Integrating the Standards for Mathematical Practice as a Natural part of Teaching Through Whole Class Mathematics Discussions.

Teruni Lamberg, Ph.D.University of Nevada, RenoTerunil@unr.edu 1

The ultimate goal of teaching is to support student LEARNING!

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Three things to keep in mind as you teach:

• What do I want my students to learn?

• What are they learning?

• Am I being effective?

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What is learning?

How do you know that learning is taking place?

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According to National Research Council learning

• When students learn concepts with UNDERSTANDING, that knowledge becomes a TOOL to solve problems in novel situations.

• Learning is an ACTIVE process!

• New knowledge builds on students’ pre existing knowledge- need to pay attention to students' prior conceptions and understandings.

• Making meaningful connections and seeing patterns is an important part of developing expertise.

• National Research Council. How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Washington, DC: The National Academies Press, 2000

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Improving teaching to support student learning:• Involves thinking about the process of planning

and teaching including setting up the classroom environment. Effective whole class discussions that support learning involves situating the discussion within the larger mathematical goals.

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No

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Students’ Role in Discussion• Students must effectively communicate their mathematical thinking

using representations.

• Listen and reflect to ideas presented.

• Ask questions if unclear about idea presented.

• Be willing to refine and revise thinking.

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Teacher’s Role in Whole Class Discussions

• The teacher’s role in whole class discussion is to select problems to pose.

• Use questions to engage students in mathematical reasoning.

• Carefully sequence discussion and questioning so that students make mathematical connections.

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Classroom discussion time must be used effectively and efficiently to Support learning

•Discussion must build a bridge between student thinking, mathematical concepts and skills.

•Students must develop new mathematical insights and make deeper mathematical connections as a result of participating in a discussion.

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Integrating the Standards of Mathematical Practice through Whole Class discussions as a natural part of teaching.

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Make Sense of Problems and Persevere in solving them (MP1)

Explore meaning of problemLook for entry points to a solutionAnalyze givens, constraints, relationships,

and goals. Think about possible solution and plan a

pathwayConsider similar problems or simpler

problemsMonitor and evaluate their progress and

change course if necessary

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Reason abstractly and quantitatively (MP2)

• Mathematically proficient students make sense of quantities and their relationships in problem situations.

• Quantitative reasoning entails habits of 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.

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Construct viable arguments and critique the reasoning of others. (MP3)

Mathematically proficient students can justify their conclusions, communicate them to others, and respond to the arguments of others.

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Model with mathematics (MP4)

They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions.

They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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Use appropriate tools strategically (MP 5)Mathematically proficient students consider the available tools when solving a mathematical problem.

These tools might include pencil and paper, concrete models … .

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Attend to precision. (MP6)Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

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Look for and make use of structure. (MP7)• Mathematically proficient students look closely to discern a

pattern or structure

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Look for and express regularity in repeated reasoning. (MP 8)Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

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Facilitating effective discussions involves thinking about the process of teaching!

• Setting up the physical space

• Classroom Routines

• Lesson Planning

• Teacher questioning (To start and facilitate discussion)

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Lamberg (2012) Framework from Whole Class Mathematics discussion book.• Can download copy from my blog:

• http://mathdiscussions.wordpress.com/whole-class-discussion-framework-checklist/

• Blog contains many resources and links to support your teaching and use the framework

• This Framework allows you to see the “big picture” of teaching and how the parts such as whole class discussion fits in. Facilitating discussions that support learning involves having all these pieces work together.

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  Not met

Work in Progress

Working Great

To do list Tools from Whole Class Discussion book

Setting up the

Classroom

        (Chapter 2)*P.36 Checklist

Setting up Physical Space

         

Cultivating Classroom

Environment/Routines 

Not Met 

 Work in Progress

Working Great 

Note: Routines for (Communicating/Listening Takes place during whole class discussion. These routines take time to develop.)

