1. Number Talks July 27, 2016 Liberty Union High School District
4/5/16
Number Talks
July 27, 2016
Liberty Union High School District
Christen Schwartz
Math Coordinator, CCCOE
Secondary Number Talks 1.0

2. Logistics Schedule Materials Number Talks Padlet 4/5/16
Secondary Number Talks 1.0

3. Outcomes Engage Reflect Collaborate 4/5/16
1)Engage participants in routines that deepen understanding of mathematics for all students
2)Reflect on how these routines can be implemented at your grade level
3)Network with others by exchanging ideas
Secondary Number Talks 1.0

4. Making Number Talks Matter Humphreys/Parker
Is This Our Current Reality?
“Students come to classrooms fearing and avoiding math and, worse, thinking they are no good at it. Believing that mathematics is mostly about using procedures correctly, they have learned to focus on getting the right answer, whether or not the process makes sense to them. Many students don't expect math to make sense at all. The result is that students learn to disengage their reasoning – and even distrust it.”
Intent: draw out teachers’ acknowledgement that students need computational support that looks different from what they’ve had in the past.
What evidence do we have that this is our current reality?
“Making Number Talks Matter is about helping students take back the authority of their own reasoning.
[It will help you] learn how to facilitate this routine so that, over time, students develop a strong sense of the meaning of quantities and operations while gaining proficiency with mathematical processes.” (p. 1)
Making Number Talks Matter Humphreys/Parker

5. Making Number Talks Matter Humphreys/Parker
4/5/16
Routines
“Number Talks turn students’ roles in math class upside down. Now they are supposed to figure out something rather than be told the steps to follow.”
Routines (vs. strategies) – the way you are getting students engaged in the mathematics; routines need to be practiced, modeled, reinforced; repeated practice to establish routines – then they can focus on the mathematics
*Routines must come before you can focus on strategies; start with number talks that are not cognitively demanding, so you can focus on the process / routine
Stand; Hand Signals; Whisper your answer; “Loud & Proud”; Turn & Talk
Teacher-record thinking w/ names; accept, respect all answers
Socially shared, scripted slices of behavior
They need to become part of the fabric of the classroom through repeated use
Practices that are crafted to achieve specific ends through the use of effective tools in an efficient and workable manner
Making Number Talks Matter Humphreys/Parker
Secondary Number Talks 1.0

6. Dot Images How many do you see? How do you see it? Do Dot image talk
Give a minute to think.
Record different strategies on papers.
Do not tell to count the dots!
Called a Dot Image or Dot Talk, can be made of symbols, holiday themed, squares or geo figures…
Connect to deeper understanding gains to apply to math tasks with repeated patterns, algebraic thinking
Cathy Humphrey’s response to the question, “Where are the 10th grade number talks?” (Shared by Chrissy): Here it is
Geometry connection, although number talks don’t have to be tied to the day’s lesson and it may be helpful if they are not
A 10th grader will respond differently to this prompt than a 6th grader. The task is self-differentiated.
“Number Talks are all about students and their ways of thinking. Many teachers, however, don’t know how to bring student thinking to the foreground in their classrooms. This book is designed to help you, the teacher, learn to enact Number Talks in ways that help you accomplish this goal.”

7. Making Number Talks Matter Humphreys/Parker
4/5/16
What is a Number Talk?
“... a brief daily practice where students mentally solve computation problems and talk about their strategies, as a way to dramatically transform teaching and learning in their mathematics classrooms.”
Connect to: Number Talks by Sherry Parish
“Simply defined, number talks are five- to fifteen-minute classroom conversations around purposefully crafted computation problems that are solved mentally.”
Brief: 5-15 minutes ​
Making Number Talks Matter Humphreys/Parker
Secondary Number Talks 1.0

8. Why Are Number Talks So Important?
Think about the questions below as we reflect using two different resources.
What is at stake for our students who lack flexibility and confidence working with numbers?
How might Number Talks help students?
A/B Partners
A: Read the first section of Chapter 1 on pages 5-6
B: Read Math Matters excerpt

9. Why Are Number Talks So Important?
With an elbow partner, chose one person to be A and one to be B.
Partner A:
Read and reflect on the intro on p. 5-6 in the book, highlight any key points.
Partner B:
Read and reflect on the excerpt from Math Matters, highlight any key points.
A/B Partners
A: Read the first section of Chapter 1 on pages 5-6
B: Read Math Matters excerpt

