so as promised: one of the most worthwhile things I’ve ever read. This piece is not typically seen as “math education research,” but I think it’s of vital importance for us.

Clever Hans

You don’t have to read the whole original book by Oskar Pfungst; there are plenty of good summaries of the research online (for instance here and here). This amazing story gets talked about in comparative psychology but I think it has special significance – often missed – for math educators.

Take-home lesson: never underestimate your ability to fool yourself into believing your students understand something when really what they are doing is watching you. To force them to engage the material it is often necessary to restrict their access to you or systematically confound the signals they get from you.

I think this is a central issue for modern math teachers. We need to explicitly develop ways of question-posing and interacting with our classes and individual students that hide or disguise our intentions for how they are supposed to respond. This needs to be part of the core training of math teachers, much more than it already is.

Clever Hans was a horse owned by Wilhelm von Osten, a high school teacher in Germany around the turn of the twentieth century. von Osten attempted to teach several animals basic arithmetic and German. The other animals failed but Hans, a Russian stallion, seemed to get it. He indicated answers by tapping his front right hoof. He was able to produce correct answers to a shocking array of questions including the four basic operations, some square roots, calculating days of the month, spelling words, etc. He was regularly displayed by von Osten in public. His feats caused an international stir. Nobody who hadn’t seen it could believe it (von Osten must be cueing the horse somehow, right?), but skeptics were converted when they saw and interacted with the horse themselves. A commission convened by the German board of education found in 1904 that no trickery was involved. von Osten didn’t even have to be present.

Psychologist Oskar Pfungst solved the puzzle: no trickery was involved, to be sure. But if the questioner didn’t know the answer to the question, then Hans couldn’t answer it. (For example, if Pfungst whispered one number to it, and von Osten another, and then asked for their sum, Hans would answer incorrectly.) What was going on was that whoever asked the question was cueing the horse totally subconsciously by tiny imperceptible body movements when the answer was reached. The horse took these movements as an indication to stop tapping that hoof.

Once Pfungst became aware of these movements, and trained himself to control them in himself, he was able to cause the horse to answer a question with any answer he desired. For example, he could say “tap 14” but then cause the horse to tap 8 by using the body language he had found. Later, in laboratory experiments, he assumed the role of the horse, and “read the mind” of his subjects. He would instruct them to think of a number, and then begin tapping his hand. If he saw the characteristic body language indicating the “right answer” had been arrived at, he would stop tapping. The subjects were amazed by his ability to guess their number and had no idea they were sending tiny signals.

In comparative psychology they’ve taken the lesson of this story, and they’re diligent about preventing the “Clever Hans effect” from interfering with their experiments. In math education we still need to. If a horse can have all of Germany and beyond believing that it can extract square roots, when what it’s really doing is taking subconscious cues from its trainer, think of what human children can do.

This is not idle speculation, but a very real dynamic that shows up at least occasionally, in one form or another, in almost every math class I have ever watched or taught. It can happen at two levels:

1) Some students are committed Clever Hanses.
2) The whole class becomes Clever Hans.

To some students, getting cues from the teacher rather than thinking about the math has become such an ingrained habit that it is their entire modus operandi when doing math with an expert around (teacher, tutor, parent, peer, etc.). They usually have lost faith in their ability to have math actually make sense to them, and this cueing has become their sole survival strategy. They regularly fail tests and have come to accept this, but they would rather fake you out than admit ignorance – understandable, since they often carry a feeling of stupidity arising from their lost faith, and admitting ignorance would mean exposing this supposed stupidity. So they watch you and produce the right answers more often than not. Not with any sense of trickery or getting over on anyone – often, they’re barely conscious of the game. But it’s how they survive.

