Pinwheel Discussion
Basic Structure: Students are divided into 4 groups. Three of these groups are assigned to represent specific
points of view. Members of the fourth group are designated as “provocateurs,” tasked with making sure the
discussion keeps going and stays challenging. One person from each group (the “speaker”) sits in a desk facing
speakers from the other groups, so they form a square in the center of the room. Behind each speaker, the
remaining group members are seated: two right behind the speaker, then three behind them, and so on, forming
a kind of triangle. From above, this would look like a pinwheel. The four speakers introduce and
discuss questions they prepared ahead of time (this preparation is done with their groups). After some time
passes, new students rotate from the seats behind the speaker into the center seats and continue the
conversation.
Variations: When high school English teacher Sarah Brown Wessling introduced this strategy in the featured
video (click Pinwheel Discussion above), she used it as a device for talking about literature, where each group
represented a different author, plus one provocateur group. But in the comments that follow the video, Wessling
adds that she also uses the strategy with non-fiction, where students represent authors of different non-fiction
texts or are assigned to take on different perspectives about an issue.
Socratic Seminar
a.k.a. Socratic Circles
Basic Structure: Students prepare by reading a text or group of texts and writing some higher-order discussion
questions about the text. On seminar day, students sit in a circle and an introductory, open-ended question is
posed by the teacher or student discussion leader. From there, students continue the conversation, prompting
one another to support their claims with textual evidence. There is no particular order to how students speak, but
they are encouraged to respectfully share the floor with others. Discussion is meant to happen naturally and
students do not need to raise their hands to speak. This overview of Socratic Seminar from the website Facing
History and Ourselves provides a list of appropriate questions, plus more information about how to prepare for
a seminar.
Variations: If students are beginners, the teacher may write the discussion questions, or the question creation
can be a joint effort. For larger classes, teachers may need to set up seminars in more of a fishbowl-like
arrangement, dividing students into one inner circle that will participate in the discussion, and one outer circle
that silently observes, takes notes, and may eventually trade places with those in the inner circle, sometimes all
at once, and sometimes by “tapping in” as the urge strikes them.
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Low-Prep Discussion Strategies
Affinity Mapping
a.k.a. Affinity Diagramming
Basic Structure: Give students a broad question or problem that is likely to result in lots of different ideas,
such as “What were the impacts of the Great Depresssion?” or “What literary works should every person read?”
Have students generate responses by writing ideas on post-it notes (one idea per note) and placing them in no
particular arrangement on a wall, whiteboard, or chart paper. Once lots of ideas have been generated, have
students begin grouping them into similar categories, then label the categories and discuss why the ideas fit
within them, how the categories relate to one another, and so on.
Variations: Some teachers have students do much of this exercise—recording their ideas and arranging them
into categories—without talking at first. In other variations, participants are asked to re-combine the ideas
into new, different categories after the first round of organization occurs. Often, this activity serves as a good
pre-writing exercise, after which students will write some kind of analysis or position paper.
Concentric Circles
a.k.a. Speed Dating
Basic Structure: Students form two circles, one inside circle and one outside circle. Each student on the inside
is paired with a student on the outside; they face each other. The teacher poses a question to the whole group
and pairs discuss their responses with each other. Then the teacher signals students to rotate: Students on the
outside circle move one space to the right so they are standing in front of a new person (or sitting, as they are in
the video). Now the teacher poses a new question, and the process is repeated.
Variations: Instead of two circles, students could also form two straight lines facing one another. Instead of
“rotating” to switch partners, one line just slides over one spot, and the leftover person on the end comes around
to the beginning of the line. Some teachers use this strategy to have students teach one piece of content to their
fellow students, making it less of a discussion strategy and more of a peer teaching format. In fact, many of
these protocols could be used for peer teaching as well.
Conver-Stations
Basic Structure: Another great idea from Sarah Brown Wessling, this is a small-group discussion strategy
that gives students exposure to more of their peers’ ideas and prevents the stagnation that can happen when a
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group doesn’t happen to have the right chemistry. Students are placed into a few groups of 4-6 students each
and are given a discussion question to talk about. After sufficient time has passed for the discussion to develop,
one or two students from each group rotate to a different group, while the other group members remain where
they are. Once in their new group, they will discuss a different, but related question, and they may also share
some of the key points from their last group’s conversation. For the next rotation, students who have not rotated
before may be chosen to move, resulting in groups that are continually evolving.