10. Why Are Number Talks So Important?
Turn & Talk
With your elbow partner (A/B), share your thoughts and highlights.
What is at stake for our students who lack flexibility and confidence working with numbers?
How might Number Talks help students?
All: discuss the question on the slide
Students “make mathematically convincing arguments, and critique and build on the ideas of their peers. – p.5
This book is about - p.6:
helping teachers learn to make NT matter for their students
helping students learn to work flexibly with numbers and arithmetic properties
helping them build a solid foundation and confident disposition for future mathematics learning
empowering both teachers and students as mathematical thinkers

11. Which One Doesn’t Belong?
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Which One Doesn’t Belong?
: link on page
Have teachers move to corners of room if possible. If not, have talk in groups about their choice and defend. Come back together and reflect and share.
Link on page to show.
wodb
Secondary Number Talks 1.0

12. Connections to Standards for Mathematical Practice
4/5/16
Connections to Standards for Mathematical Practice
Consider and test various strategies to see if they make sense (SMP 1).
Persevere in solving problems (SMP 1).
Clarify and justify thinking (SMP 1,2,3).
Investigate and apply mathematical relationships (SMP 2,3,7,8).
Attend to precision of solutions (SMP 6).
Build a repertoire of efficient strategies and tools (SMP 1,3,4, 5,7,8).
Make decisions about choosing efficient strategies for specific problems (SMP 5,7,8).
How do Number Talks engage students in the SMPs?
Look at papers in folder, connect to grouped SMPs
Depending on level of problem and students, demands more MPs.
Builds students understandings of connections in math.
Secondary Number Talks 1.0

13. Classroom Environment
4/5/16
Classroom Environment
“In number talks, wrong answers are used as opportunities to unearth misconceptions and for students to investigate their thinking and learn from their mistakes.”
-Parrish
“Your brain grows when you make a mistake because it’s the time when the brain is challenged and is struggling and those are the best times for brain growth.”
-Boaler
MATH – Mistakes Allow Thinking to Happen; positive class norms
Cohesive classroom community is essential for creating a safe, risk-free environment for effective number talks. Students should be comfortable in offering responses for discussion, questioning themselves and their peers, and investigating new strategies
Jo Boaler: From brain research, struggle and mistakes are really important.
Your brain grows when you make a mistake because it’s the time when the brain is challenged and is struggling and those are the best times for brain growth.
“Cognitive dissonance is necessary for learning – not just for our students. Mistakes – even our own – are truly sites for learning.”
-Hiebert
Secondary Number Talks 1.0

14. Subtraction Number String
4/5/16
Subtraction Number String
Why do you think this number string is set up like this?
What connections might students make in their thinking process during this talk?
Go over this, only do if there is time. Use an open number line as a model for participants’ thinking (as opposed to “of” their thinking) as appropriate. See p. 78 for note on open number line.
315 – 97 (Round the subtrahend to a multiple of ten: p. 41)
(p.44)
6 1/8 – 2 5/8 (p.44)
Mental computation is a key component of number talks because it encourages students to build on number relationships to solve problems instead of relying on memorized procedures.
Mental math encourages students to build on number relationships to develop efficient, flexible strategies with accuracy… strengthens students’ understanding of place value
Explain difference between “number talk” and “number string.”
Why do you think this number string is set up like this?
Each number string is structured to build on the previous.
Anticipate student thinking  highlight math that you want to come out by recording strategically
Link to Math Talks resource: Go to link and show resource. This is on the padlet!
Math Talks
Secondary Number Talks 1.0

15. Thoughts for Successful Number Talks
Provide ample wait time
Ask purposeful questions – “graduated pressing”
Thinking together – how else can we think about a problem?
Learn to listen and accurately record students’ thinking
Do Number Talks regularly
Work towards a community of discourse – encourage clarity of academic language
Make the most of multiple answers and of errors
Connect computation to visual models
Support students to add to their repertoire of strategies
Purposeful Questions:
“graduated pressing” (p. 18) to ask students to explain why their strategies make sense
How else could we think about this? (p. 19)
“Through our questions, we seek to understand students’ thinking.” (p. 26)
See also p. 174 in Ch 10: Managing Bumps in the Road
Thinking together: when students only seem to come up with one strategy, instead of who thought of it differently, ask - how else can we think about a problem?
Learning to Listen:
Patterns of interaction should be increasingly student/student as well as teacher/student
Supporting students’ developing use of language (clarity of academic language)
See also p. 173 in Ch 10: Managing Bumps in the Road
Make the Most of Errors:
Reference Error Posters (“MATH – Mistakes Allow Thinking to Happen” and “In Our Class…”)
Note that sometimes the errors are our (teacher’s) errors. That’s ok! Model for students a healthy response to error: productive struggle.
See also p in Ch 10: Managing Bumps in the Road
Connect computation to visual models: nudge students beyond traditional algorithms – to understand the “why”
Making Number Talks Matter Humphreys/Parker

