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CHAPTER 4  第 4 章

Subtraction Across the Grades
各年级的减法

We chose to focus first on subtraction for a couple of reasons. It is an appropriate operation to begin with when introducing older students to Number Talks. Middle and high school students sometimes think that addition problems are "too easy." Also, students typically find subtraction challenging (even though we teach it every year from first grade on), and they are often happy to learn that they can solve subtraction problems in ways that make sense to them.
我们选择首先学习减法有几个原因。在向高年级学生介绍 "数字讲座 "时,从减法开始是合适的。初中和高中学生有时会认为加法问题 "太简单了"。此外,学生通常认为减法具有挑战性(尽管我们从一年级开始每年都教减法),他们通常很高兴了解到他们可以用对自己有意义的方法解决减法问题。
There are two main meanings of subtraction: subtraction as taking away (removing) and subtraction as the difference, or distance, between two numbers. By the time they reach fourth grade, however, students usually think about subtraction as "take away." Understanding subtraction as distance is often overlooked despite its importance. In algebra, geometry, and calculus, students use formulas-for the slope of a line, the distance formula, or for finding the area under a curve-in which subtraction indicates the length of a line segment. (For a thorough discussion of the importance and uses of subtraction as distance in higher mathematics, see Harris 2011.) Therefore, in this chapter we focus on helping students develop an intuitive understanding of subtraction as distance. When students have experienced these concepts through Number Talks, they will have a solid foundation for the mathematics to come.
减法有两个主要含义:减法是拿走(去除);减法是两个数之间的差或距离。然而,到了四年级,学生通常认为减法就是 "拿走"。将减法理解为距离尽管很重要,但却经常被忽视。在代数、几何和微积分中,学生会用到一些公式--如直线的斜率、距离公式或求曲线下的面积--其中减法表示线段的长度。(有关减法作为距离在高等数学中的重要性和用途的详尽讨论,请参见 Harris 2011)。因此,在本章中,我们将重点帮助学生建立对距离减法的直观理解。当学生通过 "数说 "体验了这些概念后,他们将为接下来的数学学习打下坚实的基础。
We use as a sample problem to demonstrate five subtraction strategies that work efficiently across the
我们以 为例题,展示了五种有效的减法策略。

continuum of rational numbers-that is, from whole numbers through fractions, decimals, and percents. Even though some of these strategies might be new to you, resist "teaching" them because students often come up with these strategies on their own.
有理数的连续体,即从整数到分数、小数和百分数。即使这些策略中有些对你来说可能是新的,也不要去 "教 "它们,因为学生通常会自己想出这些策略。

A Note About Recording: The Open Number Line
关于录音的说明:开放式数字线

As you'll see, we often use an "open number line" as a recording strategy during Number Talks to give students a visual model for their thinking.
正如你所看到的,在 "数字讲座 "中,我们经常使用 "开放式数字线 "作为记录策略,为学生提供直观的思维模型。
Open number lines have no scale and thus are not meant to be accurate measures of units. Rather, the "jumps" can be roughly proportional. A nice thing about the open number line is it allows for really large or small numbers without having to worry about individual units.
开放式数列没有刻度,因此并不是用来准确度量单位的。相反,"跳跃 "可以大致成比例。开放式数列的一个好处是,它可以计算非常大或非常小的数字,而不必担心单个单位的问题。

Five Strategies for Subtraction
五种减法策略

Minuend - Subtrahend Difference
Minuend - Subtrahend 差值

1. Round the Subtrahend to a Multiple of Ten and Adjust:
1.将小数四舍五入为十的倍数并进行调整:

"I rounded 28 to 30. Then I subtracted 30 from 63 and got 33. Then I added 2 back because I had taken away 2 too many."
"我把 28 四舍五入为 30。然后从 63 减去 30,得到 33。然后我又加回 2,因为我去掉的 2 太多了"。

2. Decompose the Subtrahend:
2.分解副边:

"First I took 20 from 63 and that was 43 . Then, I saw the 8 in 28 as a 3 and 5; I took away the 3 from 43 first and that was 40 ; then I took away the 5 and that was
"首先,我从 63 中取出 20,这就是 43。然后,我把 28 中的 8 看成是 3 和 5;我先从 43 中去掉 3,就是 40;然后去掉 5,就是 40。

3. Add Instead: 3.改为添加:

There are several ways a student might get from 28 to 63 by adding.
从 28 到 63,学生可以用几种方法进行加法运算。
Way 1: First, get to a multiple of 10: "I started with 28 and added 2 to get 30 ; then I added 33 and got 63. So altogether I added 2 and 33 , or
方法 1:首先,求 10 的倍数:"我从 28 开始,加 2 得 30;然后加 33 得 63。所以,我一共加了 2 和 33 ,即
Way 2: First, get to a multiple of 10 , and then add a multiple of 10: "I started at 28 and added 2 to get to 30 . Then I added 30 to get to 60 , and then I added 3 to get to 63 . I added 2 plus 30 plus 3 to get 35 as my answer."
方法 2:先算出 10 的倍数,再加上 10 的倍数:"我从 28 开始,加上 2,算出 30。然后我把 30 加到 60 ,再把 3 加到 63。我把 2 加上 30 再加上 3,得到的答案是 35"。
or

Way 3: First, add a multiple of 10 : "I started at 28 and jumped 30 to get to 58 . Then I jumped 2 more to get to 60 and 3 more to get to 63. Altogether I jumped 35."
方法 3:首先,加上 10 的倍数:"我从 28 开始,跳 30 到 58。然后我又跳了 2 次,跳到了 60,又跳了 3 次,跳到了 63。我一共跳了 35 次"。
or

4. Same Difference: 4.相同差异:

"I added 2 to 28 and got 30 ; then I added 2 to 63 and got 65 . And 65 minus 30 is
"我把 2 加到 28,得到 30;然后把 2 加到 63,得到 65。65 减去 30 是

5. Break Apart by Place:
5.按地点拆分:

" 60 minus 20 is minus 8 is negative minus 5 is
" 60 减 20 为 减 8 为负数 减 5 为负数

63
35

Developing the Subtraction Strategies in Depth
深入开发减法策略

1. Round the Subtrahend to a Multiple of Ten:
1.将小数四舍五入为十的倍数:

Rounding the subtrahend can be useful for the removal or "take-away" meaning of subtraction. To encourage the use of this strategy, we purposely select problems with a subtrahend (the number that is taken away) that is close to a multiple of ten, one hundred, and so on, so that it "cries out" to be rounded. Taking away a multiple of ten, and then compensating/adjusting, makes subtraction easier while still maintaining the sense of quantity. This strategy is particularly useful when students hang on to the traditional algorithm and need to be coaxed to try something easier.
将小数四舍五入对减法的去除或 "去掉 "意义很有用。为了鼓励学生使用这一策略,我们特意选择了一些问题,这些问题的小头(被拿走的数)接近 10、100 等的倍数,因此 "需要 "四舍五入。去掉一个十的倍数,然后进行补偿/调整,可以使减法变得更容易,同时还能保持数量感。这种策略在学生坚持传统算法,需要哄骗他们尝试更简单的算法时特别有用。
How to choose problems that invite students to Round the Subtrahend:
如何选择问题,让学生对小数进行四舍五入:
We usually start with a few problems that subtract and from a two-digit number, such as:
我们通常从一些从两位数中减去 的问题开始,例如:




Sometimes we find that students more readily use this strategy for two-digit subtrahends that are close to a multiple of 10 , such as:
有时我们会发现,对于接近 10 的倍数的两位数的小数,学生更容易使用这种策略,例如:
Then with a three-digit number minus a two-digit number, we look for two-digit numbers that are close to 100 so that the strategy makes the problem easier and more efficient:
然后,用一个三位数减去一个两位数,我们寻找接近 100 的两位数,这样,该策略就能使问题变得更简单、更高效:
Gradually, you can move the subtrahend farther and farther away from a target multiple-for example, 54 - 28 or 17. The type of problem you choose will depend on the cognitive maturity and/or experience of your students.
逐渐地,你可以让小数点离目标倍数越来越远--例如,54 - 28 或 17。选择哪种类型的问题取决于学生的认知成熟度和/或经验。
Questions that are useful for the strategy of Rounding the Subtrahend:
对小数四舍五入策略有用的问题:
  • Why did you take [200] away instead of [198]?
    你为什么拿走 [200] 而不是 [198]?
  • Did you take away too many or too few?
    你带走的是太多还是太少?
  • Why did you add twice?
    为什么要加两次
This last question, "Why did you add twice?," can reveal soft spots in a student's thinking. Consider the brief vignette below from a fifth-grade classroom:
最后一个问题 "为什么要加两次?"可以揭示学生思维的软肋。请看下面一个五年级课堂的小故事:
Ms. Young writes the problem 43-28 on the board and waits for students to raise their thumbs, indicating that they have figured out the answer.
杨老师在黑板上写下问题 43-28,等待学生翘起大拇指,表示他们已经想出了答案。
Ms. Young: 杨女士
Is anyone willing to share the answer you got?
有人愿意分享你得到的答案吗?
Tim: 蒂姆
Ms. Young: 杨女士
Did anyone get a different answer that you would be willing to share?
有没有人得到了不同的答案,愿意与大家分享?

Jennifer: I got 11.
珍妮弗: 我得到了 11 分。

Ms. Young: Does anyone have a different answer? (No one does.)
杨女士没有人有不同的答案)。
Ms. Young: Is anyone willing to try to convince us that you have an answer that makes sense?
杨女士(以英语发言):有人愿意说服我们,你的答案是有道理的吗?
Jason: 杰森
Ms. Young: 杨女士
Jason: 杰森
Ms. Young: 杨女士
Angel: 天使
I'm defending 15. 28 was hard for me to think about, so I took 30 away from 43 and that gave me 13. But I took away too much so I added 2 back on and I got 15 .
我防守 15 分。28 对我来说很难考虑,所以我从 43 中减去了 30,得到了 13。但是我拿走的太多了,所以我又加了 2,得到了 15。
Why did you add
为什么要加上
When I took away 30, I took away 2 too much, so I had to put 2 back on.
当我拿走 30 块钱时,我拿走的 2 块钱太多了,所以我不得不再放回 2 块钱。
Thank you for getting us started, Jason. Did anyone think about it differently?
谢谢你的开场白,杰森。有人有不同的想法吗?
I did. I got 15 , too, but I started with 28 and added up. I added 2 to get to 30 , and then I added 13 to get to 43. So altogether I added 15 .
是的。我也得到了 15,但我是从 28 开始加起来的。我加了 2,得到 30,然后我加了 13,得到 43。所以我总共加了 15 。
Ms. Young: Does anyone have a question for Angel? (No one does.) Did anyone think about it a different way?
杨女士有没有人有不同的想法?
Jennifer: I know my answer is wrong, but I can't figure out why.
詹妮弗:我知道我的答案是错的,但我不知道为什么。
Ms. Young: Do you want to share what you did? (Jennifer nods.)
杨女士你想分享一下你做了什么吗?(詹妮弗点点头)。
Jennifer: I did it like Jason. I took 30 away from 43 and that was 13 . Since added 2 to 28 , I took the 2 away from 13 and got 11 .
珍妮弗: 我的做法和杰森一样。我从 43 中减去 30,得到 13。因为 在 28 的基础上加了 2,所以我又从 13 里去掉了 2, 得到了 11。
Ms. Young: Why did you add the 2 ?
杨女士为什么要加上 2?
Jennifer: I added it to 28 because 30 was easier to take away.
詹妮弗:我把它加到 28 中,是因为 30 更容易被拿走。
Ms. Young: So when you took away 30, did you take away too many or too few?
杨女士:那么当你拿走30个的时候,是拿走的太多了还是太少了?
Jennifer: I took away too many.
珍妮弗:我拿走了太多。
Ms. Young: You took away too many. So will you have to take away more, or will you have to put some back?
杨女士:你拿走的太多了。那么你是要拿走更多,还是要放回一些?
Ms. Young hoped her questions would help focus Jennifer on the action she had taken so she would know how to compensate for the change she had made. At another time she might have asked the class to try to figure out what Jennifer had done, but Ms. Young was hoping to squeeze in a quick Number Talk this day, and she was sure that this particular confusion would surface again when she would hopefully have more time to let other students talk about this.
杨老师希望她的问题能帮助詹妮弗把注意力集中到她所采取的行动上,这样她就会知道如何弥补她所做的改变。如果换个时间,她可能会让全班同学试着弄清楚詹妮弗做了什么,但杨女士希望今天能挤出时间快速进行一次 "数字谈话",而且她确信,当她希望有更多时间让其他学生谈论这件事时,这个特殊的困惑会再次浮现。
Jennifer: Well ... I have to take away what I added ... Oh, wait. No. Now I see what I did wrong. When I took away 30, I took off 2 too many, so I have to add them back. So now I agree with 15.
珍妮弗: 那么......我得把我加的东西拿走...哦,等等。不,现在我知道我做错了什么。当我去掉 30 时,我去掉了太多的 2,所以我必须把它们加回来。所以现在我同意 15
Ms. Young tucks this away to come back to another day. She knows that this idea can be counterintuitive for students and that very interesting and mathematically important discussions might ensue.
杨老师把这个问题收了起来,准备改天再来讨论。她知道,这个想法对学生来说可能是反直觉的,可能会引发非常有趣和重要的数学讨论。

Rounding the Subtrahend with Fractions and Decimals
分数和小数的小数四舍五入

Rounding the subtrahend works with decimals much like it does with whole numbers. We choose subtrahends that can easily be rounded to a whole number. When there are a different number of decimal places in the subtrahend, students have a little more to think about.
小数的四舍五入与整数的四舍五入一样。我们选择容易四舍五入到整数的小数。如果小数的小数位数不同,学生需要考虑的问题就更多一些。
Decimals Example:
小数 示例


"I rounded 1.97 to 2; then I subtracted 2 from 4.34 and that gave me 2.34. Then I had to add . 03 back because I took away too many. So I got 2.37."
"我把 1.97 四舍五入为 2,然后从 4.34 减去 2,得出 2.34。然后我又加了 0.03,因为我减去的太多了。所以我得到了 2.37"。
Problems to get you started:
问题让你开始
Fractions work the same way, although they may seem harder because of the weak understanding of fractions that some students have. Again, we want to use a subtrahend close to a whole number. We start with denominators in which one is a factor of the other. Fourths and eighths are a good place to start. Here are some examples to get you started.
分数的运算也是如此,不过由于有些学生对分数的理解能力较弱,因此分数的运算可能看起来更难。同样,我们要使用接近整数的小数。我们从一个是另一个因数的分母开始。四分位和八分位就是一个很好的开始。下面有一些例子可以帮助你入门。



2. Decompose the Subtrahend:
2.分解副边:

Decompose (or break up) the Subtrahend is a "removal" strategy that students often take up before other strategies. This is an important strategy because students learn that they can take numbers apart in order to reason in more efficient ways.
分解(或拆分)小半径是一种 "移除 "策略,学生往往先于其他策略学习。这是一种重要的策略,因为学生可以从中了解到,他们可以将数字拆分开来,从而以更有效的方式进行推理。
Decompose the Subtrahend uses students' comfort with subtracting multiples of ten and their fluency with small numbers. Decomposing the subtrahend can give students confidence as they are learning to use strategies that make sense to them.
分解小数利用了学生对 10 的倍数减法的熟悉程度和对小数的流利程度。分解小数可以给学生带来信心,因为他们正在学习使用对他们有意义的策略。

How to choose problems that invite Decompose the Subtrahend:
如何选择邀请分解小数的问题:

Decomposing the subtrahend is a strategy that students use naturally. In order to encourage this strategy, we start with two-digit-minus-one-digit problems where the subtrahend is larger than the ones digit in the minuend and not too close to 10.
分解小数是学生自然使用的一种策略。为了鼓励学生使用这种策略,我们先从两位数减一位数的问题开始,在这些问题中,小数的尾数比尾数中的一位数大,而且不太接近 10。
Problems to get you started:
问题让你开始
Once students have begun to use this strategy, they will apply it to larger problems such as:
一旦学生开始使用这一策略,他们就会将其应用到更大的问题中,如
Unfortunately, this strategy pretty quickly becomes less efficient as the numbers get larger. But using it gives students easy access to thinking about subtraction in new ways and thus makes sense as an early focus in Number Talks. Don't worry about this. You'll find that students will gravitate to strategies that work more efficiently with a broad range of problems.
遗憾的是,随着数字的增大,这种策略的效率很快就会降低。但是,使用这种方法可以让学生轻松地以新的方式思考减法,因此作为 "数说 "的早期重点是有意义的。不用担心这个问题。你会发现,学生会倾向于使用更有效的策略来解决各种问题。

Questions that are useful for Decompose the Subtrahend:
对 "分解小数 "有用的问题:

  • How did you decide what to take away?
    您是如何决定带走什么的?
-Why did you want to break the numbers apart?
-为什么要把数字分开?
  • Did anyone break the subtrahend apart in a different way?
    有谁用不同的方法拆分了副边长?

3. Add Instead: 3.改为添加:

Adding to subtract is an efficient way to do problems that don't work so easily by rounding the subtrahend. The idea that they might never have to subtract again delights many students. And, when recording on an open number line, this strategy also sets the stage for understanding subtraction as the distance between two numbers.
加法到减法是一种高效的方法,可以解决那些用四舍五入法不容易解决的问题。许多学生一想到他们可能再也不用做减法了,就非常高兴。而且,在开放的数线上记录时,这种策略还能帮助学生理解减法是两个数之间的距离。
How to choose problems that invite students to Add Instead:
如何选择能吸引学生 "反向添加 "的问题:
When students see two numbers that are close together, someone will usually find the difference by adding up. When recording, it is important to make sure students know where the answer is (see pages 39 and 40).
当学生看到两个数字相差无几时,通常会有人通过加法找出差值。记录时,一定要让学生知道答案在哪里(见第 39 和 40 页)。
When choosing problems for this strategy, we look for subtrahends that are much like those that we chose for Round the Subtrahend but are closer together.
在为这一策略选择问题时,我们要寻找与 "四舍五入 "中选择的问题很相似,但相距更近的小数。
We might start with these kinds of problems:
我们不妨从这类问题入手:



Then we move on to these kinds of problems:
然后,我们继续讨论这类问题:
Once students use this strategy, they will be ready to plunge into more complicated problems. Although they might not use the most efficient adding strategies at first, they will gravitate to more efficient moves. In the example problem , students may start with 28 and
一旦学生使用了这一策略,他们就可以着手解决更复杂的问题了。虽然一开始他们可能不会使用最有效的加法策略,但他们会倾向于使用更有效的方法。在例题 中,学生可以先算 28,然后再算 28。
skip by tens: plus 5 to get to 63 . Another student may begin at 28 and jump 2 to get to 30 , then skip by tens, 40 , 50,60 plus 3 is 63 , again adding 35 in all. But over time students will realize that once they have made a jump to a "friendly" number, they can get from that number to any number in just one jump. For example, 28 plus 2 gets them to 30 ; then a jump of 33 gets them to 63 .
跳过十位: 加 5 得 63。另一个学生可能从 28 开始,跳 2 到 30,然后跳到 10,40、50、60 加 3 是 63,总共再加 35。但随着时间的推移,学生会意识到,一旦他们跳到一个 "友好 "的数字,他们只需跳一次就可以从这个数字跳到任何数字。例如,28 加 2 可以得到 30;然后跳 33 可以得到 63。

Questions that are useful for the Add Instead strategy:
对 "相反添加 "策略有用的问题:

  • How did you decide your first move?
    你是如何决定第一次行动的?
  • Did anybody use this strategy but make different jumps?
    有没有人使用这种策略,但跳得不一样?
  • How do you know what the answer is?
    你怎么知道答案是什么?

Add Instead with Fractions and Decimals
用分数和小数代替加法

Add Instead is a great strategy for fractions and decimals because it gives students a fresh new way to think about subtraction. To choose problems, we use the same principles as we did with whole numbers, except that with fractions we
用 "加法代替 "来计算分数和小数是一种很好的策略,因为它为学生提供了一种全新的思考减法的方法。在选择问题时,我们使用的原则与处理整数问题时相同,只是在处理分数问题时,我们要

are careful to choose-initially, at least-"friendly" denominators.
小心选择--至少在开始时--"友好 "的分母。
Decimals Example: 1.03 - 96
小数 示例:1.03 - 96
A student who is Adding Instead might say, "I added four hundredths to get to one whole. Then I added three more hundredths to get to 1.03 . So altogether I added seven hundredths."
学生在做 "加法运算 "时可能会说:"我加了四个百分之一,得了 1 整数。然后我又加了三个百分之一,得 1.03。 所以我一共加了七个百分之一"。
Recording might look like this:
录音可能是这样的

Problems to get you started:
问题让你开始




Fractions Example:
分数示例:

A student using this strategy might say, "I added to get to 2 ; then I added 1 to get to 3; then I added again to get to . So altogether I added .
使用这种策略的学生可能会说:"我加上 ,得到 2;然后我加上 1,得到 3;然后我再加上 ,得到 。所以我总共加了
Recording might look like this:
录音可能是这样的
Problems to get you started (note the denominators):
让你开始学习的问题(注意分母):
After students become more flexible with these denominators, you are the best judge of what to try. Every new problem will give you information about where you might go next. The sky's the limit!
在学生对这些分母的处理变得更加灵活之后,你就是尝试什么的最佳评判者。每一个新问题都会为你提供下一个可能的方向。天无绝人之路!

4. Same Difference: 4.相同差异:

The Same Difference strategy relies on the notion of subtraction as a distance or a length that can be moved back and forth on a number line to find a convenient location for solving the problem. Because this strategy focuses on subtraction as distance, it prepares students to understand why subtraction makes sense in formulas like this when they get to algebra:
同差策略依赖于减法是距离或长度的概念,可以在数线上来回移动,找到方便解题的位置。由于这一策略侧重于将减法看成是距离,因此可以帮助学生在学习代数时理解为什么减法在这样的公式中是有意义的:
Same Difference is a truly wonderful idea for students who make sense of it, which even young children can do quite
对于能够理解 "相同与差异 "的学生来说,这确实是一个非常好的想法,即使是年幼的孩子也能做到这一点。

easily. Although this strategy works well for any numbers, it is one that students rarely invent for themselves, so a good way to introduce this strategy is through a class investigation. The second investigation in Chapter 9 has students investigate the Same Difference strategy and whether it will always work.
很容易。虽然这种策略对任何数字都很有效,但学生很少会自己发明这种策略,因此通过课堂探究是介绍这种策略的好方法。第 9 章中的第二个调查让学生研究同差策略以及它是否总是有效。
How to choose problems that invite students to use the Same Difference strategy:
如何选择能让学生使用 "相同差异 "策略的问题:
Same Difference can be used with all kinds of subtraction problems, but students have to think about how much to add or subtract to make the problem easier to compute. To nudge them toward this strategy, we choose problems whose subtrahend is closer to a multiple of ten or one hundred than is the minuend.
同差法可以用于各种减法问题,但学生必须考虑加减多少才能使问题更容易计算。为了引导他们采用这种策略,我们选择那些小数比最小数更接近十或一百的倍数的问题。
Problems like these may tempt students to round the subtrahend and then add or subtract the same number to the minuend.
这样的问题可能会诱使学生将小数四舍五入,然后将相同的数加减到小数。

Questions that are useful for the Same Difference strategy:
对 "相同差异 "策略有用的问题:

  • How do you know the distance is the same between the numbers?
    你怎么知道这些数字之间的距离相同?
  • Why did you shift to
    你为什么转向
  • Did anyone use this same strategy in a different way?
    有没有人以不同的方式使用相同的策略?

Same Difference with Fractions and Decimals
分数和小数的相同差值

Same Difference is a brand new way for students to think about subtraction of decimals and fractions.
同差》是学生思考小数和分数减法的全新方法。

Decimals Example: 3.76 - 1.99
小数 示例3.76 - 1.99

This is an example of a problem that the Same Difference strategy makes really easy. A student could say, "I added one one-hundredth to 1.99 and to 3.76. That changed the problem to 3.77 minus 2 , so the answer is 1.77 ."
这是一个 "同差策略 "使问题变得非常简单的例子。学生可以说:"我在 1.99 和 3.76 的基础上加了百分之一。这样问题就变成了 3.77 减 2 ,所以答案是 1.77 。
Problems to get you started:
问题让你开始




Fractions Example:
分数示例:

A student might say, "I added to both numbers. plus is 2 , and plus is . So, minus 2 is .
学生可能会说:"我把 加上这两个数字。 加上 是 2 , 加上 。因此, 减去 2 是
Problems to get you started:
问题让你开始




Once students get more comfortable, you can try problems like these.
一旦学生适应了,就可以尝试这样的问题。




Same Difference with Integers
整数的相同差异

Most students enter the upper grades with a mishmash of rules and tricks for subtracting integers but rarely have the opportunity to make sense of what is really happening. But if your students understand the Same Difference on the number line well enough to use it as a tool for their thinking, then this strategy can help them make sense of subtracting
大多数学生在进入高年级时,对整数减法的规则和技巧已经烂熟于心,但却很少有机会弄清其中的真谛。但是,如果你的学生能够很好地理解数线上的 "同差",并将其作为思考的工具,那么这种策略就能帮助他们理解减法的意义。

integers-maybe for the first time. The goal is to get them thinking about difference as distance.
整数--也许是第一次。目的是让他们把差当作距离来思考。
Another challenge that students have with negative numbers is their format, and it is important for students to be flexible with this. So we have written negative numbers in three different ways: with parentheses, with the negative sign raised, and with the negative sign looking exactly like a subtraction sign.
学生在学习负数时遇到的另一个难题是负数的书写格式,学生必须灵活掌握。因此,我们用三种不同的方式来书写负数:带括号、负号上扬以及负号看起来完全像减号。
  • Again, the negative numbers are purposely written in the variety of ways that students may encounter them. This helps students become more flexible with symbolic notation.
    同样,负数的写法也特意考虑到了学生可能会遇到的各种情况。这有助于学生更灵活地使用符号符号。

Example Problem: 5 - (-3)
例题:5 - (-3)

A student might say, "I added 3 to both numbers so that I would be subtracting 0 . Negative 3 plus 3 is 0 , and 5 plus 3 is 8 , so 8 minus 0 is 8 .
学生可能会说:"我把两个数都加了 3,这样就等于减去了 0。负 3 加 3 是 0,5 加 3 是 8,所以 8 减 0 是 8。
Or they might look at the number line and see that the distance between 5 and -3 is 8 units.
或者,他们可能会看数线,发现 5 和 -3 之间的距离是 8 个单位。
But is the answer positive or negative? The "Play Around with These" investigation in Chapter 9 gives students (and you) a chance to fiddle around with this and see what you can find out.
但答案是肯定的还是否定的?第 9 章中的 "玩一玩这些 "探究给了学生(和你)一个机会去玩一玩,看看你能发现什么。
Problems to get you started:
问题让你开始
After these kinds of problems, you can challenge students with problems like these:
做完这类问题后,可以向学生提出类似的问题:

5. Break Apart by Place:
5.按地点拆分:

Before they have been exposed to algorithms, which teach students to start from the right in addition and subtraction, children naturally add and subtract by starting from the left (Kamii 2000). The Break Apart by Place strategy can refocus students' attention on place value and maintain the relationship among the quantities of the minuend, subtrahend, and difference. This strategy can emerge naturally from young children, who very early develop an intuition about
算法教导学生在加减法中从右边开始,而在接触算法之前,儿童自然会从左边开始加减(Kamii,2000 年)。按位数分拆的策略可以让学生重新关注位值,并保持小数、减数和差数之间的数量关系。这种策略可以从幼儿那里自然产生,因为他们很早就对 "位置 "产生了直觉。

negative numbers. It is unfortunate, though, that this happens very rarely once students
负数。但遗憾的是,这种情况很少发生在学生身上。
have learned rules for subtraction of negative numbers (which haunt them throughout high school). As Phil Daro (2014), a principal author of the Common Core State Standards, recently observed, "Sense-making is a basic human response, and we have to be trained to suppress it." If you think it would help your students to make sense of subtraction, it is likely that you will need to introduce this strategy. Saying something like, "I saw someone solve this problem in a way I had never thought of, but I tried it on this problem. Here's what I did...." Once they are introduced to the strategy, many of your students will gravitate to it-if it makes sense to them.
他们已经学会了负数减法的规则(这些规则一直困扰着他们的高中生活)。正如《共同核心州立标准》的主要作者菲尔-达罗(Phil Daro)(2014 年)最近指出的那样:"感性认识是人类的基本反应,我们必须接受训练来抑制这种反应"。如果你认为这有助于学生理解减法,你很可能需要引入这一策略。可以这样说:"我看到有人用一种我从未想过的方法来解决这个问题,但我在这道题上试了一下。这就是我的做法...."。一旦向学生介绍了这一策略,如果他们觉得有意义,很多学生就会喜欢上这一策略。

How to choose problems that invite Break Apart by Place:
如何选择 "因地制宜 "的问题:

This strategy works efficiently for nearly any whole number subtraction problem.
几乎所有整数减法问题都能有效地运用这一策略。




Questions that are useful for this strategy:
对这一战略有用的问题:

  • Did you think about it as 6 minus 3 or ?
    你认为是 6 减 3 还是
  • How do you know that 30 minus 70 is negative 40 ?
    你怎么知道 30 减 70 等于负 40 呢?

Break Apart by Place with Decimals
用小数按位数拆分

This strategy also works effectively with decimals. Most of our students have little understanding of "where the decimal
这种策略对小数也很有效。我们的大多数学生对 "小数点在哪里 "缺乏了解。

point goes," and this strategy can help give them a better sense of the place value of the digits.
点去",这种策略可以帮助他们更好地理解数位的位值。

Decimals Example: 5.2 - 1.5
小数举例:5.2 - 1.5

"I took 1 away from 5 , and that was 4 ; then I took 5 tenths away from 2 tenths and I got negative 3 tenths. So then I took 3 tenths away from 4 and I got 3 and 7 tenths."
"我从 5 减去 1,得到 4;然后我从十分之二减去十分之五,得到负十分之三。然后我又从 4 中减去十分之三,得到十分之三和十分之七"。

Problems to get you started:
问题让你开始
Students will come up with other strategies than the five main ones we have identified. Let's go back to our example problem, , to explore some of these strategies.
除了我们已经确定的五种主要策略之外,学生们还会想出其他策略。让我们回到例题 ,探讨其中的一些策略。
Round both numbers: Some students will round both numbers to make the problem 60 minus 30 . While 60 minus 30 is easy to solve, the problem with rounding both numbers is that it is often difficult for students to sort out what they have done and how to compensate for both changes they have made. Don't worry about this strategy if it comes up, because
将两个数都四舍五入:有些学生会把两个数都四舍五入,使问题变成 60 减 30。虽然 60 减 30 很容易解决,但把两个数都四舍五入的问题是,学生往往很难理清他们做了什么,以及如何补偿他们所做的两个改动。如果出现这种情况,不要担心,因为

children will quickly gravitate to strategies that work more efficiently.
孩子们会很快倾向于更有效的策略。
Adjust the minuend: Some students will add 5 to 63 to change the problem to 68 minus 28 for an answer of 40 , then subtract the 5 that they added to the 68 for an answer of 35 . It is actually a good thing for students to learn that you can change either the minuend or subtrahend to make the problem easier. They will have to look carefully at the action taken in order to know how to compensate for the changes they make.
调整最小值:有些学生会在 63 的基础上加 5,使问题变为 68 减 28,得到 40 的答案;然后再减去他们在 68 的基础上加的 5,得到 35 的答案。 其实,让学生知道可以改变最小尾数或最小尾数来使问题变得简单是件好事。他们必须仔细观察所做的操作,以便知道如何补偿他们所做的改动。
Round the minuend: Other students will take 3 away from 63 to change the problem to 60 minus 28 . Students discover pretty quickly, though, that taking away a multiple of 10 or 100 is much easier than taking away any number from a multiple of 10 or 100 , so they often abandon this method early on.
将最小值四舍五入:其他学生会从 63 中取 3,把问题改为 60 减 28。不过,学生很快就会发现,从 10 或 100 的倍数中去掉一个数比从 10 或 100 的倍数中去掉任何数都要容易得多,所以他们往往很早就放弃了这种方法。
You won't want to discourage any methods when they come up. Instead, celebrate students' efforts to try out different ways to make sense of subtraction. Efficiency is not the goal at first. A focus on efficiency too early can put students back into remembering rather than sense-making. Instead, show them how pleased and excited you are that they are solving subtraction problems in ways that make sense to them. And eventually, as students see more and more strategies, the cumbersome ones will fall by the wayside.
当学生提出任何方法时,你都不要打击他们的积极性。相反,你应该鼓励学生尝试不同的方法来理解减法。一开始不以效率为目标。过早地关注效率会让学生回到记忆而非感悟的状态。相反,向他们展示你对他们用有意义的方法解决减法问题是多么高兴和兴奋。最终,当学生看到越来越多的策略时,那些繁琐的策略就会被淘汰。

Inside a Seventh-Grade Classroom: Digging into a Mathematical Error
走进七年级课堂:挖掘数学错误

Ms. Aho has been doing Number Talks with her seventhgrade students for several months. Today, she writes .79 on the document camera, then waits as students work to solve the problem mentally.
几个月来,阿霍女士一直在与她的七年级学生进行 "数字谈话"。今天,她在文件摄像机上写下 .79,然后等待学生们用头脑解决问题。
As they solve the problem, students quietly put their fist on their chest with a thumb up to indicate that they have a way of solving the problem. Some students
在解决问题的过程中,学生们会悄悄地把拳头放在胸前,竖起大拇指,表示自己有办法解决问题。有些学生
who have found one solution look for additional ways to solve the problem and indicate each new way by showing an additional finger. Ms. Aho waits until everyone has had time to solve the problem, knowing that those who finish quickly will dig into the problem in search of additional solutions.
找到一种解题方法的学生会寻找其他解题方法,每找到一种新方法,就多竖一根手指。阿霍女士会等到每个人都有时间解决问题,因为她知道,那些很快就完成的人会深入研究问题,寻找更多的解决方案。
When most thumbs are up, she asks, "Is anyone willing to share the answer they got?" (Previously she had talked with students about the importance of not indicating if they agree with an answer that is offered so that everyone has an opportunity to share their answer and she has the added benefit of informally assessing if students are successful at solving the problem.) Students offer two answers, 3.08 and 3.06, which she records without comment.
当大多数人都竖起大拇指时,她会问:"有人愿意分享他们得到的答案吗?(在此之前,她曾与学生们讨论过不表示是否同意所提供答案的重要性,这样每个人都有机会分享自己的答案,而且她还可以非正式地评估学生是否成功地解决了问题)。学生们给出了 3.08 和 3.06 两个答案,她将其记录下来,未作任何评论。
Ms. Aho: 阿霍女士
Who is willing to convince us that you have an answer that makes sense by telling us what you did? (She calls on Michelle.) Which answer are you defending?
谁愿意告诉我们你做了什么,让我们相信你的答案是有道理的?(她请米歇尔发言)你为哪个答案辩护?
Michelle: 米歇尔
I'm defending 3 and 8 hundredths. I took 80 hundredths from 3 and 87 hundredths and got 3 and 7 hundredths. But I took
我捍卫的是 3 又 8 个百分之一。我从 3 分之 87 提取了 80 个百分点,得到了 3 分之 7。但我从

away too much, so I added back the extra 1 hundredth that I had taken away, and my answer is 3 and 8 hundredths.
所以我又把多去的 1 个百分之一加了回来,我的答案是 3 又 8 个百分之一。

Michelle 米歇尔

(Note: When Ms. Aho first started working with decimals, students always read decimals like 3.87 as "Three point eight seven" [which is an almost universal response from middle and high school students]. But she knows that reading decimals like that can obscure the value of the digits, so the first time the issue arose, she said, "But what does that really mean?" or "Can you read the number without saying the word point?" So, at this time in the year, this habit had been happily eradicated.)
(注:阿霍女士刚开始教小数时,学生们总是把 3.87 这样的小数读成 "三点八七"(这几乎是初高中学生的普遍反应)。但她知道,这样读小数会掩盖数位的价值,所以第一次出现这个问题时,她就说:"但这到底是什么意思呢?"或者 "你能不说点这个词来读这个数吗?"所以,在今年的这个时候,这个习惯已经被愉快地根除了)。
There are several questions that Ms. Aho could ask at this moment. They include "How many of you solved it the same way as Michelle did?" This question gives her a quick assessment of who is using which strategy. Also, if there are only a few hands up, then she knows that there are
此时此刻,阿霍女士可以提出几个问题。其中包括 "你们当中有多少人的解题方法和米歇尔一样?这个问题可以让她快速评估谁在使用哪种策略。另外,如果只有几个人举手,那么她就知道有

other strategies that students have used. Or she could ask, "Who has a question for Michelle?" In this case, Ms. Aho asked both questions.
学生使用过的其他策略。或者她可以问:"谁有问题要问米歇尔?在这个案例中,阿霍女士问了这两个问题。
A few students raised their hands to show that they had used the same method as Michelle, so Ms. Aho knew that students had used other strategies. Before moving on to another strategy, though, she asked, "Who has a question for Michelle?" Jamie raised his hand.
有几个学生举手示意他们使用了与米歇尔相同的方法,因此阿霍女士知道学生们还使用了其他策略。不过,在转到另一种策略之前,她问:"谁有问题要问米歇尔?"杰米举起了手。
Jamie: I sort of did it the same way, but I didn't get the same answer, and I know what I did wrong now.
杰米:我也是这么做的,但我没有得到同样的答案,现在我知道我做错了什么。
Again, Ms. Aho could go a couple of different directions. Jamie did not have a question about Michelle's strategy; rather, he had learned somethingfrom Michelle's strategy that he wanted to share. Ms. Aho could have said, "Jamie, hold that thought, and we will come back to you. Does anyone have a question for Michelle?" But instead, Ms. Aho chose to follow Jamie.
同样,阿霍女士也可以有几个不同的方向。杰米并不是对米歇尔的策略有疑问,而是他从米歇尔的策略中学到了一些东西,他想与大家分享。阿霍女士本可以说:"杰米,先别着急,我们再回来找你。有人有问题要问米歇尔吗?但是,阿霍女士却选择了跟随杰米。
Ms. Aho toDo you want to share that with us? (Jamie Jamie: nods.)
你想和我们分享一下吗?(杰米-杰米:点头)。
Jamie: 杰米
I started just like Michelle, and I took 80 hundredths from 3 and 87 hundredths, and I got 3 and 7 hundredths. Then I took away the 1 hundredth that I added to the 79 hundredths and got 3 and 6 hundredths.
我和米歇尔一样,从 3 又 87 个百 分点中减去 80 个百分点,得到 3 又 7 个百分点。然后,我又从 79 个百分值中减去了 1 个百分值,得到了 3 又 6 个百分值。
But now I know that I should have added back the one hundredth because Michelle helped me see that I really took away too much to start with.
但现在我知道,我应该把那一百分加回来,因为米歇尔让我明白,我一开始确实拿走了太多。
Ms. Aho has worked hard to create a culture where mistakes are seen as opportunities for new learning rather than something to be ashamed of, and her students are frequently willing to talk about the mistakes they have made. She also knows that talking about Jamie's mistake can help to illuminate ideas that might be confusing for other students as well.
阿霍女士努力创造一种文化,将错误视为新的学习机会,而不是羞于启齿的事情,她的学生经常愿意谈论他们所犯的错误。她还知道,谈论杰米的错误有助于阐明其他学生可能感到困惑的想法。
Ms. Aho: 阿霍女士
Jamie, can you help us understand why you added the .01 instead of subtracting it, or would you like for someone else to try to explain?
杰米,你能帮助我们理解你为什么要加上 0.01 而不是减去它,或者你想让别人来解释一下吗?

