Animals can do ‘almost math’ | Science News for Students

Animals can do ‘almost math’

Scientists are searching for the evolutionary roots of human math in the barnyard and the zoo
Dec 12, 2016 — 7:00 am EST
cat and calculator

Cats and other animals show quantity-related abilities. Without training, kitties can pick out differences between groups of a few small objects, such as 2 versus 5. However, the felines may be using visual shortcuts.

S. Zielinski and B. Brookshire

When Christian Agrillo runs number-related experiments in his lab, he wishes his undergraduate subjects good luck. For certain tests, that’s about all he says. Giving instructions to the people would be unfair to the fish.

Yes, fish.

Agrillo works at the University of Padua in Italy. There, he studies how animals process information. He is finishing up several years of pitting humans against fish in trials. Those trials test their abilities to compare quantities. He can’t, of course, tell his angelfish to choose, say, the larger array of dots. He can’t tell them to do anything. So in recent tests he made his bemused students use trial and error too, just like the fish.

“At the end, they start laughing when they find they are compared with fish,” he says. Yet the fish versus human face-offs are eye-opening comparisons. And they are done as part of his search for the deep evolutionary roots of human mathematics. If fish and people eventually turn out to share some bits of their number sense (like spidey sense, except focused on quantities rather than danger), those elements might turn out to be older than 400 million years. At some point, that long ago, ancestors of angelfish and humans split apart to form different branches of the tree of life.

No one seriously argues that animals other than people have a symbolic numeral system. Your dog doesn’t have words for numbers like one, two or three. But emerging data are showing that some nonhuman animals — a lot of them, actually — manage almost-math without a need for true numbers.

“There’s been an explosion of studies,” Agrillo says. Reports of some quantity-related skills have come from much of the barnyard and parts of the zoo. Chickens, horses, dogs, honeybees, spiders and salamanders have some number-like skills. So do guppies, chimps, macaques, bears, lions, carrion crows and many more species. Some of these studies involve animals picking pictures of more dots instead of fewer dots. But other studies suggest that animal number sensing allows much fancier operations.

News stories on number sense often say that animals might all have inherited some basic skills from a shared distant ancestor. Some scientists think that idea is too simple, however. Instead of inheriting the same mental powers, animals might just have happened upon similar solutions to similar problems. That would be an example of convergent evolution. That is what occurred with birds and bats. Both fly, but their wings arose independently.

Chasing down those deep origins means figuring out how animals can make judgments about three fruits or five puppies or too many scary predators — all without counting. (That also includes babies who can’t speak yet and people who can estimate at a glance.) Studies to test this are not easy. The deep evolution of nonverbal number sense should be a rich and remarkable story. But putting it together is just starting.

Story continues after slideshow.


Who’s (sort of) counting?

Symbolic numbers work well for people. For millions of years, however, other animals without full powers to count have managed life-and-death decisions about magnitude (which fruit pile to grab, which fish school to join, whether there are so many wolves that it’s time to run).


Dogs treat tricks 

For a sense of the issues, consider the old and the new in dog science. Familiar as dogs are, they’re still mostly wet-nosed puzzles when it comes to their number sense.

When food is at stake, dogs can tell more from less. That is known from a string of lab studies published throughout more than a decade. And dogs may be able to spot cheating when people count out treats. Dog owners may not be amazed at such food smarts. The interesting question, though, is whether dogs solve the problem by paying attention to the actual number of goodies they see. Perhaps they instead note some other qualities.

An experiment in England in 2002, for instance, tested 11 pet dogs. These dogs first settled down in front of a barrier. The researchers moved the barrier so the animals could get a peek at a row of bowls. One bowl held a brown strip of Pedigree Chum Trek treat. The barrier went up again. The scientists lowered a second treat into a bowl behind the screen — or sometimes just pretended to. The barrier dropped again. The dogs overall stared a bit longer if only one treat was visible than if there were the expected 1 + 1 = 2. Five of the dogs got an extra test. And they also stared longer on average after a researcher sneaked an extra treat into a bowl and then lowered the barrier. It now displayed an unexpected 1 + 1 = 3. 

Dogs could in theory recognize funny business by paying attention to the number of treats. That would be the treats’ numerosity. Researchers use this term that to describe some sense of quantity that can be recognized nonverbally (without words). But the design of a test also matters. Dogs might get the right answers by judging the total surface area of treats, not their number. Many other factors might also serve as clues. These include the density of a cluster of crowded objects. Or it might be a cluster’s total perimeter or darkness.