Fishbowl
Basic Structure: Two students sit facing each other in the center of the room; the remaining students sit in a
circle around them. The two central students have a conversation based on a pre-determined topic and often
using specific skills the class is practicing (such as asking follow-up questions, paraphrasing, or elaborating on
another person’s point). Students on the outside observe, take notes, or perform some other discussion-related
task assigned by the teacher.
Variations: One variation of this strategy allows students in the outer circle to trade places with those in the
fishbowl, doing kind of a relay-style discussion, or they may periodically “coach” the fishbowl talkers from the
sidelines. Teachers may also opt to have students in the outside circle grade the participants’ conversation with
a rubric, then give feedback on what they saw in a debriefing afterward, as mentioned in the featured video.
Hot Seat
Basic Structure: One student assumes the role of a book character, significant figure in history, or concept
(such as a tornado, an animal, or the Titanic). Sitting in front of the rest of the class, the student responds to
classmates’ questions while staying in character in that role.
Variations: Give more students the opportunity to be in the hot seat while increasing
everyone’s participation by having students do hot seat discussions in small groups, where one person per group
acts as the “character” and three or four others ask them questions. In another variation, several students could
form a panel of different characters, taking questions from the class all together and interacting with one another
like guests on a TV talk show.
Snowball Discussion
a.k.a. Pyramid Discussion
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Basic Structure: Students begin in pairs, responding to a discussion question only with a single partner. After
each person has had a chance to share their ideas, the pair joins another pair, creating a group of four. Pairs
share their ideas with the pair they just joined. Next, groups of four join together to form groups of eight, and so
on, until the whole class is joined up in one large discussion.
Variations: This structure could simply be used to share ideas on a topic, or students could be required to reach
consensus every time they join up with a new group.
Ongoing Discussion Strategies
Whereas the other formats in this list have a distinct shape—specific activities you do with students—the
strategies in this section are more like plug-ins, working discussion into other instructional activities and
improving the quality and reach of existing conversations.
Asynchronous Voice
One of the limitations of discussion is that rich, face-to-face conversations can only happen when all parties are
available, so we’re limited to the time we have in class. With a tool like Voxer, those limitations disappear. Like
a private voice mailbox that you set up with just one person or a group (but SOOOO much easier), Voxer
allows users to have conversations at whatever time is most convenient for each participant. So a group of four
students can “discuss” a topic from 3pm until bedtime—asynchronously—each member contributing whenever
they have a moment, and if the teacher makes herself part of the group, she can listen in, offer feedback, or
contribute her own discussion points. Voxer is also invaluable for collaborating on projects and for having one-
on-one discussions with students, parents, and your own colleagues. Like many other educators, Peter DeWitt
took a while to really understand the potential of Voxer, but in this EdWeek piece, he explains what turned him
around.
Backchannel Discussions
A backchannel is a conversation that happens right alongside another activity. The first time I saw a
backchannel in action was at my first unconference: While those of us in the audience listened to presenters and
watched a few short video clips, a separate screen was up beside the main screen, projecting something
called TodaysMeet. It looked a lot like those chat rooms from back in the day, basically a blank screen where
people would contribute a few lines of text, the lines stacking up one after the other, no other bells or whistles.
Anyone in the room could participate in this conversation on their phone, laptop, or tablet, asking questions,
offering commentary, and sharing links to related resources without ever interrupting the flow of the
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presentations. This kind of tool allows for a completely silent discussion, one that doesn’t have to move at a
super-fast pace, and it gives students who may be reluctant to speak up or who process their thoughts more
slowly a chance to fully contribute. For a deeper discussion of how this kind of tool can be used, read this
thoughtful overview of using backchannel discussions in the classroom by Edutopia’s Beth Holland.
