When I was a graduate student, I did a teaching experiment and asked a first-year American boy to calculate 6+7=? . After reading this topic, the little boy calmly broke off a foot from under the table, took off his shoes and socks, and then counted his toes from his fingers.

Question: As a parent, how would you evaluate this little boy?

Can't you learn math without starting your fingers? This is a headache for many parents. Will children become dependent? Will it cause the calculation speed to be unable to increase? Is this too childish and elementary?

In the following years, colleagues in the project team could not help laughing every time they watched this video. From the professional perspective of mathematics education, we actually evaluate and analyze it like this:

First, he can immediately judge that the answer should be in the range of 10- 15 (because he only took off one shoe), which shows that he has a good sense of numbers;

Second, he knows that in the process of operation, he concretizes an abstract formula with the help of tools around him.

Mathematics learning in primary schools is inseparable from the cultivation of number sense, and the cultivation of number sense is inseparable from the use of tools. So,

What exactly is a sense of number?

Can I use my fingers?

How to use your fingers?

1. What is number sense?

In primary school mathematics education, whether in China or the United States, numbers and calculations are the main parts of the teaching content. And this part is based on a good sense of numbers. So, what is a sense of number?

The word number sense is directly translated from the English phrase number sense. It has a wide meaning, which means that children can use numbers flexibly. According to the definition of NCTM (National Council of Mathematics Teachers), the meaning of numbers should roughly include the following aspects:

-Understand numbers and how different numbers are represented.

-Understand the relationship between numbers and our number system (such as decimal or binary).

-understand different operations and know the relationship between different operations.

-You can use numbers in real life.

How is the sense of number cultivated? Cognitive psychology generally believes that children can accumulate intuitive feelings about some mathematical concepts, such as size, number, front and back, height, distance and so on, through daily life experience at a very young age.

How many spoonfuls does the whole family need to drink soup? We have guests today. How much more do we need? How long does it take from home to school? How much does it cost to buy a toy? Which of these two bananas is more, and how much?

These dialogues make mathematics an indispensable part of children's daily life. After entering school, on the basis of these mathematical dialogues, children should begin to learn a series of formal mathematical expressions, such as decimal, addition, subtraction, multiplication and division (vertical), division and integration of numbers, addition and association law, additive commutative law, multiplication and distribution rate, decimal, fraction, percentage, proportion and so on.

The sense of number also includes that children know that 6+ 17, 17+6 and 16+7 are all the same, and can say why they are the same in their own language; Four 15 is greater than 50, why? About how many people can fit in an elevator, so many people line up and we have to wait for several elevators, and so on.

However, it should be noted that many parents have more or less an illusion that formal operations are more advanced mathematics. Once a child begins to learn, the mathematical dialogue in daily life is backward, low-level and unnecessary. Instead, practice, proficiency, and repeated practice. In short, children's mathematics learning should be delivered to the school model.

According to this line of thinking, snapping fingers is obviously not advanced enough and formal enough. How can you master the principle of mathematics if you always flick your fingers to do arithmetic? Parents will have this concern.

2. Can I use my fingers?

The answer is simple and rude: yes! Not only can it be used, but it should also be encouraged. Why?

Let's look at the history of human counting.

First, let's do a situation substitution. If you lived in Mesopotamia in 9000 BC, you were a shepherd. You need to put the sheep out to eat grass every day and see if each sheep has come back at night. Unfortunately, however, this time is still the Neolithic Age, numbers have not been invented, and there is no counting system. In other words, you can't count yet. What should you do at this time?

You may think of several solutions:

Method 1: You find a lot of stones from all directions. When you open the sheepfold and put the sheep out to eat grass, whenever a sheep passes by you, you pile a stone at the door of the sheepfold until it becomes a pile, and the rest of the stones are thrown far away. When the sheep come back at night, remove the stones one by one from the pile. If there is no last one left, you will know that they are all back. Just count sheep with stones, day after day, back and forth.

Method 2: Find a stump. When the sheep pass by you in the morning, they see a sheep and draw a thin path on the stump. When the sheep return to the circle at night, one comes back and draws a circle next to the alley; The next morning, draw a road next to each circle and count the sheep by drawing a circle with a stump, day after day, in a cycle. Of course, when you can't write it, you have to change the stump.