Jamie: 杰米

At first I thought I had to subtract the .01 because I added it to .79 . But now I see that when I added it to .79 I subtracted too much. I was supposed to take away .79 and I took away .80. So I had to add the extra .01 that I took away to my answer.
起初我以为必须减去 0.01,因为我把它加到了 0.79。但现在我明白了,当我把它加到 0.79 时,我减去的太多了。我本该减去 0.79,却减去了 0.80。因此,我必须把多减去的 0.01 加到我的答案中。
Now Ms. Aho uses an instructional strategy that is valuable in Number Talks when there is a complicated issue.
现在,阿霍女士使用了一种教学策略,这种策略在 "数字谈话 "中遇到复杂问题时非常有用。
Ms. Aho to theWould you take a minute to talk to people class: around you about what Jamie just explained?
阿霍女士,请你花一分钟时间与你周围的同学谈谈杰米刚才的解释。
Kids huddle and talk quietly. When the talk dies down, which is only a couple of minutes later, Ms. Aho calls the class back together.
孩子们挤在一起小声交谈。几分钟后,孩子们的议论声渐渐平息,阿霍女士又把全班同学叫到一起。
Ms. Aho: Does anyone have a question for Jamie?
阿霍女士有人要问杰米问题吗?
No one does, but she knows better than to think that everyone has followed Jamie's thinking. So she perseveres:
没有人这样做,但她知道最好不要以为每个人都遵循杰米的想法。所以她坚持了下来:
Ms. Aho: 阿霍女士
Jamie, is it okay if we see if someone else can try to explain what your mistake was?
杰米,我们看看别人能不能解释一下你的错误?
Jamie: 杰米
Okay. 好的
Ms. Aho: 阿霍女士
Does anyone think they can help us understand what Jamie's mistake was?
有谁认为他们能帮助我们理解杰米的错误是什么?
Jennifer volunteers. 詹妮弗是志愿者。
Jennifer: I think Jamie added .01 to .79 and got .8 . Then he took .8 away from 3.87 and got 3.07. Then he subtracted .01 , but she should have added.
詹妮弗:我认为杰米在 0.79 的基础上加了 0.01,得到了 0.8。然后他从 3.87 减去 0.8,得到 3.07。然后他减去了 0.01 ,但她应该加上。
Why should he have added instead of subtracting?
他为什么要做加法而不是减法?
Jennifer: 珍妮弗
Because .8 is larger than .79 and he took .8 away, so she needed to add it back in.
因为 0.8 比 0.79 大,而他拿走了 0.8,所以她需要把它加回来。
Ms. Aho: 阿霍女士
Oh, now this is getting interesting! Are you saying Jamie should have added the one hundredth twice? (Turns to the class.) So Jamie and Jennifer both think that after Jamie added the .01 to the .79 , he should have added .01 again to the answer after he subtracted? Turn and talk to some people around you. See if you can figure out why it makes sense to add the .01 twice: first to the .79 and then to the 3.07 after she subtracted.
哦,这下有趣了!所以杰米和詹妮弗都认为,杰米把 0.01 加到 0.79 后,应该在减去 0.79 后再加 0.01?转过身去,与你周围的一些人交谈。看看你们是否能想出为什么要加两次 0.01:先在 0.79 的基础上加 0.01,然后在减法后在 3.07 的基础上加 0.01。
The students talk in small groups for another few minutes. Ms. Aho knew this was an important use of the Number Talk time because adding twice is counterintuitive to many students. She had seen this kind of error before and believed that untangling the issue would help students develop a greater understanding of how subtraction works.
学生们又在小组内讨论了几分钟。阿霍女士知道这是 "数字谈话 "时间的一个重要用途,因为对许多学生来说,两次加法是违反直觉的。她以前曾见过这种错误,并相信解决这个问题将有助于学生更好地理解减法的原理。
Ms. Aho: Who did this problem a different way?
阿霍女士谁用不同的方法解决了这个问题?
After this Number Talk, Ms. Aho will think about what problem to do next. Her students seemed comfortable using multiple strategies for subtracting with decimals, yet their
这次 "数字讲座 "结束后,阿霍女士将考虑下一步要解决什么问题。她的学生在使用多种小数减法策略时显得游刃有余,但他们的

thumbs did not come up as quickly as she had hoped. She decided that posing a similar problem would give additional practice that she knew they needed and would provide an opportunity for Jamie (and others) to confront his earlier mistake. She decided to pose the problem next.
大拇指并没有像她希望的那样很快出现。她决定再提出一个类似的问题,这样既可以增加她知道他们需要的练习,又可以让杰米(和其他人)有机会正视他之前的错误。她决定下一个问题是
While you may start with the same Number Talk at fifth or tenth grade, the trajectory will probably be different because you will base each subsequent Number Talk on what you have learned about students' thinking. A big part of the power of Number Talks is that students can discover things that we, as teachers, might never have thought of. In this way, Number Talks are generative for students and teachers alike in developing new understandings about how numbers and operations work. And while the traditional subtraction algorithm obscures place value, Number Talks depend on students using and understanding place value relationships. Therefore, while subtraction Number Talks help students learn to reason flexibly as they subtract, they also serve to develop students' understanding of important mathematical ideas that go beyond subtraction.
虽然你可能会从五年级或十年级开始进行同样的 "数字讲座",但其轨迹可能会有所不同,因为你以后的每次 "数字讲座 "都会以你所了解到的学生思维为基础。数字讲座的一大威力在于,学生可以发现我们教师可能从未想到过的东西。因此,"数字谈话 "对学生和教师来说都是一种生成,能让他们对数字和运算的原理有新的理解。传统的减法运算法则模糊了位值,而 "数字讲座 "则依赖于学生使用和理解位值关系。因此,减法 "数字讲座 "在帮助学生学会灵活推理减法的同时,也有助于培养学生对减法以外的重要数学思想的理解。

CHAPTER 5 第 5 章

Multiplication Across the Grades
各年级的乘法

In this chapter we focus on generalizable strategies for multiplication that are useful in helping students understand the properties of arithmetic and that provide a foundation for algebra. But first, we need to think about how Number Talks help students learn multiplication problems with single-digit factors.
在本章中,我们将重点讨论乘法的通用策略,这些策略有助于帮助学生理解算术的性质,并为代数打下基础。但首先,我们需要思考 "数字讲座 "如何帮助学生学习带有个位数因数的乘法问题。

Number Talks and Multiplication Facts
数字讲座和乘法口诀

Mastery of addition and multiplication "facts" has been a dilemma for as long we can remember. While there is widespread agreement that quick access to these facts-which we prefer to think of as number combinations-is vital for success in math, the customary approach has been to encourage rote memorization. Flash cards and timed tests have continued to make regular appearances in US classrooms as early as the second grade despite decades of evidence that, at best, they don't work very well-as any middle and high school teacher knows. Timed tests, in particular, which cause many children to dislike and avoid math, have long been associated with math anxiety (Tobias 1978). And as Jo Boaler (2014) points out, "Occurring in students from an early age, math anxiety and its effects are exacerbated over time, leading to low achievement, math avoidance, and negative experiences of math throughout life" (469). Early in our careers, we, too, were expected to use
从我们记事起,掌握加法和乘法 "事实 "就是一个难题。尽管人们普遍认为,快速掌握这些事实--我们更愿意将其视为数字组合--对于数学学习的成功至关重要,但惯用的方法一直是鼓励死记硬背。尽管数十年来的证据表明,闪存卡和定时测试充其量也只是起不到很好的作用,但它们仍然经常出现在美国二年级的课堂上,任何初高中教师都知道这一点。尤其是计时测验,它导致许多孩子不喜欢数学、逃避数学,长期以来一直与数学焦虑有关(Tobias,1978 年)。正如 Jo Boaler(2014 年)所指出的,"数学焦虑在学生幼年时期就会出现,其影响会随着时间的推移而加剧,导致低成就、数学回避以及一生中对数学的负面体验"(469)。在我们职业生涯的早期,我们也被期望使用

these methods, but, knowing what we know now, we so wish we could have those students back again!
但是,根据我们现在所了解的情况,我们真希望这些学生能再回来!
The following analogy helped us make sense of why timed tests and flash cards don't support children's proficiency with numbers and got us thinking about how to help students master number facts in a different way:
下面的比喻帮助我们理解了为什么计时测验和闪存卡不能帮助孩子们熟练掌握数字,并让我们思考如何以不同的方式帮助学生掌握数字知识:
Imagine a stack of cards like this:
想象一下这样一叠卡片:

