A**H 发帖数: 4797 | 1 1,文章里面的第一个例子
Consider a problem that asks how many total pencils there are if five people
have three pencils each. In the eyes of some educators, explaining why the
answer is 15 by stating, simply, that 5 x 3 = 15 is not satisfactory. To
show they truly understand why 5 x 3 is 15, and why this computation
provides the answer to the given word problem, students must do more. For
example, they might draw a picture illustrating five groups of three pencils
. (And in some instances, as was the case recently in a third-grade
classroom, a student would be considered to not understand if he or she drew
three groups of five pencils.)
CC如果强调这个,我赞同。作者括号里面的论述是正确的。但是如果数学课里面不强调
5x3和3x5的区别,那么考试的时候,5x3=15和3x5=15,都应该给分。如果建立这个乘数
和被乘数的精确概念,那么5x3是错误的,或者3x5是错误的(看课堂上是怎么定义乘数
和被乘数了),都可以说的过去。我认为最不迷惑的解答是:3 pencil/person x 5
person = 15 pencils。这个中国小学里也教过,但是考试的时候写5x3,或者3x5都会
得分。
2,文章里面第二个例子
“A coat has been reduced by 20 percent to sell for $160. What was the
original price of the coat?”
x = original cost of coat in dollars
100% – 20% = 80%
0.8x = $160
x = $200
解法正确,唯一的是0.8x应该写成80% x X (乘以x)
后面有个学生写的
160/x = 80/100
CC考试如果这个学生被判错。我觉得跟学生强调一下就可以了,下次记得加上100-20就
齐了,但我认为这个判错是对的,或者说这个题目学生不能得满分是正确的。因为80是
怎么来的,学生确实没有做出解释(100-20=80)
这个就有点像中间过程哪些可以省略,哪些不可以一样,是个很难判定的事情。老板给
员工布置任务,直接说80,没问题,但是学生直接从20跳到80,有点问题。如果直接写
出来
160/x=(100-20)/100 从而x=200,我觉得没有问题。至于文中提到的要画一个解题
步骤,我觉得家庭作业里面做做无妨。希望考试的时候不要出这种非客观的解释题。如
果已经写了100-20这个减法在数学表述式里面,而在文字解释的时候没有提到“100-20
to get 80”而不给学生这个解释的分,就有点象考语文,而不是考数学了。当然如果
是考教师执照,这么考倒是很好。
3,作者说
Despite the goal of solving a problem and explaining it in one fell swoop,
in many cases observed at the middle school, students solved the problem
first and then added the explanation in the required format and rubric. It
was not evident that the process of explanation enhanced problem solving
ability. In fact, in talking with students at the school, many found the
process tedious and said they would rather just “do the math” without
having to write about it.
这个同意作者,这个画个图来解释没有必要。CC如果要求解释,有一道这样的题目考一
下学生就可以了,而不必每一道题都这样。160/x=(100-20)/100或者160/(1-20%)已经
在数学上很清楚的解释了。但是如果在课堂,老师用图画来逐步解释,应该赞。
对于语言有障碍或者跟不上的学生,作为考数学知识的考试,不应该把解释或者画图这
个步骤来考察学生,这个我赞同。但是对于普通学生,靠一两道这样的题目也不为过。
家庭作业里面做几次也就够了,不用太多,浪费时间。我同意,这种考解释或者画图,
不只是考数学。但是我认为,不能把语言障碍的学生当作评判CC的标准,就像也不能用
有数学天赋的学生来评判CC一样。
4,
作者说 Measuring understanding, or learning in general, isn’t easy. What
testing does is measure “markers” or byproducts of learning and
understanding. Explaining answers is but one possible marker.