Talk Moves
a.k.a. Accountable Talk
Talk moves are sentence frames we supply to our students that help them express ideas and interact with one
another in respectful, academically appropriate ways. From kindergarten all the way through college, students
can benefit from explicit instruction in the skills of summarizing another person’s argument before presenting
an alternate view, asking clarifying questions, and expressing agreement or partial agreement with the stance of
another participant. Talk moves can be incorporated into any of the other discussion formats listed here.
Teach-OK
Whole Brain Teaching is a set of teaching and classroom management methods that has grown in popularity
over the past 10 years. One of WBT’s foundational techniques is Teach-OK, a peer teaching strategy that begins
with the teacher spending a few minutes introducing a concept to the class. Next, the teacher says Teach!, the
class responds with Okay!, and pairs of students take turns re-teaching the concept to each other. It’s a bit like
think-pair-share, but it’s faster-paced, it focuses more on re-teaching than general sharing, and students are
encouraged to use gestures to animate their discussion. Although WBT is most popular in elementary schools,
this featured video shows the creator of WBT, Chris Biffle, using it quite successfully with college students. I
have also used Teach-OK with college students, and most of my students said they were happy for a change
from the sit-and-listen they were used to in college classrooms.
Think-Pair-Share
An oldie but a goodie, think-pair-share can be used any time you want to plug interactivity into a lesson: Simply
have students think about their response to a question, form a pair with another person, discuss their response,
then share it with the larger group. Because I feel this strategy has so many uses and can be way more powerful
than we give it credit for, I devoted a whole post to think-pair-share; everything you need to know about it is
right there.
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Language Strategies for Active Classroom Participation
Kate Kinsella, San Francisco State University 10-2001
Expressing an Opinion Predicting
I think/believe that…
It seems to me that . . . I guess/predict/imagine that . . .
In my opinion . . .
Based on , I infer that . . .
I hypothesize that . . .
Asking for Clarification Paraphrasing
What do you mean? So you are saying that . . .
Will you explain that again? In other words, you think . . .
I have a question about that. What I hear you saying is . . .
Soliciting a Response Acknowledging Ideas
What do you think? My idea is similar to/related to
We haven’t heard from you yet.
Do you agree? ’s idea.
What answer did you get? I agree with (a person) that . . .
My idea builds upon ’s idea
Individual Reporting
Partner and Group Reporting
I discovered from that . . . We decided/agreed that . . .
We concluded that . . .
I found out from that . . . Our group sees it differently.
We had a different approach.
pointed out to me that . . .
Offering a Suggestion
shared with me that . . . Maybe we could . . .
What if we . . .
Disagreeing Here’s something we might try.
I don’t agree with you because . .
I got a different answer than you. Holding the Floor
I see it another way. As I was saying, . . .
If I could finish my thought . . .
Affirming What I was trying to say was . . .
That’s an interesting idea.
I hadn’t thought of that.
I see what you mean.
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Scaffolding Complex Text
• Don’t keep secrets – tell the students what they will be reading ahead of time to activate
their prior knowledge. You can share the content, the structure, anything that will help
them start reading with some knowledge of the text.
• Pre-teach the most important vocabulary. Write it on the board and work together to
define it ahead of time. Have students find it in the text before starting to read.
• Use graphic organizers to guide students through the reading. Some samples follow this
page.
• Choose just the most important parts of the text to read. Maybe it’s topic sentences, the
intro/conclusion or a few middle paragraphs. If they will struggle to read it, don’t make
them struggle through the whole thing.
• Read aloud to them. It is perfectly valid for you to do the reading, provided they are
working on answering questions, taking notes or talking to the text, etc…while you read.
It is helpful for them to hear a fluent reader read aloud once in a while.
• Always give students a purpose for reading and something to do with the reading once
it’s over. Why are they reading this? Do they need to answer questions, prove a
statement true/false, gather evidence to support a position, talk to the text or ask questions
about the reading?
• Make sure students understand what they are reading for. If there will be questions to
answer, review those questions ahead of time and give students access to them to help
focus their reading.
