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 andencourage kids to think deeply about math。
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1。Look for and make use of structureand
2.Look for and express regularity in repeated reasoning.
To be honest, unlike some of the other standards, these don’t seem asimmediately 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）。
Looking for and making use of structure means tolook for andsee 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 numberusing a pattern or structurerather 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 hasmade use of that structureto find all the possible facts.
With enough time and exposure, students may alsonotice mathematical propertiessuch as the commutative property or distributive property. They can begin to make connections and see expressions or computation inlots of different ways。
这类fluency and flexibilitywith numbers and operations is callednumber 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。
Form Generalizations Based on Patterns
I would sum this up as meaning students canform generalizations and rulesbased on what they observe in computations.
For example, it is often a light bulb moment when students realize they cancount upto find the difference between two numbers (例如，22-19）。
By seeing the subtraction as thedifferencebetween 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 iswith number talks。
Because number talks focus on mental math, students are forced toset aside the standard algorithmsthey know and finda more efficient method并在他们对数字和计算的考虑如何变得更加灵活。
Playing with a calculatoris 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?)
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 procedureto 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。
- Help kids see patterns in addition & subtraction with pumpkin seeds
- 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andThinking Deeply: How to Incorporate Meaningful Writing into Math Class。
Although all of my courses are available separately, they are truly designed towork togetherto provide a foundation for math teaching in K-8. All together, the 6 courses include8.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.
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 morekid friendly language。
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