Researchers lump those hints under the term “continuous” qualities. That’s because they can change in any amount, big or small, not merely in separate units (such as one treat, two treats or three).

Continuous qualities present a real challenge for anyone coming up with a numerosity test. By definition, nonverbal tests don’t use symbols such as numbers. That means a researcher has to show something. And those somethings inevitably have qualities that grow or shrink as numerosity does.

Sedona’s sense of math

Krista Macpherson studies dog cognition at Canada’s University of Western Ontario in London. To see whether dogs use a continuous quality — total area — to choose more food, she tested her rough collie Sedona.

This dog already had taken part in an earlier experiment. In it, Macpherson tested whether dogs would try to get help if their owners were in danger. That’s what the collie did on the old TV show Lassie. But Sedona didn’t. For example, neither she nor any dog in the test ran for help when their owners were trapped under a heavy bookcase.

Sedona did, however, prove good at lab work — especially when rewarded with bits of cheese.

dog numbers
A low-tech setup tests this dog, Sedona, to see if she can pick the cardboard box showing a greater number of geometric cutouts on its face without being distracted by size or shape.
K. MACPHERSON

To test number sense, Macpherson set up two magnetic boards. Each had different numbers of black triangles, squares and rectangles stuck to them. Sedona had to select the one that had the greater number. Macpherson varied the dimensions of the shapes. This meant total surface area wasn’t a good clue to the right answer.

The idea came from an experiment with monkeys. They had taken the test on a computer. But “I’m all cardboard and tape,” Macpherson explains. Sedona was perfectly happy to look at two magnet boards fastened to cardboard boxes on the ground. She then chose her answer by knocking over that box.

Sedona in the end triumphed at picking the box with more shapes. She could do this regardless of all the trickery about surface area. The project, though, took considerable effort from both woman and beast. Before it was over, both had worked through more than 700 trials.

For Sedona to succeed, she had to pick the larger number of shapes more than half of the time. The reason: Just picking randomly, the dog would probably choose correctly half of the time.

The tests started as simply as 0 shapes versus 1 shape. Eventually Sedona scored better than chance when dealing with bigger magnitudes, such as 6 versus 9. Eight versus 9 finally stumped the collie.

Macpherson and William A. Roberts reported their findings three years ago in Learning and Motivation.

Earlier this year, another lab highlighted the Sedona research in Behavioral Processes. Its researchers called the Sedona data the “only evidence of dogs’ ability to use numerical information.”

Dogs might have number sense. Outside of a lab, however, they may not use it, says Clive Wynne. He works at Arizona State University in Tempe. There he studies animal behavior. He’s also a co­author of that Behavioral Processes paper earlier this year. To see what dogs do in more natural situations, he designed a test along with Maria Elena Miletto Petrazzini of the University of Padua.

The pair offered pets at a doggie daycare a choice of two plates of cut-up treat strips. One plate might hold a few big pieces. The other had more pieces, all of them small. And the total of those smaller pieces added up to less of the yummy treat.

These dogs didn’t have Sedona’s training. Still, they went for the greater total amount of food. The number of pieces didn’t matter. Of course not. It’s food —and more is better.

This study shows that experiments need to check if animals use something like total amount instead of number. If not, the tests may not measure number sense at all.

Beyond dogs

Animals may choose differently in a number-related test depending on their pasts. At the University of Padua, Rosa Rugani studies how animals process information. She pioneered the study of number sense in newly hatched chicks. If Rugani motivates them, they will learn test methods quickly. Indeed, she notes, “One of the more fascinating challenges of my job is to come up with ‘games’ the chicks like to play.”

Young chicks can develop a strong social attachment to objects. Little plastic balls or lopsided crosses of colored bars become like pals in a flock. (This process is called imprinting. It normally helps a chick quickly learn to stay near its mom or siblings.)

Rugani let day-old chicks imprint on either two or three objects. She offered them either a few identical objects or a cluster of mismatched ones. The set of different pals were, for example, a small black plastic zigzag of rods dangling near a big red double-crossed t-shape. Chicks then had to choose to which flock of new and strange plastic objects they would toddle over.