同意。考解题思路解释,只能是一个测试手段。
如果CC考试能把解释的分量控制在比较低的参考水平,无关大碍。如果能以标准化考试
的形式体现出来,就没什么好抵触的,我觉得。同时,如果是计算机考试,通过大量的
标准化测试,必然能测试出学生是不是掌握了5x3=15 pencils。如果CC也是通过这种标
准化考试,就行。至于老师在课堂上怎么教5x3=15pencils的概念,只要老师不是错的
,就可以了。至于老师教的是不是最好的方法,这个就看孩子自己的理解能力和造化了。
题外话一句,我觉得让娃解释其实总体来说还是可以的,因为娃确实不是只学数学,解
释的能力很重要。而对于我来说,我几乎每一个题目都会问娃为什么是这个答案,不管
他回答得正确与否。这样才可以更好的判断孩子理解是不是正确的,而不是蒙对的。只
有真正学懂了,能举一反三了,加上说的过去的解释能力,才能深入浅出的讲解题目,
解释解题思路。
说个前两天我娃教一个小娃的除法的故事。娃是这样的教的:
15除以5,我娃开始说除法就是乘法,多少乘以5等于15,这个是多少那么答案就是多少
,3x5=15,所以15/5=3;然后娃又说除法其实就是减法,15减去5还有10,再减5还有5
,再减5就只有0,这除法就除完了。然后娃有解释了一下14除以5,14-5-5=4,也就是
等于2余4的概念给小娃。LOL | t******l 发帖数: 10908 | 2 我可以举一个对于某些 math accelerated path 的更好的例子。
同时说明我认为的,对于特定学习阶段的 math explanation 的
do or don't 的基本原则
Don't over-explain "intuition", if such intuition is
practically correct in current stage, and kids is not
ready for go beyond intuition.
下面举一个实际的例子,SBAC vs AMC 8:
SBAC:
Randomly throw 1 coin, what is the probability of head up.
And explain.
AMC 8:
Randomly throw 3 coins, what is the probability of exactly
2 heads up. (AMC 8 do not require explain, but the online
answer explains. And without correct understanding, kids
cannot solve correctly instead of remembering).
下帖讨论。 | t******l 发帖数: 10908 | 3 我看到的目前 SBAC 的 theory 多半是,如果 kids 能 well explains
SBC 1 coin problem,then kids will be ready for AMC 8 3 coins
problem。
And this probably is how SBAC's "wanted" explanation constructed:
Kid's, please make an ELA passage for 1 coin problem that reflect
the following formula:
Probability of an event happening =
(Number of ways it can happen / Total number of outcomes)
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 我可以举一个对于某些 math accelerated path 的更好的例子。 : 同时说明我认为的,对于特定学习阶段的 math explanation 的 : do or don't 的基本原则 : Don't over-explain "intuition", if such intuition is : practically correct in current stage, and kids is not : ready for go beyond intuition. : 下面举一个实际的例子,SBAC vs AMC 8: : SBAC: : Randomly throw 1 coin, what is the probability of head up. : And explain.
| t******l 发帖数: 10908 | 4 不要说这个 ELA 解释是不是对其对应的 AMC 8 problem 有没有帮助,
我觉得很多教师可能没有意识到的是,overly ELA explain 这个不仅
是没有帮助的,而且是 theoretically flawed。
这个 theoretical flaw 是在于,why you think the two different
outcomes, head-up and tail-up, are equally probable?
(我先让罗素跑一圈)。
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 我看到的目前 SBAC 的 theory 多半是,如果 kids 能 well explains : SBC 1 coin problem,then kids will be ready for AMC 8 3 coins : problem。 : And this probably is how SBAC's "wanted" explanation constructed: : Kid's, please make an ELA passage for 1 coin problem that reflect : the following formula: : Probability of an event happening = : (Number of ways it can happen / Total number of outcomes)
| t******l 发帖数: 10908 | 5 回答这个问题需要超过绝大多数初中娃的 math thinking skill --
formal thinking applied in math modelling。
这不是说娃不能 understand more or less formal thinking,
而是说,这个阶段的娃,无法去 apply 不同的问题,因此无法直接
用来解题。
比如举个例子,一种 formal thinking 的 modeling 是 re-tagging。
Let's have a thought experiment:
We throw the coin, then after it drop, we change the head to
tail, and change the tail to head.
Or, we change head-tail before throw the coin.
Question, are they the same? Or a better word, does head and
tail distinguishable/swappable in terms of outcome's probability?
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 不要说这个 ELA 解释是不是对其对应的 AMC 8 problem 有没有帮助, : 我觉得很多教师可能没有意识到的是,overly ELA explain 这个不仅 : 是没有帮助的,而且是 theoretically flawed。 : 这个 theoretical flaw 是在于,why you think the two different : outcomes, head-up and tail-up, are equally probable? : (我先让罗素跑一圈)。
| t******l 发帖数: 10908 | 6 如果 SBAC 认为六年级普通娃能够那样考虑问题,那我也没啥可说。
如果不能,我要说,why do damn extensive ELA explanation
for rather small number of problem? Let's just remember
the formula, then we apply the formula to 20 different
problems, and if any of you guys fail on any of those
problems, let's see see why somebody fail while others
got it right?
