Three-year-olds can already distinguish two non-parallel lines among many parallel ones. Perhaps they can’t explain the concept, much less name it, but they understand that there is something that makes those lines different. The same thing happens with many other geometrical concepts, the right angle, closed or open figures, the number of sides on a figure, symmetry, etc.
There are two natural ways to investigate universal aspects that are not established by education. One is by observing children before they have been overly affected by culture and the other is by travelling to places where education is very different, as a sort of anthropologist of thought.
One of the most studied cultures for investigating mathematical thought is that of the Munduruku tribe, deep in the Brazilian Amazon. The Munduruku have a very rich, ancient culture, with very different mathematical ideas from those we inherited from the Greeks and Arabs. For example, they don’t have words for most numbers. There is a composite word to refer to one (pug ma), another for two (xepxep), another for three (ebapug), another for four (ebadipdip), and that’s it. Then they have words that represent approximate amounts, like pug pogbi (a fistful), adesu (some) and ade ma (quite a few). In other words, they have a mathematical language that is more approximate than exact. Their language can distinguish between many and few but not determine nine minus two is seven, which is inexpressible. Seven, thirty and fifteen do not exist in the Munduruku language.
Nor is their language rich in abstract geometrical terms. Does that mean that geometric intuitions in the Munduruku communities are very different from those in Boston? The answer is no. The psychologist Elizabeth Spelke discovered that when geometric problems are expressed visually and without using language, Munduruku children and kids from Boston solved them with very similar results. What’s more, the things that are simple for a kid from Boston–like recognizing right angles among other angles–are also easy for a Munduruku. Harder things–like recognizing symmetrical elements among non-symmetrical ones–are difficult for both groups of children.
Mathematical intuitions cut across all cultures and are expressed from infancy. Mathematics is built on intuitions about what we see: the big, the small, the distant, the curved, the straight; and about space and movement. In almost all cultures, numbers are expressed in a line. Adding is moving along that line (typically towards the right) and subtracting is doing the same thing in the other direction. Many of these intuitions are innate and develop spontaneously, without the need for any formal instruction. Later, of course, formal education is added on top of that body of already formed intuitions.
When comparing adults in Boston with Mundurukus, the former solved geometric problems much more effectively. This is almost stating the obvious, merely corroborating the fact that, if someone spends years studying a trade, they get better at it. But what’s most interesting and revealing is that while education improves our ability to solve all the problems, there is still a hierarchy of difficulty. The problems that are most difficult for us as adults are the ones that were impossible when we were children.
To sum up, when people discover something, they analyse it according to their own conceptual framework, which is built from very early (maybe even innate) intuitions. Through time and learning we go through conceptual revolutions that change the way in which we organize concepts and represent the world. But old intuitive conceptions persist. And we can trace that childish way of solving problems through adulthood, even in proficient experts considered to be great thinkers in their field. Problems that are not very intuitive remain tedious and hard to solve throughout our educational formation. Understanding how this body of intuitions works within the human mind is one natural route to improve the way we teach our children.
Gestures and words
Earlier, I described learning as a process that transfers reasoning to the visual cortex of the brain in order to make it parallel, fast and efficient. Now we will look at the inverse process by which we acquire symbols that can describe innate visual intuitions.
Liz Spelke, Cecilia Calero and I studied how geometric intuitions turn into rules and words. Our theory was that the acquisition of knowledge has two stages. The first is a hunch; the body knows the response but cannot express it in words. Only in a second stage do the reasons become explicit as rules that can be described to ourselves and others. We also had another theory, conceived in the desert of Atacama, where Susan Goldin-Meadow, one of the great researchers of human cognitive development, told us about an extraordinary discovery she made after re-examining an old exercise done by Jean Piaget.
In the Swiss psychologist’s experiment, children were shown two rows of stones and had to choose which one had more. The trick was that while both rows had the same number of stones, in one of them they were more spaced out. The six-year-olds, driven by a ubiquitous intuition in our thinking, confused length with quantity and systematically chose the longer row.
Susan made a subtle but very important discovery about this classic experiment. While all the children answered that there were more stones in the longer row, there were remarkable differences in how they gestured their response. Some extended their arms to show that one row was much longer than the other. Other children moved their hands to establish a correspondence between the stones in each row. Those children, who were counting with their hands, had in fact discovered the essence of the problem. They weren’t able to express that knowledge with words, but their body language did. For that second group of children, the Socratic dialogue would work. The teacher only has to give them a little push to help them express the knowledge they already have. This finding is not a mere intellectual curiosity; when educators apply this information, their teaching becomes much more effective.
By this careful observation, Susan discovered that gestures and words tell different stories. We then decided to explore how the children expressed their geometric knowledge along three different channels: their choices, their explanations and their gestures.