The original imprinting objects — identical or mismatched — made a difference in that choice. Chicks used to identical pals typically moved near the larger cluster or toward the largest buddy. Something like total area might have been their clue. But chicks used to buddies with individual quirks paid attention to numerosity in the test.

Chicks that had imprinted on three plastic pals were more likely to hang out with three new ones instead of a pair. Those imprinted on a quirky plastic pair made the opposite choice. They chose the pair, not the threesome.

Some animals can deal with what people would call numerical order. Rats, for instance, have learned to choose a particular tunnel entrance, such as the fourth or tenth from the end. They could choose correctly even when researchers fiddled with distances between entrances. Chicks have passed similar tests.

Rhesus macaques react if researchers violate rules of addition and subtraction. This is similar to the dogs in the Chums experiment. Chicks can track additions and subtractions too. They can do this well enough to pick the card hiding the bigger result. They also can go one better. Rugani and colleagues have shown that chicks have some sense of ratios.

To train chicks, she let them discover treats behind cards showing a 2-to-1 mix of colored dots, such as 18 greens and 9 reds. There were no treats behind 1-to-1 or 1-to-4 mixes. Chicks then scored better than chance on picking unfamiliar 2-to-1 dot jumbles, such as 20 greens and 10 reds.

A sense of numerosity itself may not be limited to fancy vertebrate brains like ours. One recent test took advantage of overkill among golden orb-web spiders. When they have a crazy run of luck catching insects faster than they can eat them, the spiders wrap each catch in silk. They then fasten the kill with a single strand to dangle from the center of the web.

Rafael Rodríguez turned this hoarding tendency into a test. He studies the evolution of behavior at the University of Wisconsin–Milwaukee. In one test, Rodríguez tossed different sized bits of mealworms into the web. The spiders created a dangling trove of treasures. He then shooed the spiders off of their webs. That gave him the opportunity to snip the strands without the spiders watching. When they returned, Rodríguez timed how long they searched for the stolen meals.

Losing a greater volume of food inspired more strumming of the web and searching about. Rodríguez and his colleagues reported this last year in Animal Cognition.

At a glance

Nonhuman animals have what researchers refer to as an “approximate” number system. It allows for good-enough estimates of quantities with no true counting. One feature of this still-mysterious system is its declining accuracy in comparing bigger amounts that are very close in number. That’s the trend that made Sedona the collie’s struggles as important as her successes.

When Sedona had to pick the board with more shapes on it, she had more trouble as the ratio of choices moved toward nearly equal amounts. Her scores, for instance, were pretty good when comparing 1 to 9. They fell somewhat when comparing 1 to 5. And she never got good at comparing 8 to 9.

What’s interesting is that the same trend shows up in humans’ nonverbal approximate number system. This trend is called Weber’s law. And it also shows up in other animals.

Story continues below image. 


Weber's Law:

Weber’s law
Quick, which of the two circles in each pair has more dots in it? Weber’s law predicts that the answer will come easier when object numbers in a pair are very different (8 versus 2) and/or involve a small number than when two large ones (8 versus 9) are compared.
J. HIRSHFELD

When Agrillo tested guppies against people, their accuracy dropped during such difficult comparisons as 6 versus 8. But fish and people performed well for small quantities, such as 2 versus 3. People and fish could tell 3 dots from 4 about as reliably as 1 dot from 4. Agrillo and his colleagues reported their findings in 2012

subitizing counting
Take a quick glance at the clusters here before reading more. You probably saw that the box on the left had three dots. But you’d have to count the mosquitoes on the right. That immediate grasp of small quantities is called subitizing, an ability that people and other animals may share.
M. TELFER

Researchers have long recognized this instant human ease of dealing with very small quantities. They call it subitizing. That’s when you suddenly just see that there are three dots or ducks or daffodils without having to count them. Agrillo suspects the underlying mechanism will prove different from the approximate number systems. He admits, though, that his is a minority view.

The similarity between guppies and people in subitizing doesn’t prove anything about how that skill might have evolved, Argillo says. It could be a shared inheritance from some ancient common ancestor that lived several hundred million years ago. Or maybe it’s convergent evolution.

Into their heads

Studying behavior alone is not enough to trace the evolution of number savvy, says Andreas Nieder. He studies the evolution of animal brains at the University of Tübingen in Germany. Behavior in two animals may look alike. Yet the two brains may create that behavior in very different ways.