As matter of fact, I think the above is actually
challenge-response-based learning (experiential
learning), instead of elaboration-based learning.
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 回答这个问题需要超过绝大多数初中娃的 math thinking skill -- : formal thinking applied in math modelling。 : 这不是说娃不能 understand more or less formal thinking, : 而是说,这个阶段的娃,无法去 apply 不同的问题,因此无法直接 : 用来解题。 : 比如举个例子,一种 formal thinking 的 modeling 是 re-tagging。 : Let's have a thought experiment: : We throw the coin, then after it drop, we change the head to : tail, and change the tail to head. : Or, we change head-tail before throw the coin.
| t******l 发帖数: 10908 | 7 这个问题是,SBAC may think kids just remembering the formula.
But arguably, kids may build-up their intuition instead of
merely remembering. The confusion is not a surprise though,
because every piece of cognition do finally implemented down
to remembering.
Or in a deeper level, middle school-er haven't built-up
math-formal-modelling-based thinking. And arguable, correct
build intuition, either by experimenting or by remembering
formula, or both, maybe the right way lead to formal-math-
modelling-based thinking.
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 如果 SBAC 认为六年级普通娃能够那样考虑问题,那我也没啥可说。 : 如果不能,我要说,why do damn extensive ELA explanation : for rather small number of problem? Let's just remember : the formula, then we apply the formula to 20 different : problems, and if any of you guys fail on any of those : problems, let's see see why somebody fail while others : got it right? : As matter of fact, I think the above is actually : challenge-response-based learning (experiential : learning), instead of elaboration-based learning.
| t******l 发帖数: 10908 | 8 或者实战的说,你向初中娃用 formal thinking-modelling 解释
theoretical probability vs experimental probability 半天。
然后娃要求的可能是:Why not let's toss the coin 20 times and
see see? I'll toss and you write it down.
And that is the real kid, not the theoretical projection
in SBAC.
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 这个问题是,SBAC may think kids just remembering the formula. : But arguably, kids may build-up their intuition instead of : merely remembering. The confusion is not a surprise though, : because every piece of cognition do finally implemented down : to remembering. : Or in a deeper level, middle school-er haven't built-up : math-formal-modelling-based thinking. And arguable, correct : build intuition, either by experimenting or by remembering : formula, or both, maybe the right way lead to formal-math- : modelling-based thinking.
| t******l 发帖数: 10908 | 9 回到 AMC 8,AMC 8 的解释很简单:
Number of ways it can happen = C(3,2)
(combination wo/ repetition, modelling by indexing
the coin, then select the combination of indices.)
Total number of outcomes = 2^3
(permutation w/ repetition, modelling by putting
down coin 1 by 1)
So: (3*2/2) / (2^3) = 3/8
至于 why they are equal? (The implicit answer is:
Let's leave this part of question later.) | t******l 发帖数: 10908 | 10 从某个角度说,我 和 SBAC 其实都在 implement CCSS 的 understanding
beyond applying the standard algorithm。
但是问题是在于,有很多的 explanation / understanding 我无法 implement,
my kid simply not ready for that yet。
而我做出的 compromise 是:(1)我把 explanation 的 burden 暂时更多
的放在我自己的头上。(2)我暂时战术性地把娃的部分 understanding 让位给
intuition,前提是 such intuition work as expected。
