“四舍五入”的机械式学习和理解型学习

今天问起来心儿什么是“四舍五入”,回答说“凡是小于等于4的变成零,凡是大于等于五的自己变成零之后往前进一”。这个看起来似乎没错,可能书上都是这么说的。按照这个规则,我出了一个题:对\(37\)做四舍五入。在这里我特意避开了保留到哪一位的说法。

心儿回答\(40\)。我问,这个答案是否严格用了上面的四舍五入的规则。我演示,如果按照这个规则,我们来看这两个数,十位数是\(3\),小于\(4\),所以去掉变成零;个位数是\(7\)大于零,自己变成零之后往前进一,得到\(10\)或者保留\(7\)。因此,答案就是\(10\)或者\(7\),而不是\(40\)。你看对不对?心儿说,肯定不对,但是不知道怎么回事。

这个事情的根源在于,“四舍五入”的规则在这里被强行记住了,但是没有明白这个规则怎么来的,是用来解决什么问题的。实际上,这个规则的目的是给一个数找到一个接近的凑整数。这个凑整数可以是保留到任何一位的。例如,保留到各位、十位、百位,甚至十分位等等。

理解了这个之后,“四舍五入”的具体规则可以自己构建出来,并且明白,对“\(5\)”的处理其实是有点问题的,尽管问题不大。\(30-34\)更接近\(30\),\(36-39\)更接近\(40\),\(35\)去哪边看起来都有道理。

到这里,可以总结:从概念上,“四舍五入”是为了寻找接近给定数的保留给定位的凑整数;从技术上,需要看保留位之后的一位的数值,如果这个数值小于\(5\)则变成零,如果这个数值大于等于5则变成零之后往需要保留的那一位上进\(1\)。

学数学和教数学,一定要把是什么和怎么做分开,并且主要关注是什么,最好还能够从是什么把怎么做想出来。

那么,这样学会的四舍五入有什么好处呢?第一,更加系统化。需要独立记忆的内容变少了,很多事情可以组织起来,联系起来了。第二,可以很好地迁移。例如,将来就会遇到保留到两位小数这样的问题,这个时候,就可以迁移过去了。第三,可能会用到更深刻的地方,例如,将来的测量读数的问题,而不是仅仅会做标明了四舍五入的题。而且一旦考虑测量和读数这个事情,就会发现,确实遇到\(5\)的时候进位更加合理:这个时候大多数时候后面还会有一点点其他的数,因此整体来说大于\(5\)后面直接是\(0\)的可能性居多,于是更加接近进位之后的那个数。第四,养成这样的通过思考和联系来学习的习惯,受益无穷。当然,也会有更多苦恼。

这也是一个具体课堂教学层次的“基于概念地图的理解型学习”的例子。其他的例子还有“分数除法”、”小数的加法“,“语言的魅力”、“从“欲速则不达”语文理解型学习的例子

以下是这个帖子的概念地图型总结。其中,有一些连边,考虑到图的可读性,去掉了,换成了颜色对应关系。

更简化的图是这个样子的:

分数除法的教学

分数除法的计算步骤是比较容易学会的,例如
\begin{align}
\frac{3}{4} \div \frac{3}{8} = \frac{3}{4} \times \frac{8}{3} = \frac{3\times 8}{4\times 3} = 2 ,
\end{align}
也就是把后面的当做除数的分数分子分母倒过来,然后按照乘法计算,计算的过程中需要注意分子分母消掉相同的因子(分子分母同除以某个数)的技巧。

但是,这样的教学,对于为什么分数相除要这样计算是不明白的。当然,教材也注意到了这个问题。因此,大多数教材是从两个方面来解决这个问题的:分数除法的意义和问题背景,分数除法的计算。这个思路没有丝毫问题,我们也从这连个方面来解决这个问题。

  1. 我们先来看计算的问题。我写下来每一个步骤和这个步骤的理由,
    \begin{align}
    \frac{3}{4} \div \frac{3}{8}
    = (3\div 4) \div (3\div 8) (分数线可以看做除法,分数的意义) \\
    = (3\div 4) \div 3 \times 8 (除法去括号)\\
    = 3\div 4 \div 3 \times 8 (再去括号,结合律) \\
    = 3 \div 3 \times 8 \div 4(乘除法交换律,先除掉容易除的)\\
    = 1 \times 8 \div 4 (计算除法的结果)\\
    = 1 \times 2 (乘除法结合律) \\
    = 2 (计算乘法的结果)
    \end{align}
    或者更一般地,
    \begin{align}
    \frac{3}{4} \div \frac{3}{8}
    = (3\div 4) \div (3\div 8) (分数线可以看做除法,分数的意义)\\
    = (3\div 4) \div 3 \times 8 (除法去括号)\\
    = 3\div 4 \div 3 \times 8 (再去括号)\\
    = 3 \div 4 \times 8 \div 3(乘除法交换律,先除掉容易除的)\\
    = (3\div 4) \times (8 \div 3) (乘法加括号,结合律)\\
    =\frac{3}{4}\times \frac{8}{3}(除法可以看做分数线,分数的意义)\\
    = \frac{3\times 8}{4\times 3}(分数乘法计算)\\
    = 2(同除以某个数)\\
    \end{align}
    可以看到,这样的计算的道理要想得明白的话,需要学生之前就明白“分数线可以看做除法”、“整数乘除法的结合律”、“整数乘除法的交换律”、“整数乘除法的退括号”。其中,“分数线可以看做除法”和“除法去括号”最最重要,并且“除法去括号”可能需要复习或者额外补充一下。不过,原则上,这个应该是在学习整数除法的时候就已经通过理解型学习解决的事情。

    在这里,我想强调的事情是“每一个思考的步骤,都需要给出来明确的理由”。学数学,要学会这个。当然,这仅仅是形式上学会如何计算,尽管已经比前面教纯粹计算方式的要好,但是还是不够的。

  2. 我们在来看,分数除法的意义怎么教。
    1. 首先,我们需要构造一个意义场景,而且和之前已经学习过的东西比较接近的意义场景。例如,我们考虑先从一个切成\(4\)块取走其中\(1\)块,剩下的蛋糕如果再有\(3\)个人平均分会得到多少蛋糕,这样的问题。从那里面,我们可以复习一下分数除以整数,\(\left(1-\frac{1}{4}\right)\div 3 = \frac{3}{4}\div 3 = \frac{1}{4}\)。
    2. 接着,我们可以再铺垫一个类似于分数除以什么什么分之一的例子。例如,我们考虑先从一个切成\(4\)块取走其中\(1\)块,剩下的蛋糕给孩子们吃,可以喂饱几个小孩?如果我们假设8个孩子吃一个蛋糕就可以吃饱的话。于是,从假设我们先知道,每个小孩的肚子能够装这么多蛋糕
      \begin{align}
      1 \div 8 =\frac{1}{8}.
      \end{align}
      目前剩下的蛋糕呢,有
      \begin{align}
      1-\frac{1}{4} = \frac{3}{4}
      \end{align}
      块。
      那么分呢?很简单,动手切一下,就知道了,可以分成\(6\)个\(\frac{1}{8}\)的大小,也就是给\(6\)个孩子吃饱。从除法的意义也知道了,这个\(6\)相当于是下面的除法的答案(一个总量给每一个人分,已知每一个人需要多少,能够分几个人的问题,是除法。至于这个为什么,学习除法的时候应该已经解决了的:除法来自于减法的简便运算,就好像乘法是加法的简便运算。),
      \begin{align}
      6= \frac{3}{4} \div \left(\frac{1}{8}\right).
      \end{align}

    3. 这个时候,看具体情况,可以把计算过程和意义先尝试联系一下,也就是,如果用前面学到的计算过程,来验证一下的话,这个来自于“动手切分”的等式对不对。
    4. 这个时候,再把场景进一步变得复杂一点,假设孩子们成了“高中的大肚王”,每个人需要吃三片那样的小片蛋糕才能饱,问这个时候可以喂饱几个人?于是,提示一下学生们,就有可能可以注意到,是把现在的三个孩子当做一个。那么,就有了如下的计算
      \begin{align}
      2= 6 \div 3.
      \end{align}
      注意,这里我们用了除法。那么,整体来看,这个得数可以看做什么问题的答案呢?
      \begin{align}
      2= \frac{3}{4} \div \left(\frac{3}{8}\right),
      \end{align}
      因为大肚王的饭量是\(\frac{3}{8}\)。

    5. 总结,我们看到了关键的几步:第一,当分数除以整数的时候,就把整数放在最后的分母上就可以;第二,当分数除以分数,但是那个分数是什么什么分之一的之后,相当于把那个什么什么乘在分子上;第三,当分数除以一般的分数的时候,相当于在计算完了前一步什么什么分之一之后,再来计算一个除法。在这个过程中,我们都是用思考或者动手切分来得到答案的,而不是前面教过的计算过程。
  3. 最后,结合纯粹计算的教学(再次提醒,里面也有数学思想,每一个推理的步骤都要明确写下来)和意义的教学(主要体现,数学是思考和表达的语言,数学解决实际问题),来验证这个计算过程的结果和步骤,完全就是总结里面提到的结果和步骤。

顺便,如果你仔细看我这个设计,主要的思想是:教学要反映数学是什么,遇到有多个因素的问题需要把各个因素分离开来来讨论,最后再合起来,也就是分解和合成,或者说解构和建构。阅读理解题:问这里的数学是什么到底是什么,作者是如何通过具体的例子来反映的,这里的因素是什么,作者是如何拆分和合成的,作者这样做你觉得是为什么,你同意吗,对你自己的思考和教学有意义吗?接着,我为什么要出这几个思考题,我是按照什么原则(我多次提到的)来出这几个问题的?

这个例子展示了,在教学环节的细节的层次如何使用概念地图理解型学习,如何多问为什么,如何关注大图景(典型问题、典型思维方式、典型计算分析方法、典型应用的例子),如何运用WHWM(是什么、怎么构建、为什么这样构建、为什么说这个、对读者意味着什么)。其他的例子还有小数加法的计算的例子

有的老师做了实验。其中,有老师提出来,说没有必要讲分数和分数的除法,完全可以讲整数除以分数,例如\(5\div\frac{1}{4}\),只要讲清楚这个,由于放在前面的分数就不用有任何变化,推广到分数很简单。这是很有道理的。而且,确实使得理解分数除法更简单了。很不错的想法。

有另外的老师说,可以先把两个分数通分(变成相同的分母),然后再同时去掉这个分母。这个第一不是理解型学习:更关注怎么算,而不是为什么这样算。第二,这个认知难度非常大:通分需要用到最小公倍数,可是不容易;分子分母同时去掉某个因子值不变更是不容易。第三,完全没有必要啊,通什么分。

有老师帮我在学生中做了调查,发现,好多学生直接用了“分数除法计算要变成乘法,然后后面的分数需要换成倒数”,并且在提醒需要思考为什么可以这样计算的时候,给出来的理由关注的点往往是“倒数就是把分子分母颠倒啊”这样的。这说明,学生真的很多时候不关注为什么这样算,而是怎么算。当然,也有遇到了好的学生,给出了为什么这样算的思考的时候。

词汇层次的理解型学习

刚才心儿在做作业,刚好遇到一个理解型学习中文词汇的例子,改错别字:不遗余利。她改成了力,我问为什么?她没有说本来就这样,而是说,因为这个词的意思是用尽所有的力气和办法。因此,应该是力气的力而不是利益的利。我接着问,那,遗,什么意思?我提示了遗留遗憾(其实遗腹子这个词更好,不过她可能不懂)等词以后,想明白了,留下来。我接着问,余,是什么意思。这个简单,剩下,多余。那合起来什么意思?就是不“留下”“多余”的力气。这个就是字词层次的理解型学习。

我说过,语文学习,从字词的层次到意义构建的层次,都需要理解型学习。而且,理解型学习需要依靠分解(有的时候是多次,例如“遗”还可以继续拆分来说明含义)、合成、联系、类比,不管在那一个层次。那这里是字词层次的例子。有没有更高层次的例子呢?

拿我对《真理诞生于一百个问号之后》的评论当例子。更高的层次,应该是要问主要信息是什么(What)、如何构建和阐述这个主要信息(How)、为什么阐述这个为什么这样阐述(Why)和我是否喜欢对我意味着什么(Meaningful),还有强调“我想我欲想,我说我想说”。

首先,我们要搞清楚这个文章的论点和论据分别是什么,接着问论据如何说明论点,中间逻辑上有没有跳点,论据本身有什么不合理的地方,逻辑经得起怀疑吗,以及论点经得起怀疑吗?最后,还要问作者提出这个论点的可能的意图和这个意图可能的问题。这就是意义构建层次的理解型学习的例子。

具体来说,文章的三个例子都是反映从生活中提炼科学问题,然后展开科学研究的例子。因此,仅仅能够说明做科学研究需要做生活的有心人,却不能说明要科学研究不断地追问。在例子里面并没有不断地追问的情节。接着,这个故事本身的真实性也非常值得怀疑,可能是作者附会的。见《真理诞生于一百个问号之后》的评论。这个故事其实说明,科学研究需要严谨性,需要依靠实验。

如果经过这样的不放过的,层层递进和怀疑,构建起来的意义,就真的是理解型学习了。

Feynman有意思的话

从来反对记忆名言警句——想我所想,问我要问,说我要说即可,管他谁说过呢,类似就是佐证,不同就是创新,但是,刚好看到别人整理好的Feynman在各个书各种场合说的有意思的话,摘下几句留在这里:真的说了我想说的话,还说的挺好。

Science Quotes by Richard P. Feynman

goodreads上的检索——关键字Feynman——结果

关于理解型学习

“I learned very early the difference between knowing the name of something and knowing something.”
― Richard Feynman

“They didn’t put two and two together. They didn’t even know what they knew. I don’t know what’s the matter with people: they don’t learn by understanding, they learn by some other way — by rote or something. Their knowledge is so fragile!”
― Richard Feynman

“What I cannot create, I do not understand.”
― Richard Feynman

“Study hard what interests you the most in the most undisciplined, irreverent and original manner possible.”
― Richard Feynman

“Nobody ever figures out what life is all about, and it doesn’t matter. Explore the world. Nearly everything is really interesting if you go into it deeply enough.”
― Richard Feynman

“The highest forms of understanding we can achieve are laughter and human compassion.”
― Richard Feynman

“When it came time for me to give my talk on the subject, I started off by drawing an outline of the cat and began to name the various muscles.
The other students in the class interrupt me: “We know all that!”
“Oh,” I say, “you do? Then no wonder I can catch up with you so fast after you’ve had four years of biology.” They had wasted all their time memorizing stuff like that, when it could be looked up in fifteen minutes.”
― Richard Feynman, Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character

“Words can be meaningless. If they are used in such a way that no sharp conclusions can be drawn.”
― Richard Feynman, The Meaning of It All: Thoughts of a Citizen-Scientist

“You see, I get so much fun out of thinking that I don’t want to destroy this pleasant machine that makes life such a big kick.”
― Richard Feynman, Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character

“That was a very good way to get educated, working on the senior problems and learning how to pronounce things.”
― Richard Feynman, Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character

“I learned from my father to translate: everything I read I try to figure out what it really means, what it’s really saying.”
― Richard Feynman, ‘What Do You Care What Other People Think?’: Further Adventures of a Curious Character

关于批判性思维(和第一部分“理解型学习有联系”,和独创性有联系)

“The first principle is that you must not fool yourself and you are the easiest person to fool.”
― Richard Feynman

“Religion is a culture of faith; science is a culture of doubt.”
― Richard Feynman

“I would rather have questions that can’t be answered than answers that can’t be questioned.”
― Richard Feynman

“Of course, you only live one life, and you make all your mistakes, and learn what not to do, and that’s the end of you.”
― Richard Feynman, Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character

“What Do You Care What Other People Think?”
― Richard Feynman, Surely You’re Joking, Mr. Feynman

“In general, we look for a new law by the following process: First we guess it; then we compute the consequences of the guess to see what would be implied if this law that we guessed is right; then we compare the result of the computation to nature, with experiment or experience, compare it directly with observation, to see if it works. If it disagrees with experiment, it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is, it does not make any difference how smart you are, who made the guess, or what his name is — if it disagrees with experiment, it is wrong.”
― Richard Feynman

“We absolutely must leave room for doubt or there is no progress and there is no learning. There is no learning without having to pose a question. And a question requires doubt.
― Richard Feynman

“Curiosity demands that we ask questions, that we try to put things together and try to understand this multitude of aspects as perhaps resulting from the action of a relatively small number of elemental things and forces acting in an infinite variety of combinations”
― Richard Feynman, The Feynman Lectures on Physics vols 1-2

“Have no respect whatsoever for authority; forget who said it and instead look what he starts with, where he ends up, and ask yourself, ‘Is it reasonable?'”
― Richard Feynman

关于科学和艺术、生活

“I have a friend who’s an artist and has sometimes taken a view which I don’t agree with very well. He’ll hold up a flower and say “look how beautiful it is,” and I’ll agree. Then he says “I as an artist can see how beautiful this is but you as a scientist take this all apart and it becomes a dull thing,” and I think that he’s kind of nutty. First of all, the beauty that he sees is available to other people and to me too, I believe. Although I may not be quite as refined aesthetically as he is … I can appreciate the beauty of a flower. At the same time, I see much more about the flower than he sees. I could imagine the cells in there, the complicated actions inside, which also have a beauty. I mean it’s not just beauty at this dimension, at one centimeter; there’s also beauty at smaller dimensions, the inner structure, also the processes. The fact that the colors in the flower evolved in order to attract insects to pollinate it is interesting; it means that insects can see the color. It adds a question: does this aesthetic sense also exist in the lower forms? Why is it aesthetic? All kinds of interesting questions which the science knowledge only adds to the excitement, the mystery and the awe of a flower. It only adds. I don’t understand how it subtracts.”
― Richard Feynman

“A poet once said, ‘The whole universe is in a glass of wine.’ We will probably never know in what sense he meant it, for poets do not write to be understood. But it is true that if we look at a glass of wine closely enough we see the entire universe. There are the things of physics: the twisting liquid which evaporates depending on the wind and weather, the reflection in the glass; and our imagination adds atoms. The glass is a distillation of the earth’s rocks, and in its composition we see the secrets of the universe’s age, and the evolution of stars. What strange array of chemicals are in the wine? How did they come to be? There are the ferments, the enzymes, the substrates, and the products. There in wine is found the great generalization; all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did Louis Pasteur, the cause of much disease. How vivid is the claret, pressing its existence into the consciousness that watches it! If our small minds, for some convenience, divide this glass of wine, this universe, into parts — physics, biology, geology, astronomy, psychology, and so on — remember that nature does not know it! So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure; drink it and forget it all!”
― Richard Feynman

“Poets say science takes away from the beauty of the stars – mere globs of gas atoms. I too can see the stars on a desert night, and feel them. But do I see less or more? The vastness of the heavens stretches my imagination – stuck on this carousel my little eye can catch one – million – year – old light. A vast pattern – of which I am a part… What is the pattern, or the meaning, or the why? It does not do harm to the mystery to know a little about it. For far more marvelous is the truth than any artists of the past imagined it. Why do the poets of the present not speak of it? What men are poets who can speak of Jupiter if he were a man, but if he is an immense spinning sphere of methane and ammonia must be silent?”
― Richard Feynman

“Physics is like sex: sure, it may give some practical results, but that’s not why we do it.”
― Richard Feynman

“Physics is to math what sex is to masturbation.”
― Richard Feynman

“– and pompous fools drive me up the wall. Ordinary fools are alright; you can talk to them and try to help them out. But pompous fools – guys who are fools and covering it all over and impressing people as to how wonderful they are with all this hocus pocus – THAT, I CANNOT STAND! An ordinary fool isn’t a faker; an honest fool is all right. But a dishonest fool is terrible!”
― Richard Feynman, Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character

“… it is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.”
― Richard Feynman, The Character of Physical Law

“I believe that we must attack these things in which we do not believe. Not attack by the method of cutting off the heads of the people, but attack in the sense of discuss. I believe that we should demand that people try in their own minds to obtain for themselves a more consistent picture of their own world; that they not permit themselves the luxury of having their brain cut in four pieces or two pieces even, and on one side they believe this and on the other side they believe that, but never try to compare the two points of view. Because we have learned that, by trying to put the points of view that we have in our head together and comparing one to the other, we make some progress in understanding and in appreciating where we are and what we are. And I believe that science has remained irrelevant because we wait until somebody asks us questions or until we are invited to give a speech on Einstein’s theory to people who don’t understand Newtonian mechanics, but we never are invited to give an attack on faith healing, or on astrology — on what is the scientific view of astrology today.”
― Richard Feynman, The Pleasure of Finding Things Out: The Best Short Works of Richard P. Feynman

可惜没有找到关于系联性思考(这个应该算
“They didn’t put two and two together. They didn’t even know what they knew. I don’t know what’s the matter with people: they don’t learn by understanding, they learn by some other way — by rote or something. Their knowledge is so fragile!”
― Richard Feynman
)的。另外,独创性我觉得比创新性好,只要独创就行,不一定是新的,例如旧的东西的新的角度的理解和看法,完全重新发明一遍旧的东西,在学习过程中,都是很有意义的。

