To be able to think mathematically, however, means not to think in symbols but to learn to think in relationships.

然而，能夠進行數學思考，就意味著不要用符號來思考，而是要學會用關係來思考。

Math can't be taught, just like we can't learn ping pong from watching videos.

數學是不能教的，就像我們看視頻學不了乒乓球一樣。

Children can't learn math from reading textbooks or listening to a teacher.

孩子們學數學不是看課本、聽老師講就能學會的。

Instead, they learn math by doing math, ideally with riel objects, because only when they do math relationships are constructed right where math happens in their heads.

相反，他們是通過做數學來學習數學的，最好是用里爾物體來學習，因為只有做數學的時候，關係才會在他們腦海中數學發生的地方正確構建。

It happens in the head.

它發生在腦袋裡。

Whatever is on paper is merely a representation of mathematical thinking that happens in the brain just like musical notes.

不管是紙上的東西，都只是數學思維的表現，就像音符一樣在大腦中發生。

What is on paper is just a representation of music that actually happens when someone plays the piano.

紙上的東西只是對音樂的一種表現，當有人彈琴時，音樂就會實際發生。

To be a good musician, it's not enough to be able to read the notes.

要想成為一名優秀的音樂人，光會讀音符是不夠的。

We also need to practice a lot.

我們也要多加練習。

The same is true for math, which is why practicing mental arithmetic is so important.

數學也是如此，所以練習心算非常重要。

Math needs years of practice.

數學需要多年的實踐。

This becomes clear when we look at how Children learn to understand a number, say eight, not the symbol eight, but the idea of the quantity of eight to internalize this seemingly simple idea, Children need a lot of practice in two skills.

這就很清楚了，我們看一下孩子們如何學習理解一個數字，比如說8，不是8的符號，而是8的數量概念，要把這個看似簡單的概念內化，孩子們需要在兩個技能上進行大量的練習。

First, they need to learn how to create order and then, later on how to create hierarchical relationships.

首先，他們需要學習如何創建秩序，然後，再學習如何創建層次關係。

Let's look a order first, but construct order when four year olds learn to count.

我們先看一個順序，但四歲的孩子學數數時，要構造順序。

Most have trouble ordering objects in their heads.

大多數人都難以在腦海中對物體進行排序。

If the things they count are unevenly distributed, sometimes they skip objects.

如果他們所數的東西分佈不均，有時就會跳過對象。

Then they count the same ones twice.

然後，他們把同樣的人數了兩遍。

To do it right, Children have to learn how to construct order in their heads.

要做到這一點，孩子們必須學會如何在腦海中構建秩序。

This seems easy, but actually takes our brains a lot of practice.

這看起來很簡單，但實際上需要我們的大腦進行大量的練習。

Once Children learn to water objects in their heads, they can put them in relationships, hierarchical relationships.

當孩子們學會了在腦海中水物體後，就可以把它們放在關係、層次關係中。

As Children construct order, they count the objects as follows.

當孩子們構建秩序時，他們會按以下方式計算對象。

1234567 and eight.

1234567和八。

As they do that, the number eight represents the eighth place in the order.

當他們這樣做時，數字8代表了順序中的第八位。

In other words, eight always includes 1234567 The idea of eight is therefore a hierarchical relationship between the eighth object and all those preceding it.

換句話說，八總是包括1234567是以，八的概念是第八個對象與前面所有對象之間的等級關係。

If we don't learn to do this sort of abstraction by doing lots of math in our heads.

如果我們不通過在腦子裡做大量的數學運算來學習做這種抽象的事情。

We won't be able to form a solid foundation for arithmetic after building them.

我們在建立它們之後，將無法形成堅實的運算基礎。

Children need to learn to break relationships apart again.

孩子們又要學會拆散關係。

We can see how hard this is when we present a five year old an image of six dogs and two cats, and then ask, Are there more dogs, orm or animals?

我們可以看到，當我們向一個五歲的孩子展示六隻狗和兩隻貓的形象時，然後問，還有更多的狗、或米或動物嗎？

While most adults who see the full picture find this question odd, a five year old typically just answers MAWR dogs when you ask further more dogs than what the child replies than cats.

雖然大多數看到全貌的成年人都覺得這個問題很奇怪，但當你進一步問到比孩子回答的狗比貓更多的時候，五歲的孩子通常只會回答mawr狗。

In other words, if you ask other Mawr dogs, orm or animals, the child hears other mawr dogs, orm or cats.

換句話說，如果你問其他的毛狗、毛姆或動物，孩子聽到的是其他的毛狗、毛姆或貓。

At age five, most kids didn't practice enough math to break hierarchical relationships apart while still remembering the whole.

五歲時，大多數孩子的數學練習還不夠多，不能把層次關係拆開，同時還能記住整體。

This happens because once the child has to cut the hole into parts for them at that moment, the whole no longer exists.

之所以會出現這種情況，是因為一旦孩子在那一刻要為他們把洞切成零件，整體就不存在了。

They have not yet constructed the concept of eight without thinking of it as a sum of its parts.

他們還沒有建構起八的概念，而沒有把它作為部分的總和來思考。

So when they divide the animals into cats and dogs, all they can think off our two parts, of which one looks larger.

所以當他們把動物分為貓和狗的時候，他們能想到的就是我們的兩個部分，哪一個看起來更大。

The idea of eight is then for gotten to also think about all animals would require two opposite mental actions.

