Your task: Devise a way to teach an animal mathematical concepts. Now it’s true, many animals exhibit a basic understanding of numbers. But what about concepts such as zero, or that one number is greater than another, or addition and subtraction, or others you can think of? We assume animals don’t know about these and, yet, what if we wanted to teach them, and so test that assumption? How would you do it?

First, let’s remember Clever Hans, a famous German horse of the early 20th Century. Hans knew basic arithmetic, could read and write German, and was able to tell time—or so his owner, Wilhelm van Osten, claimed. Given two numbers, Hans would add them and then tap his hoof on the ground the correct number of times. An impressive feat, no doubt. But to be sure he was actually adding the numbers, he had to be tested. How was that done? A psychologist called Oskar Pfungst used various experimental controls—put blinkers on Hans, got others to ask him questions, made sure some of the questioners themselves did not know the answers. What he found was that van Osten was actually giving subtle, if involuntary, cues with his body when Hans approached the right answer. Hans had learned to recognize that body language and stopped tapping when he saw it. Clever Hans, but this is not arithmetic.

So, can you teach your pet swan, Hansi, to add numbers? Well, here’s one way you might do it. Put hungry Hansi into a room where all she sees are two large sheets of paper on the wall. One has a large black dot, the other has two. After a few moments to let her absorb that, we slide three boxes into the room. One has one dot on it, one has four and the third has three. If Hansi chooses either of the first two boxes, a gloved fist pops out and bonks her gently on the chin. But if she chooses the third, the one with three dots, it opens to reveal a juicy morsel of her favourite food on the planet, Maganlal’s peanut chikki.

That is, we reward her choice of the three dots. We do that because three is the sum of the numbers of dots she sees at the start, and we want her to recognize that. That’s why we penalize her if she chooses either of the other two boxes.

Run Hansi through this gauntlet a few times, and she may start choosing the three-dot box consistently. If so, clearly she has learned something. But can we conclude that it’s addition she has learned, that she knows 1 + 2 = 3? No, because what she has most likely divined is that where there are three dots, there’s chikki. Clever Hansi, but this is not addition.

So, we change things somewhat. Each time Hansi runs the gauntlet, we put different sheets of paper in the room. One and two dots, or four and one, or two and two. When the boxes appear, the chikki is always in the box with the sum of the dots, the fist always in the two boxes that have different numbers of dots.

Of course hungry Hansi chooses more or less at random at the start, and perhaps for quite a while afterwards. But what if she eventually does start choosing the box marked with the right sum? Well, if we can rule out the simpler explanations—Maybe the box is open? She smells the chikki? She sees it?—we’d have to conclude that she knows something about addition. Or, at least, she has learned how to add together these relatively small numbers.

Now I don’t know about swans, but some scientists from RMIT University in Australia have actually done an experiment very much like this... with bees. As long-time readers of this column know, I have a weakness for bees—e.g. (Bees, and also ants). Thus this kind of scientific venture intrigues me no end.

The paper they wrote begins by pointing out that “there are studies which demonstrate that vervet monkeys, chimpanzees, orangutans, rhesus monkeys, one African gray parrot, pigeons, spiders, and human infants have the ability to add and/or subtract" (Numerical cognition in honeybees enables addition and subtraction, published in Science Advances). No insects in that list, but they note that honeybees have shown some understanding of ideas such as “left/right, above/below, same/different and larger/smaller"—and this makes bees a good species on which to test a wider understanding of numbers. In particular, addition and subtraction.

For me, the really fascinating part of an effort like this is devising the experiment itself. Why so? Because addition and subtraction are such routine, familiar, unremarkable operations to us that it is a challenge to imagine how you would teach them to someone—and then another species—who has absolutely no idea what they mean. After all, they would need to learn both what numbers are, and how to manipulate numbers. Those are not trivial concepts, whether to teach or to learn.

You certainly don’t need me to tell you that 1 + 2 = 3, but how will I get a honeybee to understand it? Or a swan? This is why I spent several paragraphs above describing Hansi the swan’s travails—though, admittedly, I borrowed liberally from this honeybee paper.

The RMIT scientists built a Y-shaped apparatus, with a small “room" forming each limb of the Y. A bee enters the room on the leg of the Y, where she finds a “sample stimulus". This is just an image of a (small) number of shapes coloured either blue or yellow. Blue is intended to mean “add one element", and yellow means “subtract one element".

Let’s say the image has two blue squares. After the bee takes a few moments to absorb her surroundings, she flies through an opening into a “decision chamber". Here there are two images, each leading to one arm of the Y. One image has one blue square, the other three. Of course the scientists want the bee to choose the three squares, because that’s one more than the two she saw on the sample stimulus. If she does make that correct choice, she gets a drop of sugar solution, yummy for bees. If she chooses wrongly, she gets a not-quite-so-yummy drop of quinine solution.

The team tested 14 bees, sending each insect one hundred times through the apparatus, sometimes adding, sometimes subtracting, always with small numbers. There were various controls in place to ensure the results were credible.

**And what were the results?**

Startling, actually. While “each individual bee appears to learn differently", they all did learn, over their 100 trials. By the 20th time through, more than half the bees were choosing correctly, whether addition or subtraction; that is, they were doing better than if they chose by simply tossing a coin. By the 100th trial, all 14 seemed to have understood that yellow meant subtraction and blue addition, and were choosing the correct answer 80% of the time. There’s plenty more in the paper, too, to bolster the impression that the bees were learning. All in all, after this training they were performing “at a level that was significantly different from chance", the RMIT scientists said.

Besides, earlier research by the same team showed that bees understand the concept of zero . Put both studies together and you can’t help wondering if bees have intelligence substantially deeper and broader than we usually attribute to them.

But can we actually say they learn and do arithmetic, at least with the small numbers this study offered them? Well, of course there’s scepticism about that, as there should be in any scientific endeavour. For example, Clint Perry of the Bee Sensory and Behavioral Ecology Lab at Queen Mary University of London suggests that the bees might follow a “similar to" strategy. That is, locate the image that is “most similar to the one first seen". This would produce a 70% success rate, Perry estimated.

He went on: “The ability to add and subtract is a higher-level cognitive ability and to claim that an insect can do this is extraordinary and therefore requires extraordinary evidence."

Did the RMIT team have extraordinary evidence? Not for Perry, at any rate.

Still, there’s enough in all this to warrant more studies. At least we can be sure the bees didn’t learn via cues from the RMIT team’s body language.

*Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun*