Going Beyond Number Talks
Going Beyond Number Talks KEVIN LARKIN ADRIENNE DELONG DISTRICT STAFF DEVELOPERS ELEMENTARY MATHEMATICS PINELLAS COUNTY SCHOOLS, FLORIDA
Goals
Become familiar with the six types of conversations from Intentional Talk How to Structure and Lead Productive Mathematical Discussions.
Understand when each type of discussion is appropriate.
Learn how to use these conversations to support your Number Talks routine.
Our Background with Number Talks
Summer Camp for 1st and 2nd graders
Alternative Algorithms
Ah Ha Moment: Following one problem across various strategies and models
Number Talks by Sherry Parrish Purposefully
crafted and sequenced number strings
Emphasizes
one and only one strategy over time
Pulled
together everything our district was promoting
Discovering Intentional Talk
In Number Talks, Sherry Parrish discusses the need to use tools to explain/investigate why strategies work
While implementing Number Talks over the past five years, we discovered the need to investigate various strategies to increase student understanding of the strategies.
Intentional Talk by Elham Kazemi and Allison Hintz
Investigative lesson structure provided
Used to investigate Number Talks Strategies as needed
Purposeful student conversation
Principles to Action Effective Mathematics Teaching Practices
Establish mathematics goals to focus learning. 2. Implement tasks to promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions. 6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking. 1.
Standards for Mathematical Practice
Makes sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 1.
Developing Purposeful Classroom Discussions Four Principals for Classroom Discussions
Discussions should Achieve a Mathematical Goal
Students Need to Know What and How to Share
Teachers need to Orient Students to One Another and the Mathematical Ideas
Teachers Must Communicate that all Students are Sense Makers and that their Ideas are Valued.
Two Aims for Classroom Discussions
Open Strategy Sharing
Building a toolbox of strategies for students to use 5 Practices for Orchestrating Productive Mathematics Discussions
Smith and Stein
Targeted Discussions
Compare and Connect
Why? Let’s Justify
What’s Best and Why?
Define and Clarify
Troubleshoot and Revise
Targeted Discussions Compare and Connect
Why? Let’s Justify
What’s Best and Why?
Define and Clarify
Students compare similarities and differences between various strategies
Students generate their own reasons for why a particular strategy works
Students determine a best solutions strategy for problems
Students discuss appropriate use of models, tools, vocabulary, and notation
Troubleshoot and Revise Students discuss which strategies work or where a strategy goes wrong
Compare and Connect
Open Strategy sharing often leads to Compare and Connect discussions.
What are the mathematical connections between the strategies that you want your students to make?
How are the strategies similar or different?
It is important for students to see these similarities and differences and understand the value of doing so.
Compare and Connect
Instructional Decisions for planning
Choose the strategies to compare and connect.
Identify important connections to develop.
Record strategies the way you will present them.
What do you expect students to notice?
How will you respond to what the students notice?
What mathematical ideas do you want to highlight at the end of the lesson?
Compare and Connect When
do you have a “Compare and Connect” discussion?
Moving students towards a slightly more sophisticated strategy For example: An array model and Partial Products
Comparing different mathematical tools or representations
For example: Using a subtractive method to divide and using Partial Quotients
Making sense of different strategies students have generated For example: Adding Up to subtract and Removal (take away) on a hundreds chart or open number line
Compare and Connect: Second Grade Lesson: 8 + 7 using Ten Frames Counting on
Who doesn't have students who are still counting all or counting on?
Two common versions of this strategy.
Making Ten
Break apart one number into addends that allow students to make a friendly 10.
Compare and Connect Read the Compare and Connect lesson in your handout. Think
about the instructional decision that were made. Choose the strategies to compare and connect. Identify important connections to develop. Record strategies the way you will present them. What do you expect students to notice? How will you respond to what the students notice? What mathematical ideas do you want to highlight at the end of the lesson?
Discuss
one or two of these instructional decisions with a shoulder partner.
Why? Let’s Justify
Students develop, use and justify ideas about how mathematics works.
Students ask "Why does this make sense?".
Instructional Decisions
Identify the idea or generalization students will be exploring
Select a task/tasks that you will use to help facilitate the discovery
Anticipate what students might say and what you will say in return as students discuss and justify their thinking.
Why? Let’s Justify
When do you have a “Why? Let’s Justify” discussion? A rule or trick is commonly used. For Example: Partial Products (annexing zeros) You can connect a strategy students are beginning to use to a visual model or a problem context in order to make sense of how a strategy works For Example: Doubling and Halving Using Multiplication to Divide Making Ten
Why? Let’s Justify: Fourth Grade Lesson: Doubling/Halving 8 x 16 Day 1: Introduced a new multiplication strategy...Doubling and Halving 1 x 16 2x8 4x4 8x2 16 x 1 During the Number Talk, the students and I discussed how the factors are changing. The students discovered that one factor is being multiplied by 2 (doubling) and the other factor is being divided by 2 (halving)
Day 2: I facilitated a Why? Let’s Justify discussion to discover why doubling one factor and halving the other factor results in the same product each time. We explored the idea of doubling and halving with an 16 x 8 array At first the class wanted to double 8 and half 16. The girls tried this out and compared their array to the boys. Both arrays are the same.