These two methods have one thing in common: they both record the number of sheep in one-to-one correspondence. A stone, a thin road or a circle corresponds to a sheep. If you think about it carefully, you will find that this is actually the same as counting with your fingers-each finger corresponds to the number represented by the number "1".

When children are not familiar with abstract Arabic numerals, they need all convenient and accessible tools to help record the quantity and its changes. Just like this shepherd in the early days of human civilization. What is simpler and more convenient for children than fingers?

Judging from the history of human counting, counting with fingers is normal. Even many scholars who specialize in counting history believe that the earliest counting method of human beings is counting with fingers.

Later, stone carvings gradually appeared, and then these roads were divided into several groups. Then came the carry mechanism, and then the decimal system. It was not until Arabic numerals spread from India to the west that the symbolic foundation of modern mathematics gradually became stable.

If you pay attention to the observation, you will find that this grouping method is in units of 8.

Every Arabic numeral is an abstract mathematical symbol, and it takes a long time for children to gradually combine this symbol with the number represented behind it. Until then, using hand index is a natural and simple method.

Listen to the neurologist.

In a neurological study on 20 15, scholars Llaria Berteletti and James R. Booth analyzed the human brain. They found that some part of the brain can "see" our fingers. When 8- 13-year-old students do subtraction problems, this part of the brain will light up, even if there is no broken finger.

Other scholars have also found that if the first-grade children use their fingers skillfully, the second-grade children will have stronger ability to compare and estimate numbers. When researchers train 6-year-old children to have some "finger awareness", children not only improve their arithmetic ability, but also have a greater possibility of success in mathematics learning in the future.

Of course, these studies do not mean that children can turn on mysterious brain energy by snapping their fingers. But at an appropriate stage, when children need it, giving them a natural counting tool that humans have used for a long time is helpful to their math learning in the long run.

Finally, refer to the opinions of mathematics education experts.

Professor Jo Boaler of Stanford University has been devoted to improving the mathematics learning experience and effect of American primary school students all the year round. In her research, she found that finger-assisted operation can help children learn math better. She believes that it is very important to use fingers in the process of teaching children basic concepts, which is also related to their future mathematical IQ. Forcibly forbidding children to use finger-assisted operations is likely to hurt their mathematical ability development.

Finger is a bridge and transition for children who have just started to contact regular numbers. "Finger is our most useful visual aid system, which is very important for improving mathematical understanding and brain development. This role will last until adulthood. "

"When students are not good at remembering, or are not very skilled in using numbers, they often look for visual or other auxiliary tools. And this makes many of our greatest scientists, such as Einstein and Edison, considered stupid by teachers when they were young. "

To sum up: can you break your fingers?

Can children break their fingers when doing arithmetic?

Yes

Do children have to snap their fingers to learn math well?

No, but the results of cranial nerve research show that the training and use of fingers is helpful to the development of the brain and the cultivation of mathematical ability.

Will it cause dependence?

No. This is just a transition. With the deepening of understanding, you will naturally give up. Do I have to climb after I leave?

I learned it when I was a kid.

1. Just because you didn't use it as a child doesn't mean your child can't use it. You didn't use the internet when you were a child.

You may not remember the pain of learning math as a child.

Can you get the child to quit?

You can try. Have one or more of the following consequences.

1. The children secretly use it while you are away, or simply use the calculator. By the way, I personally don't encourage primary school students to use calculators unless they are specially designed classroom activities. I will answer why it is best for primary school students not to use calculators in the next Q&A column.

2. The speed and accuracy of children's operation will decrease.

3. Children will be afraid of difficulties in learning mathematics. If they are not adjusted in time, their interest in learning mathematics will be affected.

4. Parents provide other alternative tools, and children slowly give up using their fingers.

Of course, there is another possibility that your child can easily quit snapping his fingers (or not at all) because he/she is very talented in mathematics.

3. How to use fingers and other tools?

Now you shouldn't worry about whether you can let the child break his finger. The following question is how to use fingers or other tools.

Here again involves the concept of number sense. The sense of number is not a simple number (three tones) and number (four tones), but a knowledge of how the number "quantity" changes and a causal relationship of quantitative change. Simple rote memorization of the 9X9 multiplication table or the decimal place of pi 100 has nothing to do with the sense of numbers.