16. Teacher’s Role… In Number Talks: In Discourse:
4/5/16
Teacher’s Role…
I decide…
In Number Talks:
Help students articulate thinking by talking through/clarifying using academic language (don’t interrupt thinking)
Pre-select different thinking
Allow processing time (mental math, write out thoughts, hand signals – differentiate)
Recorder of students thinking
Facilitate/Discussion Guide
Formative assessment
Accept, respect, and consider all answers
Ask, don’t tell
In Discourse:
What to pursue in depth from among the ideas that students bring up
When and how to attach mathematical notation and language to students’ ideas
when to model
when to clarify
when to lead
when to provide information
when to let a student grapple
when and how to encourage each student to participate
-The Teacher’s Role in Discourse, NCTM, 1991
Teacher’s Role =
help students articulate thinking by talking through/clarifying using academic language (don’t interrupt thinking)
pre-select different thinking
allow processing time (mental math, write out thoughts, hand signals – differentiate),
record students thinking
facilitate/discussion guide
formative assessment
accepting all answers
ask, don’t tell
ask guiding questions
In both roles, teachers are facilitators/guides - encourage student thinking, conversation, clarification, connections
Secondary Number Talks 1.0

17. Types of Mathematical Discourse:
April/May 2016
Types of Mathematical Discourse:
Generalize
Justify/Analyze/Evaluate
Explain/Apply/Understand
Recall
Confirm (agree or disagree)
Handout: Building Options for Discourse (DOK)
Students have the opportunity to:
Clarify their own thinking
Consider and test other strategies to see if they are mathematically logical
Investigate and apply mathematical relationships
Build a repertoire of efficient strategies
Make decisions about choosing efficient strategies for specific problems
Number Talks 1.0 K-5

18. Convince myself Convince a friend Convince a skeptic
Increasing Rigor in Discourse
Convince myself Convince a friend Convince a skeptic
In text (Small group processing, Whole group processing, Wrap-up) p
Number Talks can spark investigation: Pose the question, “will it always work?” Looking at a student’s strategy from a previous number talk, try the strategy with other problems, talk about findings, convince us of your findings
Discuss proof. Students are often satisfied with checking a few cases.

19. What's the capacity of the tall vase?
4/5/16
What's the capacity of the tall vase?
Vase
Secondary Number Talks 1.0

20. CA ELD Standards Connections
4/5/16
CA ELD Standards Connections
Resource created from the CA ELD Standards
Take a couple minutes to reflect and fill in right hand column.
Secondary Number Talks 1.0

21. Connection to ELD Standards
4/5/16
Connection to ELD Standards
The goal in Number Talks is to support students’ participation in a mathematical discussion.
Focus on students’ mathematical reasoning, not accuracy in using the language.
“Precise claims can be expressed in imperfect language.”
Uncover the mathematics in what students say and do.
Handouts: ELD Planning & Reflection Graphic Organizer – we will use this is as a reflection tool; Math Talk Moves
Judit Moschkovich (UCSC) – article on padlet - Recommendations for meeting the challenges in developing mathematics instruction for Els that is aligned with CCSS
Focus on students’ mathematical reasoning, not accuracy in using language: teachers should first focus on promoting and privileging meaning
Teachers need to learn how to recognize the emerging mathematical reasoning learners construct in, through, and with emerging language.
“Instruction should provide opportunities for students to actively use mathematical language to communicate about and negotiate meaning for mathematical situations.”
from “Mathematics, the Common Core, and Language”, Moschkovich, J.
Secondary Number Talks 1.0

22. Classroom Discussions
4/5/16
Classroom Discussions
“The heart of number talks is classroom conversations focused on making sense of mathematics.”
- Sherry Parish
Uncovering the mathematics…
Discussions may naturally lead student investigations of strategies  making and testing conjectures;
Secondary Number Talks 1.0

23. Would you rather? 4/5/16 Do this one!
Have a stack of quarters from the floor to the top of your head OR $225?
Whichever answer you choose, justify your reasoning with mathematics. Help for teachers:
According to Wikipedia, average height for girls by age:
According to Wikipedia, average height for boys by age:
If you want a better conversation, change the $225 to suit your class needs:
Oh, thickness of a quarter might help:
Secondary Number Talks 1.0