After realizing that it was the same problem, the class decided to double 16 and half 8. They came up with 4 x 32.
Is this the same product as 8 x 16?
Then the class doubled 32 and halved 4 to get 2 x 64. Is this the same product as 8 x 16?
Finally the class doubled 64 and halved 2 and came up with 1 x 128. Is this the same product as 8 x 16? I asked students to prove that 1 x 128 results in the same product as 8 x 16. Students cut apart the last array to assemble the original array back together. At this point, the class was convinced that 8 x 16 will result in the same product as 4 x 32, 2 x 64 and 1 x 128.
What’s Best and Why?
Students become more selective about when to use a specific strategy.
Students discuss effectiveness and efficiency.
Instructional Decisions
Select the strategies you will highlight
Identify the goal of the discussion...effectiveness of a strategy, efficiency of a strategy
Select the problems with/without solutions that will be used to launch the discussion
What’s Best and Why?
When do you have a "What's Best and Why?" discussion?
Students are transitioning from one strategy to another. For Example: After a new strategy has come to light during your Number Talks.
You want students to think about when it would be best to use a specific strategy. For Example: Removal or Adding Up to Subtract Doubling and Halving or Partial Products
Define and Clarify Teachers are supporting student understand and use of new tools, representations and vocabulary. Students develop the skills needed to use tools with understanding and precision. Instructional Decisions Identify the tool, representation, symbol or vocabulary word(s) that you will be discussing Decide if the tool, representation, symbol or vocabulary is new to students or is used in a new way. Identify the problem or task students are working on and how you can best support their understanding.
Define and Clarify
When do you facilitate a "Define and Clarify" discussion? When you are introducing a new tool to students For Example: Using the Tens Frame for Making Ten Using the Hundreds Chart for Adding Up/Removal Using the Open Number Line to record the jumps for Adding Up/Removal Using an Area Model for multiplication or division instead of the Array
Define and Clarify
When you are using a familiar tool in a different way For Example: Using an Array to model Partial Products with whole numbers and using an array to model Multiplication to Divide with whole numbers, fractions and decimals
When you are clarifying the use of symbolic notation For Example: During any Number Talk
Troubleshoot and Revise Teachers and students work through partial understandings or misconceptions. This type of discussion can be initiated by the teacher or the student Instructional Decisions Identify the confusion or misunderstanding students are having and should be discussed. Identify the insight you would like students to gain/acquire. Identify the problem context, diagrams and/or questions that would help students revise their thinking and understanding.
Troubleshoot and Revise
When do you facilitate a "Troubleshoot and Revise" discussion?
Students might recognize they are stuck.
Students might realize that a strategy they thought made sense resulted in an incorrect answer.
The teacher might notice a misunderstanding with several students which is worth discussing. For Example: Partial Products (students are just adding a zero)
Partial Quotients (student break up the dividend and not the divisor)
Principles to Action Effective Mathematics Teaching Practices 1.
Establish mathematics goals to focus learning.
2.
Implement tasks to promote reasoning and problem solving.
3.
Use and connect mathematical representations.
4.
Facilitate meaningful mathematical discourse.
5.
Pose purposeful questions.
6.
Build procedural fluency from conceptual understanding.
7.
Support productive struggle in learning mathematics.
8.
Elicit and use evidence of student thinking.
Standards for Mathematical Practice
Makes sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 1.
In Summary
We have found that when teachers begin a new strategy in Number Talks, most of the students sit there silently, waiting for other students to answer. In our experience, the students do not seem to understand the strategy intended for that Number Talk. These targeted discussion structures posed by Elham Kazemi and Allison Hintz in their book Intentional Talk: How to Structure and Lead Productive Mathematical Decisions, can help build a conceptual understanding of the strategies presented in Number Talks.
Resources Kazemi, Elham and Hintz, Allison. Intentional Talk How to Structure and Lead Productive Mathematical Discussions. Portland, ME; Stenhouse Publishers, 2014. Parrish, Sherry. Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5. Sausalito, CA; Math Solutions, 2010. Stein, Margaret and Smith, Mary Kay. 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA; NCTM 2011. NCTM. Principles to Actions; Ensuring Mathematical Success for All. Reston, VA; NCTM 2014
Thank you for coming to our presentation on a beautiful Saturday morning when you could be sipping coffee by the river, sleeping in, or site seeing. Kevin Larkin
Adrienne DeLong
[email protected]
[email protected]
Goals
Become familiar with the six types of conversations from Intentional Talk How to Structure and Lead Productive Mathematical Discussions.
