Age: From birth to early 20s
Many children have said: "Math is too difficult!" - not just girls, but all children. Children's brains are accustomed to responding quickly to everyday problems, which means they are better at figuring out how to get back at the kid who just bullied them than they are at solving algebra problems. (Of course, this kind of "social calculation" also requires some kind of mathematical ability, because he needs to judge in advance how many friends of the other party and how many of his own friends are around.)
Children and many animals have nerves system to support the most primitive sense of quantity. Under normal circumstances, this sense of quantity is combined with other human abilities to enable us to create and manipulate symbols to generate formal mathematics. This kind of mathematics only exists in some social cultures and not in others. Mathematics seems not suitable for children's growth, but it is actually beneficial to children's growth.
In the past few decades, people have increasingly recognized that babies have a strong ability to "form concepts related to numbers." For example, if an object is hidden behind the screen and two objects appear later, the baby will stare at the object for a long time and show surprise. In contrast, if a Mickey Mouse toy "hides" behind the screen and a truck comes out, the baby doesn't care at all. And if he sees a Mickey Mouse coming out with another Mickey Mouse, he will show surprise and stare longer. This ability to notice newly added objects is an essential component of the concept of number.
This ability is not limited to smaller numbers. When a 6-month-old baby is shown a series of pictures, each with several objects on it (black dots, faces, anything), he will notice whether the number of objects in the picture is doubled or halved. . As we age, this sense of quantity becomes stronger and stronger. Infants can recognize ratios of 1:2 without counting (for example, comparing 4 and 8 objects, or 6 and 12 objects), but adults can recognize finer ratios such as 7:8. ratio.
Everyone has a sense of quantity (distinguishing groups of different sizes). Quantity recognition is another universal mathematical ability. It refers to the ability to immediately know how many objects there are (usually a small number) without counting them one by one. The term comes from the Latin subitus, meaning "suddenly." The sense of quantity and quantity recognition are the only abilities of all animals, and the nervous systems responsible for the two abilities in the brains of humans and animals may be the same.
These abilities can be seen in rats, dogs and even pigeons. These abilities provide these species with a survival advantage: allowing the animals to estimate quantities of things, from food sources to potential enemies. For example, generally speaking, lions will react when they hear another lion roaring nearby. However, they often react differently depending on how many other lions they hear and the size of their own group. If they feel that their side is not numerically superior, they will call in their companions to support them. Similarly, for orangutans, if the number of individuals in their own group is not dominant, they will avoid conflicts with each other.
Why did it take researchers decades to understand young children’s sense of quantity? One reason is that early researchers (such as Piaget) asked the wrong questions in their experiments. For example, arrange the balls in several rows. In some rows, the balls are far apart and appear to be longer. If researchers ask: "Which row has more objects?" then three- and four-year-old children will point to the row with fewer objects but more sparsely arranged. And if the small balls are replaced by chocolate and the children are promised that they can eat these chocolates in a while, their performance is much better. If analyzed, this experiment seems to measure two things: the sense of quantity and the ability to clearly express the sense of quantity. The 3-year-old has a sense of quantity, but he can’t speak it. Except for letting him eat chocolate, it is really difficult to know from his mouth whether he knows the correct answer.
The strange thing is that if it is a 2-year-old child, they can perceive it very well whether it is a small ball or a chocolate. This result seems to indicate that at 2 years of age, children have a clearer sense of quantity, but over the next 1 year, they lose this sense of abstraction. What happened? One possibility is that at the age of three or four, children's sense of quantity goes through a process of blending "an intuition for quantity" and "a clear, later-developed sense of abstract numbers." By age 5, it's all figured out. At this point, he can just count the balls—but maybe they'll look forward to eating chocolate.
The love for chocolate seems to be an instinct, and it is. Research evidence suggests that chimpanzees can also mentally combine quantities in a manner similar to addition. If chimpanzees are shown three plates in sequence, each containing a different amount of chocolate, they can tell whether the sum of the first two plates contains more or less chocolate than the third plate. It can be seen that from an evolutionary perspective, the most primitive mathematical ability is older than our human history, and it is part of a child’s innate ability.
The brain areas that represent the sense of quantity are the same in the human brain and the orangutan brain. Quantitative information appears to be represented by the prefrontal and posterior parietal lobes. A key brain region is the intraparietal sulcus. This is a recessed grooved area that represents a specific number concept (such as "seventeenth"). If this area of ??the brain is damaged, people's answers to quantitative questions are only close to correct, but not completely correct - equivalent to the level of chimpanzees.
The sense of quantity has been preserved during evolution, which makes scientists think that our brain's presentation of quantities may be related to their relative sizes, just like there is a mental sequence. There is some evidence to support this formulation. For example, when we judge which of two numbers is larger, if the two numbers are adjacent (such as 8 and 7), it will be better than if the two numbers are farther apart (such as 8 and 2) take longer to react, just like two adjacent numbers are adjacent in mental space. The intraparietal sulcus is activated when judging two adjacent numbers. You may think that the storage of quantities is like a computer processing numbers. Regardless of the difference between the numbers, the difficulty of distinguishing them is about the same. But in fact, in the brain, the situation may be completely different. The brain may represent quantities in a more orderly way, like the marks on a ruler.