Nieder and his colleagues have started the huge task of looking at how brains develop a number sense. So far they have studied how monkey and bird brains handle quantity. The researchers compared nerve cells, or neurons, in macaques with those in the brains of carrion crows.

Research in monkeys over the last 15 years has identified what Nieder calls “number neurons.” They may not be just for numbers, but they do respond to numbers.

He proposes that one group of these brain cells gets especially excited when it recognizes one of something. It could be a crow or crowbar, but these brain cells will react strongly. Another group of neurons gets especially excited by two of something. Among these cells, neither one nor three of the somethings kicks off such a strong response.

Some of these brain cells respond to the sight of certain quantities. Others respond to certain numbers of tones. Some, he reports, respond to both.

These brain cells lie in important places. Monkeys have them in the multilayered neocortex. This is the “newest” part of an animal’s brain — the one that developed most recently in evolutionary history. It includes part of your brain in the very front (behind the eyes) and on the sides (above the ears). These areas allow animals to make complex decisions, to consider consequences and to process numbers.

Birds don’t have a multilayered neocortex. Yet Nieder and colleagues have, for the first time, detected individual neurons in a bird brain that respond much as a monkey’s number neurons do.

The bird versions lie in a relatively newfangled area of the avian brain (the nidopallium caudolaterale). It didn’t exist in the last common ancestor shared by birds and mammals. Those reptile-like beasts had lived some 300 million years ago and they didn’t have a primate’s precious neocortex either.

Story continues below image. 

animal brains
Bird brains lack a fancy six-layered outer cortex. But carrion crows (right) have a brain area called the nidopallium caudolaterale that is rich in nerve cells that respond to quantity. In the macaque (left), number neurons are in a different area, mainly a region known as the prefrontal cortex.
A. NIEDER/NAT. REV. NEUROSCI. 2016

So birds and primates probably did not inherit their considerable skill with quantities, Nieder says. Their number neurons could have become specialized independently of each other. As such, this is probably convergent evolution, he argued in the June 2016 Nature Reviews Neuroscience.

Finding some brain structures to compare across deep time is a promising step in figuring out the evolution of number sense in animals. But it’s just a beginning. There are many questions about how the neurons work. There also are questions about what’s going on in all of those other brains that evaluate quantity. For now, looking across the tree of life at the crazy abundance of number smarts, the clearest thing to say may just be Wow

Power Words

(for more about Power Words, click here)

array     A broad and organized group of objects. Sometimes they are instruments placed in a systematic fashion to collect information in a coordinated way. Other times, an array can refer to things that are laid out or displayed in a way that can make a broad range of related things, such as colors, visible at once.

average     (in science) A term for the arithmetic mean, which is the sum of a group of numbers that is then divided by the size of the group.

avian     Of or relating to birds.

bat     A type of winged mammal comprising more than 1,100 separate species — or one in every four known species of mammal.

behavior     The way a person or other organism acts towards others, or conducts itself.

birds     Warm-blooded animals with wings that first showed up during the time of the dinosaurs. Birds are jacketed in feathers and produce young from the eggs they deposit in some sort of nest. Most birds fly, but throughout history there have been the occasional species that don’t.

breed     (noun) Animals within the same species that are so genetically similar that they produce reliable and characteristic traits. German shepherds and dachshunds, for instance, are examples of dog breeds. (verb) To produce offspring through reproduction.

carrion     The dead and rotting remains of an animal.

cell     The smallest structural and functional unit of an organism. Typically too small to see with the naked eye, it consists of watery fluid surrounded by a membrane or wall. Animals are made of anywhere from thousands to trillions of cells, depending on their size. Some organisms, such as yeasts, molds, bacteria and some algae, are composed of only one cell.

cognition     The mental processes of thought, remembering, learning information and interpreting those data that the senses send to the brain.

colleague     Someone who works with another; a co-worker or team member.

control     A part of an experiment where there is no change from normal conditions. The control is essential to scientific experiments. It shows that any new effect is likely due only to the part of the test that a researcher has altered. For example, if scientists were testing different types of fertilizer in a garden, they would want one section of it to remain unfertilized, as the control. Its area would show how plants in this garden grow under normal conditions. And that give scientists something against which they can compare their experimental data.

convergent evolution     The process by which animals from totally unrelated lineages evolve similar features as a result of having to adapt to similar environments or ecological niches. One example is how some species of ancient marine reptiles called ichthyosaurs and modern-day dolphins evolved to have remarkably similar shapes.