八的想法是，然後為得到了也想所有的動物將需要兩個相反的心理行動。

First, divide the whole and then put it back together.

先整體分割，再重新組合。

A mental process that most five year old Children precisely can't do only by age seven.

一個心理過程，大多數五歲的孩子恰恰在七歲前還做不到。

Most Children can see the whole and keep its abstraction in their heads and still divide the some in its parts.

大多數兒童能看到整體，並在腦海中保持其抽象性，仍能將部分劃分為整體。

Experiences proceeds language as we demonstrated.

經歷了我們所演示的語言的進行。

It takes a child a lot of mental training, aunt hands on experiences to form the concept of a number.

孩子需要經過大量的心理訓練，姨媽親身體驗，才能形成數字的概念。

At the age of five, we can build a simple row of eight later form eight square, then eight route Only.

在五歲的時候，我們就可以搭建一個簡單的排八以後形成八方，然後八路唯。

Once we have constructed number concepts inside our heads, can we effectively learn how to express them with images, symbols and language.

當我們在腦海裡構建了數字概念後，能否有效地學習如何用影像、符號和語言來表達。

Math can be expressed in different languages.

數學可以用不同的語言來表達。

Ah, 100,000 years ago, we used objects to express our mathematical thinking.

啊，十萬年前，我們用物體來表達我們的數學思維。

Later, we used images.

後來，我們用影像。

Around 1000 years ago, we began to reduce images to Arabic numeral symbols.

大約1000年前，我們開始將影像還原成阿拉伯數字符號。

In future, we might replace symbols with bits or express math in graphic simulations or games In other words, while math thinking always happens in our heads, the language that represents our thinking is evolving.

未來，我們可能會用比特來代替符號，或者用圖形模擬或遊戲來表達數學換句話說，雖然數學思維總是發生在我們的頭腦中，但代表我們思維的語言卻在不斷髮展。

But most people don't have math but language problems.

但大多數人沒有數學問題，但有語文問題。

We know, for example, that 11 year old unschooled street vendors are often highly proficient in complex money transactions but incapable of doing paper and pencil arithmetic.

例如，我們知道，11歲的無師自通的街頭小販往往對複雜的貨幣交易非常熟練，但卻無法進行紙筆算術。

This phenomenon, known as ST Mathematics, shows that when smart kids struggling school, they often just can't express their thinking in symbols.

這種被稱為ST數學的現象表明，當聰明的孩子在學校裡掙扎的時候，他們往往就是無法用符號來表達自己的思維。

Their brains conduce math but have language problems.

他們的大腦傳導數學，但語言有問題。

One way to solve this is to do it your way, just like nobody ever learned to speak a language just by learning the rules of grammar.

解決這個問題的方法之一就是按照自己的方式來做，就像從來沒有人僅僅通過學習文法規則就學會了說一門語言。

Nobody learns math by memorizing the rules of how to arrange numbers in symbols in order to find the right answer to a problem.

沒有人是通過記憶如何用符號排列數字的規則來學習數學的，以便找到問題的正確答案。

Whenever we do that, we stopped constructing fundamental principles inside our heads to get better and confident.

每當我們這樣做的時候，我們就會停止在腦海裡面構建基本的原則，從而變得更好、更自信。

Children should be encouraged to find their own path and use their own language to express a solution.

應鼓勵孩子自己找路，用自己的語言來表達解決問題的方法。

Which brings us back to rule one.

這就回到了規則一。

Math can't be taught.

數學是不能教的。

It has to be constructed.

它必須要建設。

If we want to learn math, we have to do math in our heads, ideally with riel life experiences.

如果我們想學好數學，就要在腦子裡做數學，最好是有里爾生活經驗。

Later, we replace the objects with abstractions such as language, symbols or whatever the future might bring.

後來，我們用語言、符號或未來可能帶來的任何東西等抽象的東西來代替對象。

Three ideas presented in this video are based on the work of Jean Piaget, a Constance Comey, Keith Devlin, Georgia to Clark and Jerome Bruner, who all contributed immensely to the body of work and research on how Children and adults learn math.

在這個視頻中提出的三個想法是基於讓-伯爵，康斯坦斯-科米，基思-德夫林，喬治亞-克拉克和傑羅姆-布魯納的工作，他們都做出了巨大的貢獻，兒童和成人如何學習數學的工作和研究。

If you want to get better at math today, join Keith Devlin from Stanford University and over 100,000 students from all around the world in Hiss free course on thinking mathematically.

如果你想在今天獲得更好的數學成績，請加入斯坦福大學的Keith Devlin和來自世界各地的10多萬名學生，參加Hiss免費的數學思維課程。

See the descriptions below for more details and links for the research sprouts videos are published under the Creative Commons license.

更多細節和鏈接請看下面的描述，研究芽菜視頻是在知識共享許可下發布的。

That means our videos are free and anyone can download, edit and play them for personal use and public schools, governments and nonprofit organizations can also use them for training online courses or designing new curriculums.

這意味著我們的視頻是免費的，任何人都可以下載、編輯和播放這些視頻供個人使用，公立學校、政府和非營利組織也可以使用這些視頻來培訓在線課程或設計新的課程。

Toe.

趾頭。

Help us stay independent and support our work.

幫助我們保持獨立，支持我們的工作。

You can join our patrons and contribute.

你可以加入我們的贊助人，並做出貢獻。

Just visit patryan dot com slash sprouts.

只要訪問patryan點com斜線屮。