When monkeys see a specific number of objects or a similar number of objects, some nerve cells in their intraparietal sulcus are activated. Generally speaking, these brain areas belong to the neural pathways in the brain that identify the location of objects, including identifying how many objects there are and which direction these objects are moving.
The parietal cortex, which is responsible for “positional information” (see Chapter 10), appears to have several different functions. The posterior parietal cortex is activated during eye movements in both monkeys and humans. At the same time, when neuroscientists asked subjects to lie in the MRI scanner and complete simple mathematical tasks, they found that this brain area also has an interesting ability: when people do addition or subtraction operations mentally, This brain area is activated even if the eyes are not moving. Other nearby brain areas connected to this brain area are closely involved in visual functions, such as sudden nystagmus (saccades), awareness of the moving direction of a certain image, etc. Therefore, the way we see space may be closely related to our mental sequence. We can even predict to a certain extent whether a person is adding or subtracting based on the activation pattern in the posterior parietal cortex, and the accuracy of the prediction is about moderate.
It may seem a little odd that brain areas involved in eye movement overlap with those involved in basic calculations, but it also suggests that our brains' ability to process abstract information is limited to a certain extent. The extent depends on how we deal with the physical world. In addition to numeracy, many of our cognitive abilities are "embedded" in other abilities in similar ways.
In this way, our brains are able to think abstractly using more concrete actions developed over the course of evolution (such as finding prey or finding a way home in the jungle). To convert these estimation capabilities into precise mathematical expressions requires symbolic representation capabilities.
This ability comes with language. Language is an efficient and sophisticated way of representing information. Parrots, dolphins, rhesus monkeys, and chimpanzees all use symbols to represent quantities. For example, there are two chimpanzees, one named Abel and the other named Baker. They can choose the larger of the two numbers to get a larger number of candies. In most cases, animals cannot yet add or subtract symbols. But one chimpanzee is an exception. It is called Shebach. After several years of training, it can complete some simple addition calculations.
Even though humans have the mental ability for calculation and mathematics, people don’t often use it. French scientists Stanislas Dehaene and Pierre Picza studied the Munduruku people in the Amazon jungle of South America. The people of this tribe did not know how to count, and the words used to express quantities in their language did not Very little. Only a few of the words are exact quantities (pug ma represents 1, xep xep represents 2), and most of them are approximate numbers (eg, ebapug represents between 2 and 5, and ebadipdip represents between 3 and 7). Munduruku people are pretty good at doing rough additions of large groups of objects, as well as Westerners. But this does not work when making precise calculations of small quantities; for example, if you put 6 beans into the jar and then take out 4, and ask them how many beans are left, they will say "0" or "1", which is very easy. I seldom say "2".
A child's early estimation ability can predict his future calculation ability. This suggests that individual differences in children's general ability to process quantities begin well before they begin to be able to count. So, can this ability be improved through training? Perhaps training children in early estimation skills can improve their future numeracy skills. Although this idea has not been tested, it is still very possible.
Based on the basic sense of quantity, we can construct more complex concepts, such as complex numbers, imaginary numbers, real numbers, etc. Based on these abilities and other aspects of brain function, we can discover more complex mathematical abilities: multiplication, trigonometric functions, calculus, etc.
Research on how the brain generates abstract mathematics is just beginning, but researchers have already made some discoveries. For example, higher levels of mathematical knowledge require additional concepts and more areas of the brain to be involved. Algebra requires children to combine their quantitative abilities with the ability to manipulate abstract symbols. For students just starting to learn algebra, there are many ways to get started. For example, solving word problems is relatively easier than solving equations. These different processing methods use different brain areas.
To examine which areas of the brain are involved when solving problems, researchers asked people to lie in an MRI scanner while solving word problems and similar formulas. (For example: 1. The waiter Cassie works 4 hours a day, earns 10 yuan per hour, and gets a tip of 12 yuan when she gets off work. So how much does she earn in a day? 2. If 10H 12=E, and H=4, then, what is E? ) Scanning results show that when solving word problems, the left prefrontal cortex is preferentially activated. This brain area is related to working memory and quantitative processing; when solving formulas, the brain responsible for representing mental sequence Areas are activated, such as parts of the parietal cortex, including the precuneus (an area on the inside of the parietal lobe) and parts of the basal ganglia (critical for action and movement).
Such differences suggest that beginning algebra students can try different ways to solve the same problem. When the problem is difficult, in addition to the cortical areas we mentioned earlier, more left hemisphere brain areas will be involved.
As for advanced mathematics, such as trigonometry or calculus, they have not been fully studied, but researchers believe that these abilities may also be related to the neural systems in the brain responsible for symbolic representation and spatial manipulation.
To some extent, these findings support Euclid’s famous saying about geometry: “There is no shortcut to the palace of knowledge.” Mathematics is a very complex system and is the greatest human being. One of the inventions. That we were able to discover that the neural circuits responsible for telling stories and controlling eye movements are also involved in generating, understanding, and applying mathematics is a remarkable discovery in itself. Adapting our brains to our environment is an amazing thing that our ancestors never imagined.