24. 4/5/16
If time, do this one!
Flip 3 coins and win if they all match OR roll 3 dice and win if none of them match?
Whichever option you choose, justify your reasoning with mathematics.
Secondary Number Talks 1.0

25. 4/5/16
Just show this one…
Lease a $19,000 vehicle with a $2,000 down payment OR buy a $19,000 vehicle with $0 down payment.
Some links that may be helpful:
Secondary Number Talks 1.0

26. Multiplication Number Talk
4/5/16
Multiplication Number Talk
16 x 35
Pre Number Talk Prompt: What is the teacher’s role during this number talk?
Possible closure questions:
Which of these strategies have a connection between them? What is that connection?
Which of these strategies is one that you would like to use in the future? Why?
Which strategy is the most efficient? Why?
Connect to Poster: 5 key components of a number talk
Mental computation is a key component of number talks because it encourages students to build on number relationships to solve problems instead of relying on memorized procedures.
Mental math encourages students to build on number relationships to develop efficient, flexible strategies with accuracy… strengthens students’ understanding of place value
Secondary Number Talks 1.0

27. Investigating a Strategy
4/5/16
Investigating a Strategy
How did Molly’s strategy of “halving and doubling” work?
Will Molly’s strategy always work?
Bullet 1: pair/share
Bullet 2: Table Groups: Have a participant come up and demonstrate “how Molly would do this problem.”
Teach students how to “tinker:”
With a partner, make up three other problems and try them out using Molly’s strategy. Emphasize the importance of keeping track – in a hopefully organized way – of their findings. Then after a little while, “What did you find out?” Share out a bit.
Teach students to wonder: (this is probably a question to share but not to pose, as we might have an adult “spoiler” in the room)
“Hmmm…there has to be a reason why this keeps happening over and over again. Does anybody have a theory about why this works?”
What might be homework: “Will Molly’s strategy always work? Be prepared to bring your thinking to class tomorrow. Your group will try to come up with a mathematically convincing argument for whatever answer you decide on.”
Secondary Number Talks 1.0

28. Moving Beyond Rote Procedures
Look at the teacher vignette.
How does the teacher press beyond rote procedures while still honoring what students know?
Note that “Ruth” in the vignette is the teacher (one of the authors of the book).
If time allows, ask participants to read and discuss the excerpt. (Otherwise, suggest they tab.)
Ask teachers what they notice and whether they anticipate their students will get stuck in the standard algorithms (what evidence?).
Suggest they pre-plan how they will respond in this scenario. Note p. How will you respond when students wonder?

29. Fluency Levels Multiplication and Division
4/5/16
Level 1: Making and counting all of the quantities involved
Level 2: Repeated counting on by a given number
Level 3: Use the associative property/distributive property to compose and decompose
We also see a progression through fluency levels in multiplication and division;
If a student is reaching for the answer to 7x8, we might give the input 6x8=48. Some students will know how that is relevant and will help them compose 7x8=56. For those students, we might do some work specific to memorization. Other students may require intervention that addresses the meaning of multiplication and how multiplication works.
Secondary Number Talks 1.0

30. Learn Your Facts Timed Quiz Later!
Making Number Talks Matter, page 60
We can practice over and over to memorize but it’s hard to remember them all! And imagine if someone timed us to see how fast we could say them!
Connect to Faster Isn’t Smarter Article on Padlet
Example of 7th grade timed test and one student did poorly when timed facts and then completed puzzle with same facts untimed successfully. He knew his facts just couldn’t do the timed pressure… Perception is he isn’t good at math even though he is. Next he was put into a remediation group to learn his facts when in fact he knew them.
Alternate option – have students compete against themselves. Have student record beginning and end time trying to beat their own last time.
Overemphasizing fast fact recall at the expense of problem solving and conceptual experiences gives students a distorted idea of the nature of mathematics and of their ability to do math.

31. Embedded Rehearsals of Facts
One of SCOE colleagues facilitated the 7x7 number Talk in an upper elementary class. These charts show the results.
Note again the multiple “rehearsals” of facts embedded throughout…for example, students have contact with 7x2=14 embedded in almost every prompt within the string.
Contrast this with flash cards and timed tests which address facts as discrete and disconnected.
Note how visual models (array, number bond, tape diagram, open number line) support students understanding number relationships. In particular, the array supports students’ understanding that the area of a rectangle is the product of its dimensions.