cortex     The outermost layer of neural tissue of the brain.

crow     The characteristic loud cry of a rooster. (in biology) A type of large black bird with a complex social structure that perches in trees and is known for its boisterous call.

density     The measure how condensed an object is, found by dividing the mass by the volume.

diversity     (in biology) A range of different life forms.

evolution     (v. to evolve) A process by which species undergo changes over time, usually through genetic variation and natural selection. These changes usually result in a new type of organism better suited for its environment than the earlier type. The newer type is not necessarily more “advanced,” just better adapted to the conditions in which it developed.

evolutionary     An adjective that refers to changes that occur within a species over time as it adapts to its environment. Such evolutionary changes usually reflect genetic variation and natural selection, which leave a new type of organism better suited for its environment than its ancestors. The newer type is not necessarily more “advanced,” just better adapted to the conditions in which it developed.

hypothesis     A proposed explanation for a phenomenon. In science, a hypothesis is an idea that must be rigorously tested before it is accepted or rejected.

insect     A type of arthropod that as an adult will have six segmented legs and three body parts: a head, thorax and abdomen. There are hundreds of thousands of insects, which include bees, beetles, flies and moths.

macaque     A monkey with cheek pouches and a short tail that lives mainly in the forest.

magnet     A material that usually contains iron and whose atoms are arranged so they attract certain metals.

mammal     A warm-blooded animal distinguished by the possession of hair or fur, the secretion of milk by females for feeding the young, and (typically) the bearing of live young.

mealworm     A wormlike larval form of darkling beetles. These insects are found throughout the world. The ever-hungry wormlike stage of this insect helps break down — decompose, or recycle — nutrients back into an ecosystem. These larvae also are commonly used as a food for pets and some lab animals, including chickens and fish.

mechanism     The steps or process by which something happens or “works.” It may be the spring that pops something from one hole into another. It could be the squeezing of the heart muscle that pumps blood throughout the body. It could be the friction (with the road and air) that slows down the speed of a coasting car. Researchers often look for the mechanism behind actions and reactions to understand how something functions.

nerve     A long, delicate fiber that communicates signals across the body of an animal. An animal’s backbone contains many nerves, some of which control the movement of its legs or fins, and some of which convey sensations such as hot, cold, pain.

neuron     The impulse-conducting cells that make up the brain, spinal column and nervous system.

neuroscience     The field of science that deals with the structure or function of the brain and other parts of the nervous system. Researchers in this field are known as neuroscientists.

nonverbal     Without words.

numerical     Having to do with numbers.

numerosity    The term scientists give to a quantity recognized nonverbally.

perimeter     The outer border or edge of some defined area. For instance, the perimeter of some people’s property is set off by a fence.

prefrontal cortex     A region containing some of the brain’s gray matter. Located behind the forehead, it plays a role in making decisions and other complex mental activities, in emotions and in behaviors.

primate     The order of mammals that includes humans, apes, monkeys and related animals (such as tarsiers, the Daubentonia and other lemurs).

psychologist     A scientist or mental-health professional who studies the human mind, especially in relation to actions and behavior. 

ratio     The relationship between two numbers or amounts. When written out, the numbers usually are separated by a colon, such as a 50:50. That would mean that for every 50 units of one thing (on the left) there would also be 50 units of another thing (represented by the number on the right).

reptile     Cold-blooded vertebrate animals, whose skin is covered with scales or horny plates. Snakes, turtles, lizards and alligators are all reptiles.

reward     (In animal behavior) A stimulus, such as a tasty food pellet, that is offered to an animal or person to get them to change their behavior or learn a task.

rhesus monkey     (also called rhesus macaque) A small brown macaque with red skin on the its face and rump, native to southern Asia. It is often kept in captivity and is widely used in medical research.

savvy    The quality of possessing useful and clever knowledge.

silk     A fine, strong, soft fiber spun by a range of animals, such as silkworms and many other caterpillars, weaver ants, caddis flies and — the real artists — spiders.

social     (adj.) Relating to gatherings of people; a term for animals (or people) that prefer to exist in groups. (noun) A gathering of people, for instance those who belong to a club or other organization, for the purpose of enjoying each other’s company.

species     A group of similar organisms capable of producing offspring that can survive and reproduce.