In the process of cultivating number sense, two abilities are very important, that is, observing the pattern of number and grouping or dividing logarithm.

For example, there are four tennis rackets in the picture below. Parents can ask their children, how many rackets are there in this picture?

They will say four, so you can ask them, how do you know these are four rackets?

They might say, 2 +2 =4.

It may also be said that this is 1+1+1 and+1.

Then comparing the two methods, we will find that 2+2 is obviously more advanced than1+1+1+. Why is it more advanced Because this method is more efficient. It can be seen at a glance that there are two 2' s in the picture, and a direct connection can be established between the visual picture and the quantity. Simply put, this is the mode of observing quantity.

For example, in the picture below, what relationships does the child see from the picture? At first glance, the most intuitive may be 5+3 = 8; If you spend more time and give some guidance, children may see 8+2= 10 or even 10-2=8.

After exposure to more quantitative changes, children will know 5+5= 10, 7+3 =10; Even a series of other methods can be derived to get the number 10. We can flexibly divide 10 into 8 and 2, or 7 and 3, or even 5+3+2, which is what we call grouping.

The establishment of initial number sense needs to go through the process from concrete to abstract, and mathematics learning can not be separated from the construction of mathematical image. Instead of letting children express it in abstract formulas? +? = 8, why don't you give eight bananas (or eight or something) and ask him/her how to divide these eight things between two children, how many ways are there, which one do you like best and why?

Asking more open questions and giving children a chance to tell their own reasoning and thinking process can stimulate their enthusiasm for mathematics learning more than forcing children to do 10 arithmetic problems every day.

Then, what kind of quantitative patterns can fingers help children see and what kind of quantitative segmentation can they do?

Question: Have you ever thought about how our decimal system came about?

Tip: Just snap your fingers.

Yes, our hands are full of the history of human mathematics development. Maya's 20-bit system, other ancient civilizations' 12-bit system, and the decimal system we use now. Using 10 finger can not only clearly express the different numbers of 1- 10, but also help children quickly identify the finger rules corresponding to each number. For example, 5 is all the fingers of a hand, and 6 is a hand plus an extra finger.

Can demonstrate with two hands 10=? +? Various methods. Each method is a re-division of the number 10, and we can demonstrate this division process with our fingers.

More advanced, have you ever thought about what to do if the number exceeds 10? How to express 18? The easiest way, we can count our toes together, just like the little boy who took off his socks at the beginning of our article.

Different from him, we can make an agreement with our children. Whenever we collect 10 fingers, we will mark them with 1 0 toes. With both hands and feet, there can be various number games: compare the size, guess the number, and make up 100. In the process of such a game, you are transmitting a more important mathematical concept, 10.

When children have a deeper understanding of numbers by snapping their fingers and establish a stronger connection between specific numbers (several fingers) and corresponding Arabic numerals, they will naturally reduce their dependence on fingers and learn faster and more efficient methods. Who will stick to the stupid method?

In this process, parents can of course provide their children with other alternative tools to assist their children's calculations. For example, daily necessities at home, toothpicks, rice grains, matches, or some simple and easy-to-use mathematical tools, such as detachable plastic particles.

The advantage of plastic particles is that we can assemble 5, 8, 10 and their different shapes at any time, so that children can have a further understanding of the representation and law of quantity.

In the next article about math education, we will discuss more finger games and other family math games that help to cultivate children's sense of numbers.

Write it at the end

I talked about fingers many times today, and even toes appeared several times. I say this not only because it is a common problem, but more importantly, the ideas behind it have an influence on children.

From the shepherd of early human civilization to the children of high-tech era, it is consistent and natural to understand and use numbers with images and convenient tools. We need to see that there is still a long way to go between the mathematical dialogue in daily life and the formal mathematical symbols learned in school. In this process, depth is more important than speed. Asking children why and how to ask questions is more important than knowing the answers.

After entering junior high school, under normal circumstances, all children can master four operations skillfully. Looking back at the primary school stage, parents and children were angry about math, and those became clouds. However, although all knowledge can be mastered, how children master the four operations and what the four operations mean to them often determines their attitude towards mathematics in the future.

For parents, the first step to help their children learn math well is to get rid of their math myths and provide them with the help and transition they need, starting with the pleasure of seeing their children break their fingers or even take off their shoes and toes.