*Mathematics is the science of patterns. Encourage your students to look for and make sense of math patterns & structure in order to deepen understanding and retain what they learn.*

Welcome back to my series on the Standards for Mathematical Practice! I have enjoyed reading and thinking through different ways we can foster these habits in our classrooms and**encourage kids to think deeply about math**。

***请注意**: This post contains affiliate links which support the work of this site. Read our full本文披露。*

**今天我想专注于我们希望学生努力工作的最后两种数学实践：**

1。*Look for and make use of structure*and

2.*Look for and express regularity in repeated reasoning.*

To be honest, unlike some of the other standards, these don’t seem as*immediately clear*他们指的是什么。但希望能够对他们重新描述一下并看一些例子将有助于清除任何混乱（*although maybe you’re not as confused by these statements as I was, LOL*）。

**寻找模式与结构**

Looking for and making use of structure means to**look for and****see patterns**在数字，图片，计算或属性中。

对于小孩子来说，可能会出现不同的方式**decompose the same number**using a pattern or structure*rather than random guessing.*

Take the number 8 for instance–you can start with 0+8, and then increase one addend while decreasing the other–> 1+7, 2+6, 3+5, and so on.

由noticing the pattern or structure of the math facts, the student has** made use of that structure**to find all the possible facts.

With enough time and exposure, students may also**notice mathematical properties**such as the commutative property or distributive property. They can begin to make connections and see expressions or computation in**lots of different ways**。

这类**fluency and flexibility**with numbers and operations is called**number sense**它肯定不是你可以在一天中教授的东西，然后继续前进到下一件事。

这些是习惯和技能，需要在您选择的任务方面需要时间和意向性**给孩子们有机会看到模式和结构**。

**Form Generalizations Based on Patterns**

让我们考虑一下现在的最后一个标准，我认为与第一个有关：*在重复推理中寻找和表达规律性*。

I would sum this up as meaning students can**form generalizations and rules**based on what they observe in computations.

For example, it is often a light bulb moment when students realize they can**count up**to find the difference between two numbers (*例如，22-19*）。

By seeing the subtraction as the**difference**between the two, they have discovered a**更高效的方法**求解而不是向后计数。

Another example is multiplying by multiples of 10. Rather than telling students, “When you multiply a number by 10 you add a 0, when you multiply a number by 100 you add 2 0’s, etc.” let them work out several examples either on paper or with place value tiles. After enough examples, they will begin to**看重复**和模式出现，但是**实际上理解**why the product now ends in a zero based on place value.

I know I’ve already mentioned this, but one of the best ways to consistently provide the time and exposure kids need is**with number talks**。

Because number talks focus on mental math, students are forced to*set aside the standard algorithms*they know and find**a more efficient method**并在他们对数字和计算的考虑如何变得更加灵活。

还有其他方法可以提供模式和概括的实践。

**Playing with a calculator**is a fun way for kids to see what happens with repeated operations (for example, what happens with you multiply by -1 over and over again? What happens when you multiply by 2 over and over? etc.)

鼓励孩子寻找模式或结构的另一种方法是**通知和奇迹提示**。For this activity, you simply show a picture, math problem, visual pattern, etc. to students and have them record a few things they notice and a few things they wonder (what questions do they have about what they see?)

这可以帮助他们至少开始注意到他们可能从未思考过的模式和结构。

我与a相关联**free notice and wonder journal**下面您可以使用，以及其他可能对您提供这些标准的其他资源，网站和文章。

最终，目标是**students feel comfortable deviating from a known procedure**to look for a pattern and find a shortcut or more efficient strategy. Getting kids to that place of being comfortable trying new ideas can be a challenge (*especially for older kids who have been trained to think that there is one method, one procedure*), but**提供正确的任务**并允许时间**productive struggle**将帮助他们发展这些技能。

**帮助您探索模式和结构的资源：**

**准备开始？**以下资源将帮助您找到学生探索数学模式和结构的想法，以及一个提供可视模式的额外网站。我希望这是有帮助的！

- Help kids see patterns in addition & subtraction with pumpkin seeds
- 帮助孩子们看到乘法和划分是如何与此滚动和写挑战有关的
- Read more about using a calculator to explore & discover
- Use tables to explore multiplication & division
- Another example of using tables to find a pattern
- Find visual patterns to generalize at VisualPatterns.org
- Notice & wonder journal|
**立即下载！**

**Ready to Dig Deeper?**

如果您正在享受此谈话，并且希望更多地了解可视化模式并形成一般规则或快捷方式，我比在我的课程中的时间更深度，*制作数学视觉那**有意义的数字会谈*and*Thinking Deeply: How to Incorporate Meaningful Writing into Math Class*。

Although all of my courses are available separately, they are truly designed to*work together*to provide a foundation for math teaching in K-8. All together, the 6 courses include**8.hours of professional development**那which you have lifetime access to. If you work through one course per week, you could easily complete all of them.

**>>****Learn more about all the courses HERE**

**免费数学练习标准海报套装！**

Want to explain these math habits to your students? This**有趣和免费的海报**为全年提供讨论以及视觉提醒的跳板。

There are**两个版本**included in this pdf file. One includes a cursive font and the other includes only manuscript fonts so you can use these with younger students.

I’ve also adjusted the wording of some of the standards so that is uses more**kid friendly language**。

**通过在下面输入您的电子邮件来获取此免费海报！**

### 永远不会出现乐趣的数学想法

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