(Chapter 3)*P.60 Strategies for Your Classroom, Ideas for Developing classroom Routines

Routines for Preparing for Discussion

        Standards of Mathematical Practice 1,4,5,7,8

Routines for Communicating

        Standards of Mathematical Practice 2,3

Routines for Listening/Reflecting

        Standards of Mathematical Practice 1 24

Lesson Planning 

      Note: Third level of planning takes place during lesson/discussion. The purpose of the first 2 levels of planning is to situate the discussion in larger goals to support deeper learning.

(Chapter 4)*P.91 Strategies for

Your Classroom (Three Levels of

Planning)

First level Planning(Long term & Short Term Goals)Concepts (big ideas)Unit Plan (Sequencing/learning trajectory)    

        *P.92 Concept Map*P.93 Rubric for Unit Planning

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Second Level of Planning5 E-Lesson Plan- (Anticipating Student Reasoning/Misconceptions Errors,Format for using a problem solving approach to teaching and structuring time) 

        *P.94 Rubric for 5E Lesson Plan: Level 2

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Takes Place During the Lesson

Third Level of Planning(Adapting discussion to support student understanding/needs)Making decisions on what to talk about based on student reasoning during lessons

Not met 

 Still Working

Working Great 

  *Rubric for Planning the Discussion: Level 3

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The Whole Class Discussion

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Three Levels of Analysis and Sense making

Phase 1: Making Thinking Explicit

(Explaining Reasoning)

Phase 2: Analyzing Each Other's Solution

(Analyzing Low Level to More Sophisticated Reasoning)

Phase 3: Developing New Mathematical Insights

(Abstract Mathematical Concepts)

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Continuum: Levels of Understanding and Student Strategies

Inefficient strategies

Efficient Strategies

11111 11111 11111 11111 5+5+5+5

5x4=20 20 ÷5=4

 

Simpler Representations (Concrete)

Abstract Representations

** + **

2 + 2

2 apples and 2 apples two groups of two apples

two plus two

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.

Mathematical Connections

Teacher Questioning/ Supporting Mathematical

Connections 

Not Met   Still Working  Working Great

Note: These levels of Sense Making make up the Whole discussion. The teacher poses a problem and issue for class to discuss. The teacher uses questions to help students make mathematical connections. Students communicate their ideas; reflect on their own ideas and others being presented to make connections. (See classroom routines section).

   (Chapter 5)See p. 69Figure 4.1(Identify topic for discussion based on goals)

Three Levels of Sense Making         *P.116 Strategies for You Classroom: The Three Levels of Sense Making

Phase 1: Making Thinking explicit         Standards of Mathematical Practice 2, 3, 4

Phase II: Analyzing Each other’s solutions 

        Helping students make connections from low level strategies to sophisticated strategies See p. 102-103Address Errors/MisconceptionsStandards of Mathematical Practice 1,3,4,6,7,8

Phase III:  Developing New Mathematical Insights

   

                  

See Case Study p.103-107 Identify “big ideas” in Lesson and create a recordStandards of Mathematical Practice 1,2,4,5,6,7,8

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        Improving Teaching Through Reflection

 

Reflecting on Your Teaching

(Making Teaching Visible)

(Chapter 6)

What are you currentl

y doing?

What is working/what is not?

      

Making teaching VisibleWhat are you currently focusing on?

        See: Reflecting on Practice Questions throughout chapters &*Reflecting on Your Practice Worksheets in End of Chapter Study Guides

 

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Working Smarter not Harder!Integrate discussion as a natural part of teaching to support

mathematical learning.

Use time efficiently

Remember facilitating effective discussions is a journey and a process.

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Whole Class Mathematics Discussions: Improving in-depth Mathematical thinking and Learning

Slides, Video Clips and downloadable worksheets are available in PDToolkit

http://pdtoolkit.pearsoncmg.com/login

Video clip 1.4 Video Clip 1. 2

Blog: www.mathdiscussions.wordpress.com

Teruni Lamberg, Ph.D.University of Nevada, Reno

Terunil@unr.edu34