而我看到的 SBAC 的方案,貌似是(1)降低 problem 的难度,(2)降低 problem
的数量(同时不可避免的降低了 variety 和 repetition 的次数),(3) Promote
extensive-but-naive explanation。
当然我对 SBAC 的感觉不一定对就是了。 | | | A**H 发帖数: 4797 | 11 我看了一圈,怎么觉得SBAC考试题目基本上都是客观题?一个解释都没有要求。
比如说,下面这个八年级的样题
http://www.smarterbalanced.org/wordpress/wp-content/uploads/201
里面唯一一个不是简单答案的是一个图形怎么翻转得到新坐标图形,基本上也还是客观
题。
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 从某个角度说,我 和 SBAC 其实都在 implement CCSS 的 understanding : beyond applying the standard algorithm。 : 但是问题是在于,有很多的 explanation / understanding 我无法 implement, : my kid simply not ready for that yet。 : 而我做出的 compromise 是:(1)我把 explanation 的 burden 暂时更多 : 的放在我自己的头上。(2)我暂时战术性地把娃的部分 understanding 让位给 : intuition,前提是 such intuition work as expected。 : 而我看到的 SBAC 的方案,貌似是(1)降低 problem 的难度,(2)降低 problem : 的数量(同时不可避免的降低了 variety 和 repetition 的次数),(3) Promote : extensive-but-naive explanation。
| A**H 发帖数: 4797 | 12 这个七年级的另一类样题(Performance task vs Items样题)
http://www.smarterbalanced.org/wordpress/wp-content/uploads/201
有一点文字性的答案要求,属于概括总结图表类型的问答。 | t******l 发帖数: 10908 | 13 另外 by the way,mathsisfun.com 上对于 probability 的解释还是不错的。
当然问题是,在学习的阶段,我们到底需不需要娃 replay 那些解释,还是首先
要能做题 problem-solving,解释以后慢慢跟进。
当然,我娃的风格是,撞上南墙做错题还看不懂解释以前,是绝不看解释类网页的。
我如果墙裂要求,她就给我上有政策下有对策,应付一下。所以这个 mathsisfun.com
虽然是我娃介绍给我的,但我估计娃多半没看下面这个 probability 的网页。
这里也有个问题,如果娃混日子型的,大人如果没有 create need (比如错了题
还看不懂解释,或者干脆屡败屡战),那强迫娃看解释搞成鸡飞狗跳,真还不如等
娃做错了题让娃自己去看。
(当然,这也是现实的、老师不盯着明天就立马逃课的不靠谱初中娃,而不是 SBAC
想象的、您让娃干嘛、娃就屁颠屁颠地干嘛的、理想天使型娃)。
上面这个其实还不是数学教育的问题,上面这个其实是教育心理学的问题。但目前
这个问题,貌似更多在 assignment 上(娃觉得 boring 也不知道咋整),上课
讲课娃反而说比以前更喜欢。所以我个人无端怀疑 SBAC 的问题,可能比老师的
问题,更大一些。
mathsisfun.com 有关 probability 的解释 link:
http://www.mathsisfun.com/data/probability.html | t******l 发帖数: 10908 | 14 你是不是 agree 这更多的是一个 reading test 而不是 math test? :-)
但另一方面,这也许是个 trend。也许 SBAC 的意思是,在电算年代,如果
人群的 average 是无法掌握 math-formal-thinking 的,then why bother?
或者政治不正确的说,人群的 average 只要把数学部分 translate 成 machine
readable,是不是就能读很多本科专业了?
另一方面,math accelerated path 从来也不会 fail 在 SBAC state test 上,
都是 exceed 的 band 里。
将来的趋向,确实可能是对于 math accelerated path,AMC 变成 required。
而且确实数学州考的难度过去的年代里,逐年都在下降。而 AMC 8 的总体难度目测
没降低过,好像还略有升高。
所以,这可能也是个不错的解决方案。当然可能不应该仅仅用 SBAC 来指导 math
accelerated track,或者应该把 SBAC 和 AMC 给 blend 一下,营养更全面?
【在 A**H 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 这个七年级的另一类样题(Performance task vs Items样题) : http://www.smarterbalanced.org/wordpress/wp-content/uploads/201 : 有一点文字性的答案要求,属于概括总结图表类型的问答。
| t******l 发帖数: 10908 | 15 你看到的这些主要不是新增的部分,当然跟从前一致。其实就是从以前继承过来的。
新增的部分是 Constructed-response items 和 Performance tasks。
http://www.smarterbalanced.org/sample-items-and-performance-tas
Selected-response items prompt students to select one or more
responses for a set of options.
Technology-enhanced items take advantage of computer-based
administration to assess a deeper understanding of content
and skills than would otherwise be possible with traditional
item types. Technology-enhanced items capitalize on technology
to collect evidence through a non-traditional response type,
such as editing text or drawing an object. Selected-response
and technology-enhanced items can be scored automatically.
Constructed-response items prompt students to produce a text
or numerical response in order to collect evidence about their
knowledge or understanding of a given assessment target.
