# Standards for Mathematical Practice

### Summary

TLDR本视频脚本探讨了数学共同核心(Common Core)中的八个数学实践标准，强调了解决问题、抗挫折能力（也称为“韧性”）、抽象和量化推理、以及数学交流等关键能力的重要性。通过提倡在班级中解决实际问题，而非仅仅传授知识，视频强调了这些实践对于培养学生深入理解数学概念的重要性。它还讨论了通过合作、批判性思考和有效使用数学工具等方式，如何在学生中培养这些技能。通过具体实例，如高斯求和故事，视频展示了如何在日常教学中融入这些实践，以培养学生的数学习惯，为他们的未来学习和生活打下坚实基础。

### Takeaways

- 📝 数学共同核心标准中的八大数学实践是数学领域的核心理念。
- 💪 问题解决和坚持不懈是数学实践中的首要准则，与教育领域外的“坚毅”概念息息相关。
- 🌟 坚毅不是天生的性格特质，而是可以通过不断解决困难问题来学习和发展的。
- 📚 课堂环境应该是一个安全的学习空间，让学生在面对挑战时能够尝试、沟通和协作。
- 🤔 教学不仅仅是传授知识，更重要的是让学生通过亲身体验来理解数学并建立数学思维。
- 🔄 数学家解决问题时能够在具体情境与抽象概念之间灵活转换，这种能力对学生同样重要。
- 🎬 教学中应鼓励学生通过故事问题来建立直观理解，并将数学知识与现实世界联系起来。
- 🗣️ 培养学生构建有效论证和批判他人推理的能力，这包括自我表达和倾听他人观点。
- 🤝 合作学习和同伴评审是建立积极学习氛围和深化学生理解的重要手段。
- 🔢 教师应鼓励学生解释自己的思考过程，这有助于发现和纠正误解。
- 📈 通过真实世界的问题来教授数学，让学生理解数学的实际应用和背后的目的。

### Q & A

### 数学共同核心标准中的八项数学实践是什么？

-数学共同核心标准中的八项数学实践包括：1. 问题解决和坚持不懈；2. 抽象和量化推理；3. 构造可行的论证并批判他人推理；4. 模型建立；5. 使用恰当的工具；6. 寻求精确度；7. 运用结构和重复推理；8. 寻找数学之美。这些实践旨在帮助学生深入理解数学概念，并在解决问题时发展数学思维习惯。

### 为什么在课堂上需要给学生足够时间来解决问题？

-在课堂上给学生足够时间来解决问题是为了让他们能够发展坚持不懈的心态。如果所有的课堂活动都在30秒或更短的时间内完成，学生就没有机会培养在遇到困难时坚持解决问题的心态。

### 教师如何创造一个安全的学习环境？

-教师通过设计课堂活动，让学生在遇到困难时有机会尝试、沟通、协作和分享想法，从而创造一个安全的学习环境。教师提供适当的挑战，鼓励学生尝试解决问题，并在学生遇到困难时提供支持和引导。

### 数学教育中如何培养学生的抽象思维能力？

-数学教育中培养学生的抽象思维能力可以通过让学生在解决实际问题时，学会从具体情境中抽离出来，建立数学模型，并使用数学工具来解决问题。例如，通过故事问题让学生在理解情境的同时，学会用数学方式表达和解决问题。

### 为什么学生需要学会批判性地分析问题？

-学生需要学会批判性地分析问题，因为这能帮助他们更好地理解问题的本质，评估不同解决方案的优劣，并能够提出自己的见解和论证。这种能力在未来的学习和工作中都是非常宝贵的。

### 数学教育中如何平衡追求精确度与接受错误？

-在数学教育中，追求精确度是重要的，因为我们需要学生得到正确的答案。但同时，也要让学生理解错误是学习过程的一部分，从错误中可以学习和成长。教师应该鼓励学生接受并分析错误，而不是因为犯错而感到沮丧。

### 数学共同核心标准如何帮助学生为未来做好准备？

-数学共同核心标准通过提供一系列基本的数学实践，帮助学生发展解决问题的能力、批判性思维、合作和沟通技巧等。这些技能不仅对学习数学有用，也对学生未来的学习和职业生涯有着重要的影响。