spider     A type of arthropod with four pairs of legs that usually spin threads of silk that they can use to create webs or other structures.

subitizing    An instant ease in recognizing quantities without having to count them, such as when someone sees a trio of ducks and knows that there are three of them without having to count them.

surface area     The area of some material’s surface. In general, smaller materials and ones with rougher or more convoluted surfaces have a greater exterior surface area — per unit mass — than larger items or ones with smoother exteriors. That becomes important when chemical, biological or physical processes occur on a surface.

theory     (in science) A description of some aspect of the natural world based on extensive observations, tests and reason. A theory can also be a way of organizing a broad body of knowledge that applies in a broad range of circumstances to explain what will happen. Unlike the common definition of theory, a theory in science is not just a hunch. Ideas or conclusions that are based on a theory — and not yet on firm data or observations — are referred to as theoretical . Scientists who use mathematics and/or existing data to project what might happen in new situations are known as theorists.

tissue     Any of the distinct types of material, comprised of cells, which make up animals, plants or fungi. Cells within a tissue work as a unit to perform a particular function in living organisms. Different organs of the human body, for instance, often are made from many different types of tissues. And brain tissue will be very different from bone or heart tissue.

tree of life     A diagram that uses a branched, treelike structure to show how organisms relate to one another. Outer, twiglike, branches represent species alive today. Ancestors of today’s species will lie on thicker limbs, ones closer to the trunk.

undergraduate     A term for a college student — one who has not yet graduated.

vertebrate     The group of animals with a brain, two eyes, and a stiff nerve cord or backbone running down the back. This group includes amphibians, reptiles, birds, mammals and most fish.

NGSS: 

  • MS-LS1-8
  • MS-LS4-4
  • MS-LS4-6
  • HS-LS1-2
  • HS-LS2-8
  • HS-LS4-4

Citation

Journal: R. Rugani et al. Ratio abstraction over discrete magnitudes by newly hatched domestic chicks (Gallus gallus)Scientific Reports. Vol.6, published online July 28, 2016, p. 30114. doi: 10.1038/srep30114.

Journal: A. Nieder. The neuronal code for number. Nature Reviews Neuroscience. Vol. 17, June 2016, p. 366. doi: 10.1038/nrn.2016.40.

Journal: M.E. Miletto Petrazzini et al. Do humans (Homo sapiens) and fish (Pterophyllum scalare) make similar numerosity judgments? Journal of Comparative Psychology. Vol.130, November 2015, p. 380. doi: 10.1037/com0000045.

Journal: H. Ditz and A. Nieder. Neurons selective to the number of visual items in the corvid songbird endbrain. Proceedings of the National Academy of Sciences. Vol. 112, June 23, 2015, p. 7827. doi: 10.1073/pnas.1504245112.

Journal: R.L. Rodríguez et al. Nephila clavipes spiders (Araneae: Nephilidae) keep track of captured prey counts: Testing for a sense of numerosity in an orb-weaverAnimal Cognition. Vol. 18, January 2015, p. 307. doi: 10.1007/s10071-014-0801-9.

Journal: K. Macpherson and W.A. Roberts. Can dogs count? Learning and Motivation. Vol. 44, November 2013, p. 241. doi: 10.1016/j.lmot.2013.04.002.

Journal: L. Cantrell and L.B. Smith. Open questions and a proposal: A critical review of the evidence on infant numerical abilitiesCognition. Vol. 128, September 2013, p. 331. doi: 10.1016/j.cognition.2013.04.008.

Journal: C. Agrillo et al. Evidence for two numerical systems that are similar in humans and guppiesPLOS ONE. Vol. 7, February 2012, p. e31923. doi: 10,1371/journal.pone.0031923. 

Journal: K. Macpherson and W.A. Roberts. Do dogs (Canis familiaris) seek help in an emergencyJournal of Comparative Psychology. Vol. 120, May 2006, p. 113. doi: 10.1037/0735-7036.120.2.113.

Journal: R. Rugani et al. Arithmetic in newborn chicks. Proceedings of the Royal Society B. Vol. 276, July 7, 2009, p. 2451. doi: 10.1098/rspb.2009.0044.

Journal: R. West and R.J. Young. Do domestic dogs show any evidence of being able to countAnimal Cognition. Vol.5, September 2002, p. 183. doi: 10.1007/s10071-002-0140-0.