附几个小品

“After reading the salary, I’ve decided that I must refuse. The reason I have to refuse a salary like that is I would be able to do what I’ve always wanted to do- -get a wonderful mistress, put her up in an apartment, buy her nice things.. . With the salary you have offered, I could actually do that, and I know what would happen to me. I’d worry about her, what she’s doing; I’d get into arguments when I come home, and so on. All this bother would make me uncomfortable and unhappy. I wouldn’t be able to do physics well, and it would be a big mess! What I’ve always wanted to do would be bad for me, so I’ve decided that I can’t accept your offer.”
― Richard Feynman

“I think the educational and psychological studies I mentioned are examples of what I would like to call cargo cult science. In the South Seas there is a cargo cult of people. During the war they saw airplanes land with lots of good materials, and they want the same thing to happen now. So they’ve arranged to make things like runways, to put fires along the sides of the runways, to make a wooden hut for a man to sit in, with two wooden pieces on his head like headphones and bars of bamboo sticking out like antennas—he’s the controller—and they wait for the airplanes to land. They’re doing everything right. The form is perfect. It looks exactly the way it looked before. But it doesn’t work. No airplanes land. So I call these things cargo cult science, because they follow all the apparent precepts and forms of scientific investigation, but they’re missing something essential, because the planes don’t land.”
― Richard Feynman, Surely You’re Joking, Mr. Feynman! Adventures of a Curious Character

“It was a kind of one-upmanship, where nobody knows what’s going on, and they’d put the other one down as if they did know. They all fake that they know, and if one student admits for a moment that something is confusing by asking a question, the others take a high-handed attitude, acting as if it’s not confusing at all, telling him that he’s wasting their time… All the work they did, intelligent people, but they got themselves into this funny state of mind, this strange kind of self-propagating “education” which is meaningless, utterly meaningless.”
― Richard Feynman, Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character

展示Feynman的思考方式的视频

机械式和理解型学习

我一直在实践和推广理解型学习和教学。有人让我给一些对比的例子。一方面,有些材料确实很难用理解型学习的。另一方面,所有的能够用好理解型学习的地方都需要对内容有深刻的理解并且在具体讲解的环节需要有创造性才行。今天刚好撞到了一个比较容易讲明白两者的区别并且还能够体现理解型学习的好处的例子。

心儿在学校学过了面积单位之间的关系,1平方米=100平方分米,1平方分米=100平方厘米。当然,两个连起来,就是1平方米=100平方分米=100*100平方厘米。但是,遇到类似这样的问题,1平方千米=100平方米是否正确,就不好办了——课上(应该)没教过。

但是,如果是是通过理解型学习来明白上面的面积单位转换,就会非常清楚1平方千米=100平方米是否正确。为了实现面积转换这个问题上的理解型学习,需要理解和运用好下面几件事情:

  1. 一个1平方什么代表的是一个边长为1什么的正方形的面积(这个内容本身仍然可以做一定程度上的理解型学习,在学习面积这个概念的时候)
  2. 把正方形的每一个1什么的边划分成边长为另一个1什么什么单位,从而得到很多个小格子
  3. 运用第一条,看看小格子的面积
  4. 计算有多少个小格子
  5. 总结这样的大格子分小格子的方式和一般性的联系:1平方什么 = 1什么 × 1 什么

有了这个理解,
\begin{align}
1 \mbox{unit}^2 = 1 \mbox{unit} \times 1 \mbox{unit},
\end{align}
那很多事情就非常简单了。例如,1平方千米 = 1千米 × 1千米 = 1000 米 × 1000 米,那肯定不是100平方米。

因此,在这里,最关键的联系就是平方什么的单位,不是独立定义的,而是通过没有加上平方的那个什么的单位来定义的。这一联系建立以后,当然,给大格子划分小格子也是有意义的,就实现了理解型学习。理解型学习的目的是让学习者明白点什么,更具体来时候,就是把一个新的概念和之前学习过的概念通过其知识内部的关系联系起来,从而能够活学活用,还能够降低记忆的成本。

当然,我所提倡的“以概念地图为基础的精简教育体系”除了这个细节上联系的建立,还有更高的要求——我们需要选择哪些最少量的概念和联系来学来教,为什么。也就是说,不仅仅要在学习和传授概念的具体教和学的环节注意建立联系,还要按照学科思想学科基本问题培养学生对学科的情感和学习方法的角度,先建立整个学科知识之间相互联系的大图,然后通过这个大图来挑选少量的概念和联系,做这些选择了的例子的理解型学习。