Understand when each type of discussion is appropriate.
Learn how to use these conversations to support your Number Talks routine.
Our Background with Number Talks
Summer Camp for 1st and 2nd graders
Alternative Algorithms
Ah Ha Moment: Following one problem across various strategies and models
Number Talks by Sherry Parrish Purposefully
crafted and sequenced number strings
Emphasizes
one and only one strategy over time
Pulled
together everything our district was promoting
Discovering Intentional Talk
In Number Talks, Sherry Parrish discusses the need to use tools to explain/investigate why strategies work
While implementing Number Talks over the past five years, we discovered the need to investigate various strategies to increase student understanding of the strategies.
Intentional Talk by Elham Kazemi and Allison Hintz
Investigative lesson structure provided
Used to investigate Number Talks Strategies as needed
Purposeful student conversation
Principles to Action Effective Mathematics Teaching Practices
Establish mathematics goals to focus learning. 2. Implement tasks to promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions. 6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking. 1.
Standards for Mathematical Practice
Makes sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 1.
Developing Purposeful Classroom Discussions Four Principals for Classroom Discussions
Discussions should Achieve a Mathematical Goal
Students Need to Know What and How to Share
Teachers need to Orient Students to One Another and the Mathematical Ideas
Teachers Must Communicate that all Students are Sense Makers and that their Ideas are Valued.
Two Aims for Classroom Discussions
Open Strategy Sharing
Building a toolbox of strategies for students to use 5 Practices for Orchestrating Productive Mathematics Discussions
Smith and Stein
Targeted Discussions
Compare and Connect
Why? Let’s Justify
What’s Best and Why?
Define and Clarify
Troubleshoot and Revise
Targeted Discussions Compare and Connect
Why? Let’s Justify
What’s Best and Why?
Define and Clarify
Students compare similarities and differences between various strategies
Students generate their own reasons for why a particular strategy works
Students determine a best solutions strategy for problems
Students discuss appropriate use of models, tools, vocabulary, and notation
Troubleshoot and Revise Students discuss which strategies work or where a strategy goes wrong
Compare and Connect
Open Strategy sharing often leads to Compare and Connect discussions.
What are the mathematical connections between the strategies that you want your students to make?
How are the strategies similar or different?
It is important for students to see these similarities and differences and understand the value of doing so.
Compare and Connect
Instructional Decisions for planning
Choose the strategies to compare and connect.
Identify important connections to develop.
Record strategies the way you will present them.
What do you expect students to notice?
How will you respond to what the students notice?
What mathematical ideas do you want to highlight at the end of the lesson?
Compare and Connect When
do you have a “Compare and Connect” discussion?
Moving students towards a slightly more sophisticated strategy For example: An array model and Partial Products
Comparing different mathematical tools or representations
For example: Using a subtractive method to divide and using Partial Quotients
Making sense of different strategies students have generated For example: Adding Up to subtract and Removal (take away) on a hundreds chart or open number line
Compare and Connect: Second Grade Lesson: 8 + 7 using Ten Frames Counting on
Who doesn't have students who are still counting all or counting on?
Two common versions of this strategy.
Making Ten
Break apart one number into addends that allow students to make a friendly 10.
Compare and Connect Read the Compare and Connect lesson in your handout. Think
about the instructional decision that were made. Choose the strategies to compare and connect. Identify important connections to develop. Record strategies the way you will present them. What do you expect students to notice? How will you respond to what the students notice? What mathematical ideas do you want to highlight at the end of the lesson?
Discuss
one or two of these instructional decisions with a shoulder partner.
Why? Let’s Justify
Students develop, use and justify ideas about how mathematics works.
Students ask "Why does this make sense?".
Instructional Decisions
Identify the idea or generalization students will be exploring
Select a task/tasks that you will use to help facilitate the discovery
Anticipate what students might say and what you will say in return as students discuss and justify their thinking.
Why? Let’s Justify
When do you have a “Why? Let’s Justify” discussion? A rule or trick is commonly used. For Example: Partial Products (annexing zeros) You can connect a strategy students are beginning to use to a visual model or a problem context in order to make sense of how a strategy works For Example: Doubling and Halving Using Multiplication to Divide Making Ten
Why? Let’s Justify: Fourth Grade Lesson: Doubling/Halving 8 x 16 Day 1: Introduced a new multiplication strategy...Doubling and Halving 1 x 16 2x8 4x4 8x2 16 x 1 During the Number Talk, the students and I discussed how the factors are changing. The students discovered that one factor is being multiplied by 2 (doubling) and the other factor is being divided by 2 (halving)
Day 2: I facilitated a Why? Let’s Justify discussion to discover why doubling one factor and halving the other factor results in the same product each time. We explored the idea of doubling and halving with an 16 x 8 array At first the class wanted to double 8 and half 16. The girls tried this out and compared their array to the boys. Both arrays are the same.