### 数学教育中如何使用真实世界的问题？

-数学教育中使用真实世界的问题可以让学生理解数学的实际应用和目的。教师可以通过引入真实数据和情境，让学生探索和解决与现实世界相关的问题，从而增强学生对数学的兴趣和理解。

### 为什么学生需要学习基础的计算和数学事实？

-学生需要学习基础的计算和数学事实，因为这为他们提供了解决更复杂问题的基础。掌握基础技能可以帮助学生建立数学思维，发展心算能力，并为使用更高级的数学工具和技术打下基础。

### 数学教育中如何培养学生的结构和重复推理能力？

-数学教育中培养学生的结构和重复推理能力可以通过引导学生发现数学模式、关系和结构，以及鼓励他们使用这些发现来解决新问题。例如，通过介绍著名数学家高斯求和的故事，让学生理解如何通过观察数字的模式来简化问题的解决过程。

### 数学共同核心标准如何促进学生之间的合作和交流？

-数学共同核心标准鼓励学生通过讨论、解释和批判性分析他们的思路和解决方案来相互合作和交流。这种互动不仅有助于学生理解彼此的想法，还能够促进他们批判性思维和沟通技巧的发展。

### Outlines

### 📐数学实践标准的重要性

此段介绍了数学共同核心(Common Core)中的八项数学实践标准，特别强调了解决问题和坚持不懈的重要性。这不仅是一种性格特征，而且是通过解决困难问题可以学习到的技能。教室环境应提供超过30秒的挑战，以帮助学生发展这种心态。强调了家长需要理解，学校为学生提供了一个安全的环境来探索和解决问题，即使这意味着面对挑战。此外，通过与他人合作和分享想法，孩子们可以模仿成人在面对问题时的行为，从而建立更深层次的理解和数学习惯。

### 🤝促进数学理解和交流的策略

第二段强调了在数学学习中鼓励学生之间的交流和批评性思维的重要性。通过展示工作并邀请反馈，学生们不仅能够享受数学，并且能从他人的思考方式中学习，这有助于建立对数学的积极态度。在教室中创造一个安全的空间，让学生可以相互协作和评价是至关重要的，教师通过让学生解释他们的思考过程来识别误解或误区。此外，通过实际应用和探索现实世界问题，学生能够看到数学的实际用途，从而增加了他们对学习的兴趣和目的。

### 🔢发现数学结构和推理的力量

最后一段通过卡尔·弗里德里希·高斯的故事强调了识别数学结构和重复推理的能力的重要性。高斯在小学时就通过观察数字的结构快速解决了一个数学问题，展现了数学思维的力量。这一段指出，所有学生都能学会像数学家那样看待世界，识别和利用数学中的结构和模式。共同核心标准提供的数学实践是帮助学生发展这些终身技能的关键，无论他们未来的专业是人文、艺术、音乐还是科学。