Eight different letters are randomly paired with each of the other letters to produce 64 different cards; each combination has an "answer" written on the back of the card (for ? by the way, the answer is ).
8 个不同的字母与其他每个字母随机配对,产生 64 张不同的卡片;每种组合的卡片背面都写有 "答案"(对于 ? ,答案是 )。
Suppose we are asked to memorize the answers to all 64 combinations. We know the names of the letters, and with a little practice we can remember that ? . We might even notice that ? also is equal to . We practice, over and over. But it's hard to remember them all. And imagine if someone timed us to see how fast we could say them!
假设要求我们记住所有 64 种组合的答案。我们知道字母的名称,稍加练习就能记住 ? 。我们甚至会注意到 ? 也等于 。我们反复练习。但要记住所有的字母是很难的。想象一下,如果有人给我们计时,看我们能以多快的速度说出它们!
This scenario is not unlike what learning basic facts is like for many children. One could argue that this analogy isn't fair-that letters and numbers are different because combinations of letters are unrelated to their answer, while numbers have relationships that can help the answers make sense. We agree completely! Without inherent relationships,
这种情况与许多孩子学习基本事实的情况并无二致。有人可能会说,这种类比并不公平--字母和数字是不同的,因为字母组合与答案无关,而数字则有关系,可以帮助答案变得有意义。我们完全同意!没有内在联系

letter combinations can only be learned by rote memorization or mnemonic devices - hard to learn and easy to forget.
字母组合只能通过死记硬背或记忆法来学习--难学易忘。
We have come to realize, however, that flash cards and timed tests treat the number combinations as if they are, like letter combinations, unrelated to their answers. But number combinations do have inherent patterns and relationships that, when explored and understood, help students learn and use the multiplication facts with flexibility and confidence.
然而,我们逐渐意识到,闪卡和计时测验将数字组合视为与字母组合一样,与答案无关。但是,数字组合确实有其内在的规律和关系,只要加以探索和理解,就能帮助学生灵活、自信地学习和运用乘法口诀。
We are sometimes asked, "Does it matter if students learn multiplication facts through Number Talks or with flash cards and timed tests, just as long as they learn them?" Yes, it matters! We might think flash cards and timed tests can't hurt, but they can. They give students a false idea about what mathematics is and about what it means to be good at math. (For further information about the damage done by timed tests, see Boaler 2014.)
有时我们会被问到:"只要学生学会了乘法口诀,是通过数字讲座还是通过闪卡和计时测验来学习乘法口诀,这重要吗?是的,这很重要!我们可能会认为闪卡和计时测验不会有什么坏处,但其实是有坏处的。它们会让学生对数学是什么以及学好数学意味着什么产生错误的认识。(有关定时测试所造成伤害的更多信息,请参阅 Boaler 2014)。
Number Talks can give every student the chance to master-and understand-the multiplication facts. Here is a brief glimpse into a seventh-grade class as students are engaged in a Number Talk:
数字讲座可以让每个学生都有机会掌握并理解乘法口诀。下面是一个七年级班级的简短片段,学生们正在进行 "数字谈话":

As soon as is written on the document camera, a bunch of thumbs go up. After waiting for everyone's thumb, the teacher calls on a student who says, "56."
写在文件摄像机上时,一群人竖起了大拇指。等所有人都竖起大拇指后,老师叫一个学生说:"56"。
Mr. Hoffman: Did anyone get a different answer? (No one admits to it.)
霍夫曼先生:有人得到不同的答案吗?
Mr. Hoffman: Who can explain how you got it?
霍夫曼先生:谁能解释一下你是怎么得到它的?
Susanne: I just knew it.
Susanne: 我就知道。
Mr. Hoffman: Did anyone think about 7 times 8 in a different way? (Again, no one. But there are probably a few students in the room who counted by 7 eight times, keeping track on their fingers under the desk, and others who just waited for someone else to respond.)
霍夫曼先生:有没有人以不同的方式思考过 7 乘 8 的问题?(还是没有。不过,教室里可能有几个学生用 7 乘 8 的方法数了 8 次,他们在桌子下面用手指记数,还有一些学生只是等着别人回答)。
Mr. Hoffman: It sounds like everyone just knows that 7 times 8 equals 56 . But let's explore this a little bit and think about how you could work it out if you didn't know. So, pretend you don't know. What would be an easy way to figure out 7 times 8 quickly? (The teacher waits for what seems too long until enough hands are up.)
霍夫曼先生:听起来好像每个人都知道 7 乘以 8 等于 56 。但是,让我们来探讨一下这个问题,想一想如果你不知道,该怎么算出来。那么,假装你不知道。有什么简单的方法可以快速算出 7 乘 8 呢?(老师等了很久,直到有足够多的人举手)。
Marta: 玛塔
I know 7 times 7 is 49 , so I added one more 7 and got 56 .
我知道 7 乘以 7 等于 49,所以我又加了一个 7,得到了 56。
Mr. Hoffman: Why did you add one more 7?
霍夫曼先生(以英语发言):为什么要增加一个 7?
Marta: 玛塔
Mr. Hoffman: 霍夫曼先生
Marta: 玛塔
Mr. Hoffman: 霍夫曼先生
Jacob: 雅各布
Teresa: 特雷莎
I needed eight 7s, but I only had seven.
我需要八个 7,但我只有七个。
So you added one more 7 to 49 . How did you do that?
所以你在 49 的基础上又加了一个 7 。你是怎么做到的?
I know 7 times 7 is 49 , so I added 1 to 49 to get 50 , then added 6 more.
我知道 7 乘以 7 等于 49 ,所以我在 49 的基础上加 1 得到 50 ,然后 再加 6。
Who thought about it a different way? (No one.) Well, let's think about this. How else could we do it if we didn't know what 7 times 8 is?
谁有不同的想法?(没有人。)好吧,让我们想一想。如果我们不知道 7 乘以 8 是多少,还能怎么做呢?
Well, 4 times 7 is 28 , so if you add 28 and 28 , that would be the same.
那么,4 乘以 7 等于 28 ,所以如果把 28 和 28 相加,结果是一样的。
10 times 7 is 70 , and you could take away two , or 14 , and that's 56 .
10 乘以 7 是 70 ,再去掉两个 ,即 14,就是 56。
When you engage your students in a Number Talk like this, continue asking, "How else?" and "How else?" And don't forget to ask students why their strategies make sense when you think it will help others understand.
当你让学生参与这样的 "数字谈话 "时,继续问 "还能怎样?"和 "还能怎样?"当你认为学生的策略有助于他人理解时,别忘了问他们为什么这样做有意义。
But what are the students learning during this Number Talk that they don't learn through flash cards and timed tests? They are learning that they have mathematical ideas worth listening to-and so do their classmates. They are learning not to give up when they can't get an answer right away
但是,学生们在 "数字谈话 "中学到了什么,是他们通过闪存卡和计时测验所学不到的呢?他们在学习,他们有值得倾听的数学想法,他们的同学也有。他们学会了在无法立即得到答案时不要放弃

because they are realizing that speed isn't important. They are learning about relationships between quantities and what multiplication really means. They are using the properties of the real numbers that will support their understanding of algebra.
因为他们意识到速度并不重要。他们正在学习数量之间的关系以及乘法的真正含义。他们正在利用实数的性质来帮助理解代数。
And what about the Mathematical Practices? Here are just a few that students used during this one brief Number Talk:
那么数学实践又是什么呢?以下是学生们在这次简短的 "数字讲座 "中使用的一些方法:
  • Make sense of quantities and their relationships [MP2]
    理解数量及其关系 [MP2]
  • Justify their conclusions
    证明其结论的合理性
  • Communicate precisely to others [MP6]
    与他人准确沟通 [MP6]

Multiplication Across the Grades
各年级的乘法

Multiplication Number Talks are brimming with potential to help students learn the properties of real numbers (although they don't know it yet), and over time, the properties come to life in students' own strategies.
乘法数字讲座 "在帮助学生学习实数的性质(尽管他们还不知道)方面潜力无穷,而且随着时间的推移,这些性质会在学生自己的策略中栩栩如生地展现出来。
Before this can happen, though, we have a delicate problem, especially for many middle and high school students. If, after 8 or more years in school, a student has little experience reasoning about arithmetic and only the traditional algorithms to rely on, then we need to give patient attention to helping them break free.
不过,在此之前,我们还有一个棘手的问题,尤其是对许多初高中学生而言。如果学生在学校学习了 8 年或更长时间后,几乎没有推理计算的经验,只能依靠传统的算法,那么我们就需要耐心地帮助他们摆脱困境。

Liberating Students from Their Dependence on Rote Procedures
让学生摆脱对死记硬背程序的依赖

When students have had little experience thinking with numbers, it is natural that they resort to traditional algorithms. We have devoted a small section of this chapter to exploring how to help students move into sense-making if they get stuck in the algorithm. We have also devoted a section to this in Chapter 10.
当学生缺乏用数字思考的经验时,他们自然会求助于传统的算法。在本章中,我们用了一小部分的篇幅来探讨如果学生在算法中陷入困境,如何帮助他们进入感性认识。在第 10 章中,我们也有一节专门讨论这个问题。
There is no direct route here; every class is different. Students come with varying understandings, experiences, and confidence in themselves as mathematical thinkers. And they don't often come to us with a disposition to work on multiplication in ways other than the standard algorithm. It can help to just talk with your students honestly about this. For a glimpse at what this might look like, we offer the following excerpt from a Number Talk that Ruth did when she was visiting an eighth-grade class in California. We enter about ten minutes into a Number Talk where these eighth graders are doing mental computation to solve the problem 12 . Number Talks were brand new to this group of students.
这里没有直接的途径,每个班级都是不同的。学生的理解、经验和对自己作为数学思考者的信心各不相同。而且,他们往往并不愿意用标准算法以外的方法来学习乘法。与学生坦诚地讨论这个问题会有所帮助。下面是露丝在加利福尼亚州访问一个八年级班级时进行的 "数字谈话 "节选,我们可以从中一窥究竟。我们进入 "数字讲座 "大约十分钟,这些八年级学生正在进行心算,以解决 12 这个问题。对这群学生来说,"数字讲座 "是全新的。