After realizing that it was the same problem, the class decided to double 16 and half 8. They came up with 4 x 32.
Is this the same product as 8 x 16?
Then the class doubled 32 and halved 4 to get 2 x 64. Is this the same product as 8 x 16?
Finally the class doubled 64 and halved 2 and came up with 1 x 128. Is this the same product as 8 x 16? I asked students to prove that 1 x 128 results in the same product as 8 x 16. Students cut apart the last array to assemble the original array back together. At this point, the class was convinced that 8 x 16 will result in the same product as 4 x 32, 2 x 64 and 1 x 128.
What’s Best and Why?
Students become more selective about when to use a specific strategy.
Students discuss effectiveness and efficiency.
Instructional Decisions
Select the strategies you will highlight
Identify the goal of the discussion...effectiveness of a strategy, efficiency of a strategy
Select the problems with/without solutions that will be used to launch the discussion
What’s Best and Why?
When do you have a "What's Best and Why?" discussion?
Students are transitioning from one strategy to another. For Example: After a new strategy has come to light during your Number Talks.
You want students to think about when it would be best to use a specific strategy. For Example: Removal or Adding Up to Subtract Doubling and Halving or Partial Products
Define and Clarify Teachers are supporting student understand and use of new tools, representations and vocabulary. Students develop the skills needed to use tools with understanding and precision. Instructional Decisions Identify the tool, representation, symbol or vocabulary word(s) that you will be discussing Decide if the tool, representation, symbol or vocabulary is new to students or is used in a new way. Identify the problem or task students are working on and how you can best support their understanding.
Define and Clarify
When do you facilitate a "Define and Clarify" discussion? When you are introducing a new tool to students For Example: Using the Tens Frame for Making Ten Using the Hundreds Chart for Adding Up/Removal Using the Open Number Line to record the jumps for Adding Up/Removal Using an Area Model for multiplication or division instead of the Array
Define and Clarify
When you are using a familiar tool in a different way For Example: Using an Array to model Partial Products with whole numbers and using an array to model Multiplication to Divide with whole numbers, fractions and decimals
When you are clarifying the use of symbolic notation For Example: During any Number Talk
Troubleshoot and Revise Teachers and students work through partial understandings or misconceptions. This type of discussion can be initiated by the teacher or the student Instructional Decisions Identify the confusion or misunderstanding students are having and should be discussed. Identify the insight you would like students to gain/acquire. Identify the problem context, diagrams and/or questions that would help students revise their thinking and understanding.
Troubleshoot and Revise
When do you facilitate a "Troubleshoot and Revise" discussion?
Students might recognize they are stuck.
Students might realize that a strategy they thought made sense resulted in an incorrect answer.
The teacher might notice a misunderstanding with several students which is worth discussing. For Example: Partial Products (students are just adding a zero)
Partial Quotients (student break up the dividend and not the divisor)
Principles to Action Effective Mathematics Teaching Practices 1.
Establish mathematics goals to focus learning.
2.
Implement tasks to promote reasoning and problem solving.
3.
Use and connect mathematical representations.
4.
Facilitate meaningful mathematical discourse.
5.
Pose purposeful questions.
6.
Build procedural fluency from conceptual understanding.
7.
Support productive struggle in learning mathematics.
8.
Elicit and use evidence of student thinking.
Standards for Mathematical Practice
Makes sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 1.
In Summary
We have found that when teachers begin a new strategy in Number Talks, most of the students sit there silently, waiting for other students to answer. In our experience, the students do not seem to understand the strategy intended for that Number Talk. These targeted discussion structures posed by Elham Kazemi and Allison Hintz in their book Intentional Talk: How to Structure and Lead Productive Mathematical Decisions, can help build a conceptual understanding of the strategies presented in Number Talks.
Resources Kazemi, Elham and Hintz, Allison. Intentional Talk How to Structure and Lead Productive Mathematical Discussions. Portland, ME; Stenhouse Publishers, 2014. Parrish, Sherry. Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5. Sausalito, CA; Math Solutions, 2010. Stein, Margaret and Smith, Mary Kay. 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA; NCTM 2011. NCTM. Principles to Actions; Ensuring Mathematical Success for All. Reston, VA; NCTM 2014
Thank you for coming to our presentation on a beautiful Saturday morning when you could be sipping coffee by the river, sleeping in, or site seeing. Kevin Larkin
Adrienne DeLong
[email protected]
[email protected]