### Mindmap

### Keywords

### 💡数学实践标准

### 💡问题解决

### 💡坚持不懈

### 💡抽象推理

### 💡定量推理

### 💡故事问题

### 💡话语

### 💡合作

### 💡精确性

### 💡结构

### 💡重复推理

### Highlights

数学共同核心标准中的八大数学实践是数学领域的核心思想。

问题解决和坚持不懈的能力在教育领域受到越来越多的关注，被称为'坚毅'。

在课堂上，如果所有任务都在30秒或更短时间内完成，学生就没有机会培养坚持解决问题的心态。

教师在课堂上创造一个安全的环境，让学生在困难中挣扎，这是可以学习的过程。

学生需要通过解决困难问题来学习坚持，而不仅仅是通过重复性的任务。

数学教学不仅仅是告诉学生答案，而是让学生体验数学，自己理解数学。

数学家在解决问题时能够在具体情境和抽象概念之间灵活转换。

共同核心标准强调非传统的故事问题，让学生在心中构建故事情境，并将数字转化为方程。

学生应该能够构建可行的论证并批判他人推理，这是数学交流的重要组成部分。

教师鼓励学生解释他们的思路，以便发现和纠正误解。

在数学课堂上，学生通过合作和同伴评审来创造一个安全的学习空间。

数学建模关注实际应用，将现实世界问题转化为数学问题进行解决。

共同核心强调使用适当的工具，但要战略性地使用，避免学生过早依赖技术。

精确性对于数学学习至关重要，但也要避免因为错误而让学生感到沮丧。

数学家卡尔·弗里德里希·高斯的故事说明了结构和重复推理的实践标准。

共同核心的数学实践是终身技能，对学生未来的任何领域都非常重要。

### Transcripts

thank you

one of the really

Central ideas or maybe Cornerstone ideas

of of the Common Core in mathematics is

these eight standards for mathematical

practices these are things that people

who do mathematics do the first one

problem solving and persevering is this

is something that I think has been

getting a lot of attention recently in

in fields other educational disciplines

known as grit so how many did she have

to begin with

we don't know so let's write that down

to represent that this sort of

intangible almost idea of being able to

persevere in a problem or a task when

the going gets rough to stick with it

and

just it's not a character trait that

folks have we might say that some have

it more than others but it's it's really

something that can be learned by

persevering through difficult problems

if everything in the classroom takes 30

seconds or less to complete there's no

opportunity for students to develop that

mindset I think I really need to remind

parents that in the classroom

environment that teacher is making it a

safe environment and it's a safe place

for students to struggle that struggle

isn't the same struggle as giving a

child something that is developmentally

inappropriate we're giving a child

something that we know they can tackle

we want them to try and communicate with

a neighbor and collaborate with somebody

and share ideas and talk through it

because as adults that's what we do we

don't sit at our desk and bang our head

and and and you know sit alone and are

unable to solve something we reach out

to people for help we talk about it we

try and then we try again and if we want

the if we want our children to exhibit

those same kinds of behaviors we really

need to Foster an environment where it's

okay to do that one of the things that

we know is if we want students to

develop the kind of deep understanding

you're talking about that we do need to

engage them in actually solving problems

in class not just standing up and

telling or as one of my colleagues would

say teaching isn't telling

right told isn't taught we really need

students to experience the mathematics

themselves make sense of it

so that they're making the connections

and so the extent to which we can look

at materials that both have a careful

development of the content but also

engage kids in the kind of problems that

will help them build those mathematical

habits of mind along with understanding

the content will much better prepare

them to be successful in the future a

second practice is reasoning abstractly

and quantitatively when mathematicians

solve problems they

easily move back and forth between the

situation of the problem the real life

thing

how many gallons of gas is the United

States going to consume in the next year

and then they find ways to be abstract

about that to model that in a way using

mathematics they're able to connect

what they're doing mathematically to

what the what the purpose of it is in

the world one of the things that is

emphasized in the common core standards

is an idea of story problems that aren't

traditional and this is where I think

it's really important for kids to kind

of what I say to my own kids at school

is to make a movie in your mind what's

happening in this story

can you explain to me what you're doing

here and then also take them back to the

story so they're not just getting mixed

up with the numbers and make that

picture in your mind and then try and

translate it to

an equation if you can another practice

standard in the common core is the

practice standard about discourse

it is construct a viable argument and

critique the reasoning of others it's

really being able to think about your

own solution to a problem and be able to

defend it to another person

subtract the 12 so I'll have to subtract

a 10 and then two more

but how is an easier way that I can

represent it very importantly it's also

about being able to listen to someone

else's strategy to make sense of it and

to agree or disagree that's a big Focus

for us we talk about our math we present

our math we even have a procedure in

presenting students will present their

work and then turn the set procedure

that I've taught turn and say do you

have any questions or comments talk to

your friends

last because

um zero and seven cents is greater than

um

and that is for everyone else to

actually learn from them so that gives

them this positive disposition towards

mathematics and they they begin to enjoy

it they enjoy you know talking about the

ways that they are thinking about things

and the other students get to critique

them and the other students and it's

okay I respectfully disagree with Cliff

because