Performance tasks measure a student’s ability to integrate
knowledge and skills across multiple standards—a key component
of college and career readiness. Performance tasks will be
used to better measure capacities such as depth of understanding,
research skills, and complex analysis, which cannot be adequately
assessed with selected- or constructed-response items. Some
constructed-response items and performance tasks can be scored
automatically; many will be hand-scored by professionally
trained readers.These tasks will be delivered as part of the
interim and summative components of the assessment system,
with more extended tasks available in the digital library
as part of the Smarter Balanced-developed exemplar instructional
modules and inventory of currently available resources. The
time and resource constraints for each performance task will
be guided by its placement in the overall assessment system.
Smarter Balanced has a comprehensive research agenda. During
the spring 2013 pilot, the Consortium will research the effects
of including a brief classroom interaction phase in the summative
performance task on accessibility and standardization. The
classroom interaction phase will then be appropriately
constrained for performance tasks depending on the intended
use and placement within the overall Smarter Balanced
assessment system.
【在 A**H 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 我看了一圈,怎么觉得SBAC考试题目基本上都是客观题?一个解释都没有要求。 : 比如说,下面这个八年级的样题 : http://www.smarterbalanced.org/wordpress/wp-content/uploads/201 : 里面唯一一个不是简单答案的是一个图形怎么翻转得到新坐标图形,基本上也还是客观 : 题。
| t******l 发帖数: 10908 | 16 您这个至少是算看对了。衡量一考试,显然要看拉风题而不是送分题。
其实整个学区都 shift to math explanation,这显然不是说州考
送分题都是 math explanation。你就说老师上课的主要发力点是州
考的送分题还是拉风题嘛?
【在 A**H 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 这个七年级的另一类样题(Performance task vs Items样题) : http://www.smarterbalanced.org/wordpress/wp-content/uploads/201 : 有一点文字性的答案要求,属于概括总结图表类型的问答。
| A**H 发帖数: 4797 | 17 阅读理解能力是不可避免的要考的。这个对于老中的娃们是吃亏的,比如我娃还是在题
目里面第一次见到consecutive, exclusive, inclusive 这些个词。但是,既然是art
of problem solving, 那也自然还是有一个问题(problem)怎么呈现出来的问题。
比如说考题A, 小明8岁,小军6岁,小明比小军大几岁;B, 8-6=?
考题A,针对于B来说,是阅读考试还是数学考试,很难讲。有个度吧。正常范围里面就
可以。
想问一下,以你的经验,AMC8 难度稳定,从哪个年份开始的?我参考下,给娃做做题
目。
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 你是不是 agree 这更多的是一个 reading test 而不是 math test? :-) : 但另一方面,这也许是个 trend。也许 SBAC 的意思是,在电算年代,如果 : 人群的 average 是无法掌握 math-formal-thinking 的,then why bother? : 或者政治不正确的说,人群的 average 只要把数学部分 translate 成 machine : readable,是不是就能读很多本科专业了? : 另一方面,math accelerated path 从来也不会 fail 在 SBAC state test 上, : 都是 exceed 的 band 里。 : 将来的趋向,确实可能是对于 math accelerated path,AMC 变成 required。 : 而且确实数学州考的难度过去的年代里,逐年都在下降。而 AMC 8 的总体难度目测 : 没降低过,好像还略有升高。
| t******l 发帖数: 10908 | 18 这其实或多或少还真是个问题,就好比 AMC 8 的部分 pre-algebra 啰嗦题,不会
transferable to AMC 10 algebra 阶段。
这就好比我们那时初中平几狂人,不一定高中数学都行。(当然也不要过度解读成平几
狂人高中数学一定不行,不同的娃不一样)。
AMC 8 的难度,我其实就是听我娃乱嚷嚷。我自己没做过整张卷子,都是以前答疑才
做的。而且我娃又不靠谱。所以无所谓,你从最新的往前做就好了。
art
【在 A**H 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 阅读理解能力是不可避免的要考的。这个对于老中的娃们是吃亏的,比如我娃还是在题 : 目里面第一次见到consecutive, exclusive, inclusive 这些个词。但是,既然是art : of problem solving, 那也自然还是有一个问题(problem)怎么呈现出来的问题。 : 比如说考题A, 小明8岁,小军6岁,小明比小军大几岁;B, 8-6=? : 考题A,针对于B来说,是阅读考试还是数学考试,很难讲。有个度吧。正常范围里面就 : 可以。 : 想问一下,以你的经验,AMC8 难度稳定,从哪个年份开始的?我参考下,给娃做做题 : 目。
| A**H 发帖数: 4797 | 19 我觉得吧,主要看考试大纲
至于老师用的教学大纲,用什么方法教,没什么好办法,毕竟老师个体差异大。平庸的
解题思路里面好老师也能讲出个一二三四五来。
如果考试大纲里面主要都是客观题,咱老中也就没什么好怕的了。而且教学大纲终究是
要跟考试大纲走的。
回答你的问题,作为一个普通学校的普通老师来说,当然是着重在送分题上。
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 您这个至少是算看对了。衡量一考试,显然要看拉风题而不是送分题。 : 其实整个学区都 shift to math explanation,这显然不是说州考 : 送分题都是 math explanation。你就说老师上课的主要发力点是州 : 考的送分题还是拉风题嘛?