Most of us have had students like this in our class whether or not we recognized it. I’ve worked with several such students as a private tutor and this context puts their habit into very stark light. The first day I meet such a student, the very first problem I have them work on, they’ll make a guess quickly and watch my reaction to see if they’re on the right track. I’ve trained myself to respond noncommittally to everything:
“Is it 1/3?”
“Why do you say so?” (Even if 1/3 is correct.)
“Oh, no no, 1/4?”
“Why do you say so?”
I also often hide my face when posing a problem, or (even better) move out of my student’s visual field. These are not the only moves in this situation, but something similarly radical is needed: such a student will never engage the math itself unless all recourse to his or her standby survival method has been methodically denied. I believe that a necessary part of the repertoire of every math teacher is a set of moves designed to hide or confound the cues we send. The above are just a few suggestions.

In Pfungst’s research, when Hans was fitted with blinders and posed a question by a questioner outside his visual field, he made strenuous efforts to see the questioner. Similarly, when we begin to work on not telegraphing the answer we are hoping for – developing a good poker face, for example, or responding with “what makes you say so?” whether the proffered answer is right or wrong – students who are committed Clever Hanses will make every effort to get the answer out of us anyway. They will ask questions, make guesses, keep the game going till we let something slip. One of the lessons of the story is that it’s virtually impossible not to let something slip. Pfungst found that people communicated answers with minute physical signals they weren’t aware of. Even people who were made aware had trouble not sending them. So when working with a committed Clever Hans you can’t just trust yourself not to give up the goods. It’s often necessary to physically vacate the space between the student and the problem. When working with an individual kid this can even mean leaving the room.

Now even kids who are not in the situation described above, who have not come to rely solely on observation of the expert, can be drawn into a Clever Hans game if the teacher is telegraphing the answers too intensely. In most classrooms, for example, the question “Is this enough information to solve the problem?” does not receive a straight answer. Since this question is not usually asked when there is enough information, it is generally safe to conclude that the answer is “no.” Furthermore, this conclusion is a lot less work than actually thinking about the problem. The teacher obviously wants to hear “no” so let’s say it and be done with it.

Similarly, any yes-or-no question posed by a teacher to a class is a setup for a Clever Hans game. It is a lot less work to guess one answer blindly (or wait for someone else to do so) than to actually think about the question. The teacher’s response to that one blind guess gives away the game and the class moves on with the question “answered,” but the very real possibility that no one in the room actually thought about it.

With this in mind I believe it’s a very productive exercise to scrutinize our lessons (working with the whole class, a group, or an individual) with the question “how far was it possible to get by just following my lead?” It’s especially powerful to videotape class and then watch the videotape with this question in mind – you see a lot more that way. It’s also worthwhile to observe other teachers with this same question in mind, because the dynamic is much easier to recognize from the outside than in the heat of it.

I have one other suggestion related to all this: it helps to be legitimately open to students’ thought processes whether or not they initially sound like what we had in mind. This is something I’ve had to work on. I have throughout my career been repeatedly surprised by the discovery that nearly every time a student offers an idea authentically (i.e. not as just a random guess), it makes some sort of sense. Maybe not complete sense, and maybe it’s not at all where I was headed. But if I can curb my initial reaction of “this kid is totally confused” long enough to actually take in the train of thought, there is almost uniformly some worthwhile reasoning inside it. Then even if I need to say “we’re going to stick to the topic,” I can do so after acknowledging the reasoning. The connection to Clever Hans is that if we want them actually thinking, we have to make sure our questions are legit. This gets communicated by acknowledging people for treating them as legit. If the only answers we acknowledge are ones that fit our preexisting image for what the answer is supposed to be, this communicates that the question wasn’t authentic, and it’s probably easier to try to guess what the teacher is up to than to engage it authentically.

To summarize: the lesson of Clever Hans is of central importance for our profession. We want our students thinking about math, not watching us for cues. But it is natural to subconsciously cue them as to what we want to hear. So a necessary part of becoming a math teacher is developing techniques to deny access to or confound these cues. Vacating the students’ visual field while they work is one important method. Another is responding the same way (e.g. “why do you say so?”) whether we believe the student is on the right track or not. But all of us need to be thinking about this.