Students have given four answers that are listed on the board: 116, 206, 216, and 204.
学生们给出了黑板上列出的四个答案:116、206、216 和 204。
Keanon: I did 10 times Well ... I broke the 12 into 10 plus 2, and then I did 10 times 18 and got 180 . Then I did 2 times 18.
基农:我做了 10 次 好吧......我把 12 分解成 10 加 2,然后做了 10 乘以 18,得到了 180 .然后我又做了 2 乘以 18。
Ruth: 露丝
Which answer are you defending?
你在为哪个答案辩护?
Keanon: 216.
Ruth: 露丝
Okay, so when you multiplied 2 by 18 , what did you get?
好吧,那么当你用 2 乘以 18 时,你得到了什么?
Keanon: 基农
  1. So I added 180 and 36.
    所以我又加了 180 和 36。
Ruth: 露丝
How did you add them?
你是如何添加它们的?
Keanon: 基农
I added 0 and 6 . Then I added 8 and 3, then
我加了 0 和 6 。然后加上 8 和 3
I put the 1 by the other 1 , and I got 216 .
我把 1 放在另一个 1 旁边,结果是 216。
Ruth notices that Keanon slipped right back into the traditional algorithm when adding the partial products. She knows that this happens frequently when students are becoming familiar with Number Talks. Initially, they have a tendency to think creatively about the topic at hand-in this case, multiplication-but fall back on the familiar traditional algorithms without seeming to notice. She decides not to mention this now because Keanon had at first been reluctant to share his thinking.
露丝注意到,基农在进行部分积的加法运算时又回到了传统算法。她知道,当学生开始熟悉 "数说 "时,这种情况经常发生。起初,他们倾向于对手头的主题进行创造性思考--本例中就是乘法--但似乎没有注意到,他们又回到了熟悉的传统算法上。她决定现在不提这个问题,因为基农起初并不愿意分享他的想法。

Ruth: 露丝

How many of you used the same method as Keanon?
有多少人使用了与基农相同的方法?
(One other hand goes up.)
(另一只手举起)。
That means there are more strategies out there. Who is willing to share a different one? Elizabeth, what answer are you defending?
这意味着还有更多的策略。谁愿意分享不同的策略?伊丽莎白,你在捍卫什么答案?
Elizabeth: Well, I moved the 12 under the 18. And I did 2 times 8 and got 16 , so I put down the 2 and put the 1 above the 1 .
伊丽莎白我把 12 移到了 18 的下面。我把 2 乘以 8,得到 16 ,所以我把 2 放下来,把 1 放在 1 的上面。
Ruth writes , with the 12 under the 18 , and records what Elizabeth has said so far.
路得在 上写下了 18 岁以下的 12 个孩子,并记录了伊丽莎白迄今为止所说的话。
Ruth, turning toElizabeth has used what we call the the class: "traditional algorithm." How many of the
露丝和伊丽莎白使用了我们所说的 "类"传统算法"有多少

rest of you used the traditional algorithm? (Most of the students raise their hands.)
你们还有谁用过传统算法?(大多数学生举手)。
Most of you-that's the way I was taught to multiply, too. And I was already teaching before I learned that there are easier ways to multiply. So, I have some bad news for you: we were all taught to work way too hard. Number Talks help us learn to work smart and efficiently, and I know you'll all learn to do that. As soon as the problems get bigger, the traditional algorithm is going to become almost impossible to do mentally. Did anybody do it a different way?
你们中的大多数人也是这样教我乘法的。在我知道有更简单的乘法之前,我已经在教书了。所以,我要告诉你们一个坏消息:我们都被教得太辛苦了。数字讲座帮助我们学会聪明、高效地工作,我知道你们都会学会这样做的。一旦问题变得越来越大,传统算法就会变得几乎无法在头脑中完成。有人用过其他方法吗?
Sean: 肖恩
Well, I did it differently, but I don't know if it's right.
我换了一种方法,但不知道对不对。
Ruth: 露丝
Thanks for being willing to share when you aren't sure! Which answer are you defending, Sean?
谢谢你在不确定的时候愿意分享!你在为哪个答案辩护,肖恩?
Sean: 216.
Ruth: 露丝
Can you explain what you did and why it makes sense?
你能解释一下你的做法以及为什么这样做有意义吗?

added 144 and 72.
增加了 144 和 72。
Ruth: 露丝
How did you add 144 and
您是如何添加 144 和
Sean: 肖恩
I did it like Keanon. I moved the 72 under the 144.4 and 2 is 6 , and 7 plus 4 is 11 . So I carried the 1 and I got 216 .
我的方法和基农一样。我把 72 移到 144.4 下面,2 是 6,7 加 4 是 11。所以我把 1 移到了 216 .
Ruth: Oh-you used the traditional algorithm for addition. Does anybody have a question for Sean? (No one does.)
露丝: 哦--你用的是传统的加法算法。有人有问题要问肖恩吗?(没人有)。
Ruth: 露丝
Who would like to tell how they got a different answer and why it makes sense? (No one does.)
谁愿意告诉大家他们是如何得到不同答案的,以及为什么这个答案是有道理的?(没人想)。
This doesn't surprise Ruth because she has seen many students change their mind after becoming convinced by others' explanations. She knows that as students become comfortable with Number Talks, they begin to share mistakes that they made.
露丝对此并不感到惊讶,因为她见过很多学生在被别人的解释说服后改变了主意。她知道,随着学生对 "数字讲座 "的适应,他们会开始分享自己犯过的错误。
Ruth: 露丝
Thank you for sharing your thinking today. Maybe tomorrow more of you will have a chance to share.
感谢你们今天分享了自己的想法。也许明天会有更多的人有机会分享。
Ruth now is thinking about what problem to do tomorrow. She decides that might be a good way to go, because it is enough like that students can build on the methods that were shared today.
露丝现在正在考虑明天要做什么问题。她认为 可能是一个不错的方法,因为它与 很相似,学生们可以在今天分享的方法基础上继续学习。
In the rest of this chapter, we use the problem to demonstrate four multiplication strategies, most of which work efficiently across the rational numbers. Several of these strategies are ones that students usually come up with on their own.
在本章的其余部分,我们将利用问题 来演示四种乘法策略,其中大多数策略在有理数中都能有效地运用。其中有几种策略通常是学生自己想出来的。

Four Strategies for Multiplication
乘法的四种策略

Factor Factor Product
因素 因素 产品

1. Break a Factor into Two or More Addends:
1.将一个因数分解成两个或多个加数:

"I broke the 16 into 10 and 6. First I multiplied 10 times 12 and got 120 . Next I multiplied 6 times 12 and got 72 . Then I added 120 to 72 and got
"我把 16 分解成 10 和 6。首先,我把 10 乘以 12,得到 120 。然后我把 6 乘以 12,得到 72。然后我把 120 加到 72,得到

2. Factor a Factor:
2.因素一因素:

"I know 16 equals 4 times 2 times 2. First I did 4 times 12, and that was 48 . Then I did 48 times 2 , and I got 96 . And then I did 96 times 2, and I got 192."
"我知道 16 等于 4 乘以 2 再乘以 2。我先做了 4 乘以 12,得到 48 。然后我做了 48 乘以 2,得到了 96。然后我又做了 96 乘以 2,得到了 192"。

or
If students are ready for an explicit connection to more symbolic recording, we might also record like this to emphasize the associative property of multiplication:
如果学生已经准备好与更多的符号记录建立明确的联系,我们也可以这样记录,以强调乘法的联立属性:
  1. Round a Factor and Adjust:
    四舍五入系数并调整:
"I rounded 16 to 20, and I did 12 times 20 and got 240. Then I took away four 12s, or 48. I took 40 from 240 and got 200; then I took away 8 more and got an answer of