In our experiment, the children were asked to choose the odd man out among six cards, the only one that didn’t share a geometric property with the others. For example, five of the cards had two parallel lines drawn on them and the other had two oblique lines in the shape of a V. More than half of the children under four years of age chose the only card that showed non-parallel lines. The others chose wrongly, but not randomly.
Some chose the card that had the most space between the two lines. Or the one in which the lines were the longest. They were focusing on an irrelevant aspect of the problem. Most of those children explained their choice in a consistent way, using words that referred to size. Their actions were coherent with their words. However, their hands told a completely different story. They moved them to form a wedge shape and then in parallel. Which is to say, their hands clearly expressed that they had discovered the pertinent geometric rule. Let’s just say that if it were an exam, their spoken answer would have failed them, but if they were scored on their hands they’d have passed.
We do not know yet the brain mechanisms that explain why information about geometry may be expressed through gestures or choice but not through language. Or what exactly happens in the brain in the moment in which children can have a more consistent grasp of these geometric intuitions and are able to express them in words.
But the experiments in which knowledge is measured through words, actions and gestures help us understand how we learn to forge concepts. Some concepts, like shape, form part of a core set of intuitions that are accessible to implicit knowledge and only later in development can be conveyed explicitly. Younger children can easily identify an odd shape even when they cannot express (to others and probably also to themselves) the geometric reasoning that justifies these choices.
The development of other geometric concepts, such as angles, follows a different path. They are first expressed through gestures at a time in which children cannot use this information for solving specific problems or for describing these conce
pts in words.
Why different concepts build up differently may be due to our innate biological predisposition, but most certainly, as well, they are due to how we relate to geometry in schools and homes. Most children grow up playing frequently with shapes, but have very little practical experience with angles, a dimension that can be much more naturally expressed by gestures. Above and beyond this, the more general point is that there are different precursors that serve to consolidate explicit knowledge.
Cecilia’s study showed how rudimentary children are when they have to express geometrical concepts with words. And in fact it’s not just children that this is true for. The Menon dialogue, which I described at the beginning of Chapter 5, shows that it’s also the case for adults. Developing notions of geometry is different from many other concepts such as number or theory of mind, because geometrical concepts aren’t composed in the same way that numerical and mental state concepts are. This is why it may be so hard for children and adults to express them verbally or learn them from others’ verbal expressions.
And here is where the real pertinence of these results for educational practice becomes evident. First, they suggest that geometry (and many other concepts) may not be taught well using words. This might be the essence of the failure of the Menon dialogue. Second, they also tell a teacher that language may not be a good vehicle to inquire about students’ knowledge of these matters.
The body is a consortium of expressions. Our words represent only a small fragment of what we know. And they are incredibly effective in conveying certain concepts and quite clumsy in expressing others. This may seem trivial in other domains. Imagine a football player being examined through a verbal description of how to take a free kick. As absurd as this may seem, it may be, to some extent, what we do with millions of children when we ask them to explain in words what they know about geometry.
Good, bad, yes, no, OK
Luis Pescetti, an Argentinian novelist, musician and actor, wrote a song in which a parent asks a teenage son a long series of questions. They all have the same responses: yes and no. This, of course, doesn’t mean that the son has no replies to the questions; just that he doesn’t want to answer. The song touches on an important lesson for developmental science: the best way to discover a teenager’s or a child’s inner thoughts is not through direct questioning, not in real life and not in the realm of scientific experimentation.
By exploring various procedures for investigating what children know, we found that the best way was not to ask anything but just to let them speak. This reveals an important principle of social beings: nothing has meaning in and of itself, but, rather, meaning is acquired when someone can share it. The need to share and communicate is a very natural predisposition.
What began as a technical resource for investigating explicit knowledge became something much more interesting, since we discovered that children have a sort of teaching instinct. They are natural teachers. A child with any sort of knowledge has a very strong propensity to share it.
The teaching instinct
Antonio Battro studied with Piaget in Geneva in 1967. Over time he became the standard-bearer of technological transformation in the classrooms of Nicaragua, Uruguay, Peru and Ethiopia. Just as we were exploring children’s innate desire to share their knowledge, Antonio came to our laboratory in Buenos Aires with an idea that was to transform our work, protesting that it was absurd that all neuroscience was dedicated to studying how the brain learns while completely ignoring how it teaches. And he argued that this was particularly strange because the ability to teach is one of the things that distinguishes us as a species, that makes us human. It is the seed of all culture.
We share the capacity to learn with all other animals, including the Caenorhabditis elegans, a worm less than a millimetre long, and the Aplysia sea slug, with which the Nobel laureate Eric Kandel discovered the molecular and cellular mechanics of memory. But we have something distinctive and particular that takes this ability and both communicates and propagates knowledge; those who have learned something have the capacity to transmit it. It is not a passive process of assimilating knowledge. Culture travels like a highly contagious virus.