um he didn't um put the numbers by place

value and it's perfectly fine that other

students are critiquing them because

they know that they will be critiqued as

well I have them explain their problems

to their peers so that their peers can

also see that some of their other

students in the classroom may not have

solved it the same way that they did but

they got the same answer and what I see

in kindergarten in first grade and

second grade in Middle School are

teachers who are creating that safe

space where

students are collaborating with each

other where there's peer review of what

you're doing and and so it's a safe

space well it's important as a teacher

for the children to explain themselves

to me so I understand how I can help

them we're we're where they're confused

possibly it also they're Sometimes the

best teacher for the other students in

the classroom because the way they'll

phrase something or um put it out there

it helps children oh say oh that's a

great idea or that's not a good idea and

let me help her understand why that's

not a good idea I feel like it's really

really important for us to

um you know help them to explain their

thinking as often as possible because

this is where we draw on and find

misconceptions or misunderstandings and

we're able to clarify those modeling

with mathematics I think goes back to to

the sort of practical

uses in what ways can situations in in

the real World be modeled and maybe more

easily

figured out or dealt with by treating

them as mathematical problems and by

representing them in ways using

mathematics a lot of what you'll see in

classrooms is kind of a reversal of what

you saw in the past a lot of times you

saw just kind of the procedure being

done repeatedly and then maybe a few

rule problems tacked on at the end and

maybe they weren't very authentic real

world problems but now you'll see where

teachers are really approaching the

content from The Real World to begin

with how does this look in the real

world there's real data that I can

collect and then we explore the math

that unfolds from that real world

problem so there's really that

understanding I'm not just doing math to

do math but there's a purpose behind it

the students have been asking for years

why why am I doing this why why why now

we're starting with the why the way the

Common Core talks about the use of

appropriate tools is with the word

strategically

and that's an important way to think

about this because you don't want

technology

being used to teach kids basic facts

well the reality is

calculators computers and all kinds of

Technology are out there and students

just like adults are going to use those

for complex problems what you don't want

is students grabbing for a calculator to

multiply something by 10. and so we want

students to learn some computation to

know some facts to know some procedures

and the truth is that developing mental

math skills that's the best tool we can

give them in their toolbox Precision

matters we want students to get the

right answer and everything we do is

aimed toward giving students tools that

will help them consistently get the

answers to problems the trick is there's

a delicate balance between striving for

precision and making students feel badly

every time they make a mistake and what

we now know about learning and about

intelligence is that it's from the

mistakes that we really grow and that's

where the learning happens we don't want

students to you know leave the classroom

thinking oh there are so many different

answers there there's not one right

answer I I don't know what the right

answer is no we want them to be precise

in their answer the last two standards

for mathematical practice involve the

use of structure and repeated reasoning

a good illustration of these practices

is a a pretty well known story about the

famous German mathematician named Carl

Friedrich Gauss so when Gauss was in

Primary School his teacher gave students

a problem of summing up the integers

from 1 to 100

and he just wanted to keep them busy I

guess for a while I had some grading to

do or something but to the surprise of

his teachers and classmates Gauss almost

immediately wrote down the correct

answer of 5050 on his slate and he

explained how he realized that the first

and last number 1 and 100 add up to 101

but the second number and the second to

last number 2 and 99 also had the sum of

101 and so he realized that this sum

just repeats itself 50 times all the way

up to 50 plus 51 the last pair of

numbers so we just multiplied 101 by 50

to get the answer

5050. with

experiences and um and with an

environment where kids can start to

learn to make these connections

um it's really something that that all

students can learn to do and to begin to

see the world of quantities and numbers

as mathematicians do by seeing structure

and repeated reasoning the Common Core

lays out a lot of these fundamental math

practices which are lifelong skills that

are important for kids to have no matter

where they go if they major in the

humanities

or art or music or

chemistry those kinds of skills are

really really important and ones that we

I think again that the Common Core gives

us the opportunity to make sure our

students that are graduating from our

schools today have that opportunity to

learn those skills

thank you

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