32. Which is Greater?
Number String – different number talks that connect to each other
Engage participants in this number talk; have them think/write/pair share; model recording and questioning strategies
Mental computation is a key component of number talks because it encourages students to build on number relationships to solve problems instead of relying on memorized procedures.
The need for students to have a sense of the value of fractions in relation to whole numbers.
The fraction procedures we often emphasize have students operating separately on numerators and denominators, so they see a fraction as two separate numbers.
Fourth prompt: always, sometimes, or never? “a” might be….?
A study was done where students were asked to place the number 6/7 on a number line. The majority of students, whether “low achievers” or “high achievers” place it between 6 and 7 on the number line.

33. Geometry Number Talk How do you see this growing?
4/5/16
Geometry Number Talk
How do you see this growing?
Time 0
Time 1
Time 2
Time 3
***Video is hyperlinked to Geometry Number Talk – Have poster with problem for participants to see. Play to 4:30, 5:30-7:30, 10 on… Skip over student work time.
Have participants reflect on ways to grow from this number talk – constructive feedback
What will it look like at Time 10? How do you know?
Secondary Number Talks 1.0

34. Ways to Develop Accountability with Students
4/5/16
Ways to Develop Accountability with Students
Ask students to use finger signals to indicate the most efficient strategy.
Keep records of problems posed and the corresponding student strategies.
Hold small-group number talks throughout each week.
Create and post class strategy charts.
Require students to solve an exit problem using the discussed strategies.
Give a weekly computation assessment.
Talking points:
Finger signals for students to select their most efficient strategy (strategy 1, 2, 3). Students who don’t give the signal for “I agree” could be taken to have a different idea or strategy.
Use a roster and tally marks to keep track of which students are contributing. This will enable you to target non-contributors, whether by circulating more closely during pair-share or inviting them to share their thinking to the class.
Another way to target non-participants.
May be difficult in a secondary setting
Some ideas: Ask students to record a strategy they wish they had used. Ask students to use a particular strategy on a new problem.
Khan Academy, for example. Not recommended to include this in the students’ grading, but as an effective way to provide differentiation and targeted feedback. Be ready to “park” this if needed.
***Video is hyperlinked to Geometry Number Talk – Have poster with problem for participants to see. Play to 4:30, 5:30-7:30, 10 on… Skip over student work time.
Have participants reflect on ways to grow from this number talk – constructive feedback
See for research supporting random selection
Secondary Number Talks 1.0

35. 4/5/16
Number Talk vs. Lesson
Understanding how numbers work, rather than learning various skills.
Empowers students to examine problems in their own way. (Life–long Learner)
Short term practice toward long term goals.
Increased difficulty levels - encourages students to find more efficient ways to solve problems.
Never expect students to see the problem the “teacher’s” way.
Not predictable.
Don’t replace current curriculum or lesson; only minutes of each day.
Connect to article
Students make sense of numbers and how they work...
Secondary Number Talks 1.0

36. Arithmetic in High School?
“Students ability to reason with numbers is the bedrock of their understanding of algebra and therefore is in everyone’s curriculum.”
A lot of rational number arithmetic that arises during algebra, geometry, and beyond.
Anytime numbers are involved in a formula, expression, or equation = opportunity for a number talk
Geometry has many ways to approach problems – volume, perimeter, area, angle measures
Making Number Talks Matter Humphreys/Parker

37. Starting with 5 Small Steps
4/5/16
Starting with 5 Small Steps
Start with smaller problems to elicit thinking from multiple perspectives.
Be prepared to offer a strategy from a previous student.
It is all right to put a student’s strategy on the back burner.
As a rule, limit yourself to 5-15 minutes.
Be patient with yourself and your students as you incorporate number talks.
Students must learn the routine through practice
Start with smaller problems
Be prepared to offer a strategy from a previous students
It is all right to put a student’s strategy on the back burner
As a rule, limit yourself to 5-15 minutes
Be patient with yourself and your students as you incorporate number talks
Refer to routines chart paper and graphic organizer
Six Ways to Develop Accountability with Students p. 25 1) finger signals, 2) keep records of strategies, 3) Hold small group number talks, 4) Create and post class strategy charts, 5) Exit problem, 6) Weekly computation assessment
Secondary Number Talks 1.0

38. 4/5/16
Feedback
How are Number Talks connected to the 4 C’s (Communication, Collaboration, Critical Thinking, Creativity)?
Poll Everywhere
Secondary Number Talks 1.0

39. 4/5/16
Contact Info & Resources
Christen Schwartz –
Number Talks Padlets
Padlet image is a hyperlink to the K-5 padlet.
– some resources are the same.
On the padlets are many resources for teachers to use in support of their number talks.
Secondary Number Talks 1.0