Our hypothesis was that this voracity to share knowledge is an innate compulsion, like drinking, eating or seeking pleasure. To be more precise, it is a programme that develops naturally, with no need to be taught or explicitly trained. We all teach, even when no one has ever taught us how. Just as Noam Chomsky suggested we have an instinct for language, my colleague and friend Sidney Strauss and I emulated his idea and proposed that we all have a teaching instinct. The brain is predisposed to spread and share knowledge. This hypothesis is built upon two premises.
(1) Prototeachers
Long before learning to speak, children communicate. They cry, they request, they demand. But do they communicate information with the sole objective of remedying a gap in knowledge? Do they teach before starting to talk?
Ulf Liszkowski and Michael Tomasello came up with an ingenious game to answer these questions. An actor let an object fall off a table in full view of a one-year-old child. The scene was composed in such a way that the children saw where it fell but the actor didn’t. Later, the actor diligently and fruitlessly searched for the object. The little ones spontaneously acted as if they recognized this gap in knowledge and wished to remedy it. And they did so with the only resource available to them (since they could not yet speak), which was by pointing to the location of the object. This could be merely automatism. But the most revealing element of this experiment was that, if it was made clear in the staging that the actor knew where the object had fallen, then the one-year-olds would no longer point to it.
This is almost pedagogy, in that:
(1) The infant does not gain anything (evident) from it.
(2) It denotes a clear and precise perception of a gap in knowledge.
(3) It isn’t automatism but rather expresses a specific action in order to transmit knowledge to someone who does not have it.
In some sense, the one-year-olds have an economic perspective of knowledge; that is to say, the effort of transmitting it is only worthwhile when it is useful to the other person.
What their action lacks to make it fully teaching is for the transmission of knowledge to empower the student to continue on their own. In this case, the baby shows the actor where the object has fallen, but–ungenerously–doesn’t show them how to find it when it falls again.
Before learning to speak, children can also proactively intervene by warning an actor when they anticipate their making a mistake. Which is to say, they try to close the communication gap even when dealing with actions they presuppose will happen but which have yet to occur. This ability to foresee others’ actions and act accordingly is at the core of teaching and is expressed even before a baby starts to talk and walk.
(2) Teaching, naturally
No one taught us to teach as children. We obviously didn’t go to teachers’ college or pedagogical workshops. But if we indeed possess an innate teaching instinct, we should teach naturally and effectively. At least as children, before that instinct atrophies. Here we see a problem: the teaching quality depends on how much the teacher knows about the topic. To know whether children communicate effectively, independent of their specific knowledge about the subject, we have to observe their gestures, not their words. Here, what is not said is more important than what is.
There are universal aspects to human communication. Beyond words, semantics and content, one of the virtues of effective speeches–like those of the great leaders in history–is that they work on an ostensive level. Ostensive communication is a concept that has been visited and revisited by philologists and semiologists like Ludwig Wittgenstein and Umberto Eco. It refers to the ability to use gestures to amplify the speech and use as few words as possible. It uses an implicit key that is shared between the speaker and the interlocutor. If we lift one hand with a salt shaker in it and ask someone: ‘Want some?�
�, there is no need to be explicit about what we are offering them. It’s the salt. This is a precise dance of gestures and words that happens in a fraction of a second without us even knowing that we are dancing. A robot versed in language would have asked: ‘Excuse me, what is it you are asking me whether I want some?’
The key to this method of communication is pointing. When we say: ‘That one’, and point, others understand what those words and that hand are indicating. It is a highly efficient way of communicating. Monkeys, who are able to do a countless number of sophisticated things, don’t understand this code that is so simple for us. It is a way of relating to each other that defines us, that makes us who we are.
By adorning our speech with prosody, gestures and signs, ‘ostensive communication’ also serves to label and parse out relevant moments of discourse. With this, the emitter ensures that the listener does not get distracted during the essential part of a message, which would result in a major communication failure.
Ostensive keys are easily recognizable. One is looking into the other person’s eyes and directing one’s body towards them. Aiming one’s gaze or body at the listener functions as a magnet for their attention. Other ostensive cues are using the receiver’s name, lifting our eyebrows or changing our tone of voice. These all make up a system of gestures, which we recognize as natural but that were never taught to us, and that determine the efficiency with which a message is communicated. Perhaps the most spectacular demonstration of how gestures come naturally without needing to be taught is that they are used by the congenitally blind even when in many cases they have never perceived them through other sensory modalities. We can think of it as a channel of communication. The transmission of the message is effective if we tune that channel in well, and it becomes static-y, confused or ineffective if we don’t find the exact frequency of this natural channel of human communication.
The Secret Life of the Mind Page 23
