Mathematical Reasoning: Developing, promoting and expanding thinking in whole group and small group

Posted on January 21, 2012. Filed under: Assessment, Common Core, During the Guided Math Lesson | Tags: , , , , , , , |

Mathematical Practice 2 is about students learning to “make sense of quantities and their relationships in problem situations.” It involves students being able to “contextualize” and ” decontextualize” numbers.  Are your students able to put words to numbers?  For example, can they tell you a story that makes sense about 2.5 x 10? Also, can they take apart and discuss the numbers in a problem?   Do they reason quantitatively  by “creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects[?]” —CCSS
NCTM (2000) says that reasoning involves:
  • Recognizing reasoning and proof as fundamental aspects of mathematics
  • Make and investigate mathematical conjectures
  • Develop and evaluate mathematical arguments and proofs
  • Select and use various types of reasoning and methods of proof

What does all this really mean?  Well in the next few posts I would like to explore this.   I believe reasoning is taught over time and reinforced through our daily practices.  It involves questioning, prompting, listening, challenging, proving, giving evidence, thinking out loud, thinking with a partner, thinking by yourself, thinking in a group, and thinking publicly to name a few things.  Teachers set up environments where this happens.  We set students up to reason or not to reason.  We set them up to publicly prove what they know or not.  We teach them specific reasoning skills or we don’t.

My point is students don’t just start reasoning.  This is a “habit of mind” that is developed, practiced, promoted and expanded over time and throughout the years.  Schools have to think about how they develop a culture of reasoning.  Grade levels should come up with some benchmark routines that promote reasoning.  You should think about how reasoning is interwoven throughout the day, week, the month and the year.

Here are some sample weekly routines:

What’s the problem routine?  You give the students some numbers, say 10 x 30 and ask the students to make up a story about this.

Convince me paper:  You give the students something that they have to convince you about:  Convince me that ….. (1/2 is greater than 1/3)

True or False essays:  You give the students a statement and they have to decide and defend whether it is true or false.  For example: Anytime you multiply a number by 0, you get 0.  True or False?  Defend your answer with 3 examples.  Show your thinking.  Explain your thinking.

Teachers can do these at any grade level with the appropriate content.  These are great thinking prompts that work across grade levels.  It’s important to think about access for all. I encourage teachers to develop different graphic organizers for the responses so that they are tiered and give different entry points. I recommend that you  process them in small guided math groups so that you can hear the thinking.  I also think there should be some whole group sharing, so that the entire mathematical community in your class can engage together.  There are benefits to facilitating both types of conversations and there should be a balanced approach.

Happy Mathing,

Dr. Nicki


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Stop, Look, Think and then Go! Teaching the CCSS Practices in Guided Math Groups

Posted on December 14, 2011. Filed under: Assessment, Common Core, During the Guided Math Lesson, Elementary math, Graphic Organizers, Mathematical Proficiency | Tags: , , , , , , |

The Math CCSS practices are about teaching children how to stop and think before they start doing math problems.  They work hand in hand with the content.  They should be taught explicitly alongside with the content.  You should make posters of the practices and talk about them as you teach the content.  Dr. Phil Daro (one of the writers of the Math CCSS) talks about “delaying answer getting” so that students can learn the math.  Dr. Bill McCallum tells the students  “Don’t just do something, stand there” in the same spirit of getting them to THINK first!  So, in all of our teaching, we have to get them to Stop, Look at the problem, Think about it and make a plan and then go for it!  Do the math and be sure to explain their thinking!  Also, we have to get them to check it in a different way than they solved it!

To do all of this, scaffolds work great.  How are you scaffolding the thinking in your classroom?  Do you have posters heralding these ideas.  Do you give the students graphic organizers so they can show their  thinking in a clear and orderly manner?  How do you delay answer getting and get students to lay out a pathway.  (The other day in one of my schools we actually drew out a pathway and drew pictures showing all the steps along the pathway to solving the problem…modeling it, writing an equation, explaining our thinking, checking a different way and writing the answer).

Happy Mathing,

Dr. Nicki

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