| t******l 发帖数: 10908 | 20 美帝的 National / State 的 Average Math 的定义不同。照围棋哥的说法,
就是高中毕业不会用一元二次方程解鸡兔同笼的那种。搞不好真的没夸大。
所以只要 7 分以上学校就不可能着重在送分题上。
【在 A**H 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 我觉得吧,主要看考试大纲 : 至于老师用的教学大纲,用什么方法教,没什么好办法,毕竟老师个体差异大。平庸的 : 解题思路里面好老师也能讲出个一二三四五来。 : 如果考试大纲里面主要都是客观题,咱老中也就没什么好怕的了。而且教学大纲终究是 : 要跟考试大纲走的。 : 回答你的问题,作为一个普通学校的普通老师来说,当然是着重在送分题上。
| t******l 发帖数: 10908 | 21 现在回到这个 re-tagging 上。
从 space-time & topological symmetric / congruence 的角度
个人理解,re-tagging basically is also a strong test, which test
the topological symmetric / congruence of the decision tree
for the 1 coin toss probability problem.
Or, it is more like derive the equality of the faces of a cube,
from it symmetric / congruence in 3-D continuous space. The
only difference is the toss coin probability is a topological
structure (decision tree) in the topological space.
But, do we expect our poor 11 years old little kids, be able
to understand the above strong test (and any form of any
strong test)?
If not, when we ask our kids copy the word "alike" from the
white board to their answer sheet, how can we assure this
"explaining-by-copying" is better than the following proven
problem-solving teaching strategy:
Throw kids 20 relevant problems after teacher's explanation,
review the results, and hope they can build their at least
correct intuitions from experiential-learning (if they are
not ready for any theoretical strong test yet), without writing
the damn word "alike" 20 times.
【在 t******l 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 回答这个问题需要超过绝大多数初中娃的 math thinking skill -- : formal thinking applied in math modelling。 : 这不是说娃不能 understand more or less formal thinking, : 而是说,这个阶段的娃,无法去 apply 不同的问题,因此无法直接 : 用来解题。 : 比如举个例子,一种 formal thinking 的 modeling 是 re-tagging。 : Let's have a thought experiment: : We throw the coin, then after it drop, we change the head to : tail, and change the tail to head. : Or, we change head-tail before throw the coin.
| a*****g 发帖数: 19398 | 22 真正要问的问题是——为神马搞出这种解释题
我提两个情况——
1, 船上有10头羊,5头猪,请问船长的年纪是多少?
美国小朋友很多就开始加减乘除乱算。
2, 老师的考卷还没有发,学生开始涂答卷圈圈……
解释题,基本上就是用战术办法解决战略问题,最后结果就是惹出更多的麻烦。
people
the
pencils
drew
【在 A**H 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 1,文章里面的第一个例子 : Consider a problem that asks how many total pencils there are if five people : have three pencils each. In the eyes of some educators, explaining why the : answer is 15 by stating, simply, that 5 x 3 = 15 is not satisfactory. To : show they truly understand why 5 x 3 is 15, and why this computation : provides the answer to the given word problem, students must do more. For : example, they might draw a picture illustrating five groups of three pencils : . (And in some instances, as was the case recently in a third-grade : classroom, a student would be considered to not understand if he or she drew : three groups of five pencils.)
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