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Reading for Meaning Graphic Organizer
Evidence For (True) Statement Evidence Against (False)
*You create true or false statements from the text and fill them into the “statements” column
ahead of time. While reading, students then find evidence either supporting or refuting the
statement write that evidence in the appropriate column.
Say, Mean, Matter Graphic Organizer
SAY MEAN MATTER
What does the text say? What does this mean? Why does this matter?
*Students fill this out while they read. They record important quotes under “say”, put it in their
own words under “mean” and write why this quote is important under “matter”.
*You can pre-fill the “say” column if there are specific points you want to make sure they
understand.
Before, During, After Reading Questions
• Develop “before reading” questions for students to answer before they read that will
activate their background knowledge and get them thinking about ideas in the text
• Develop “during reading” questions for students to answer while reading. These
questions should help their comprehension and keep them on track with their reading.
• Develop “after reading” questions for students to answer when they finish. These should
move beyond comprehension and require students to think critically, form an opinion
about what they read, ask questions and/or analyze parts of the text.
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Something Old, Something New Graphic Organizer
*This is best used if you want students to just grasp the gist of an article but they don’t need to
understand details. A reproducible copy follows.
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Name
Date
Period
Something Old, Something New
One sentence final reaction:
Something New I Learned . Something Old I Already Knew
.
.
.
.
.
.
.
.
.
Summary of What I Learned
------------------------------ ----------------------------
Something Surprising Something Confusing
.
.
.
.
.
.
.
.
.
.
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58
K-W-L Chart
Name Date
Passage
Read the title of the passage. Write information that you already know in the “K”
part. Write questions that you would like to find the answers to from the passage
in the “W” part. Read the passage. Write answers to the questions and new
information that you found out from the passage in the “L” part.
What I learned
What I want to
know
What I know
Ready-to-Use Nonfiction Graphic Organizers with Before-During-After Activities – Grades 2 -5 © 2010 readingwarmupsandmore.com
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Graphic Organizer for Essay Preparation
*You may add to this page as much as you wish. You will probably need to make another copy of this as a blank sheet once you fill
up the available space or you may also create your own graphic organizer to continue working on.
QUOTE FROM BOOK MY REFLECTION PAGE #
Keep track of what you read that you Why are you marking this part? What Make sure you cite the page number so
might be able to refer to when writing are your thoughts about it? you can return to this part later.
your final essay!
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LDC Task Templates for Grades 6–12
Cognitive ARGUMENTATION INFORMATIONAL/EXPLANATORY
Demand
Task IE1: [Insert optional question] After
1
Definition (reading/researching) (literary text/s and/or
informational text/s on content), write
(product) in which you define (concept or term)
and explain (content). Support your discussion
with evidence from the text/s.
Task IE2: [Insert optional question] After
2 (reading/researching) (literary text/s and/or
Description
informational text/s on content), write
(product) in which you describe (content).
Support your discussion with evidence from the text/s.
Task IE3: [Insert optional question] After
3 (reading/researching) (literary text/s and/or
Explanation
informational text/s on content), write
(product) in which you explain (content).
Support your discussion with evidence from the texts.
Task A4: [Insert optional question] Task IE4: [Insert optional question] After
After (reading/researching) (reading/researching) (literary text/s and/or
4 (literary text/s and/or informational informational text/s on content), write
Analysis
text/s on content), write (product) in which you analyze (content).
5
Comparison (product) in which you argue Support your discussion with evidence from the text/s.
6 (content). Support your position with
Cause-
Effect evidence from the text/s.
Task A5: [Insert optional question] Task IE5: [Insert optional question] After
After (reading/researching) (reading/researching) (literary text/s and/or
(literary and/or informational text/s on informational text/s on content), write
content), write (product) (product) in which you compare (content).
in which you compare (content) Support your discussion with evidence from the text/s.
and argue (content). Support
your position with evidence from the
text/s.
Task A6: [Insert optional question] Task IE6: [Insert optional question] After
After (reading/researching) (reading/researching) (literary text/s and/or
(literary text/s and/or informational informational text/s on content), write
text/s on content), write (product) in which you examine cause/s of
(product) in which you argue the (content) and explain the effect/s (content).
cause/s of (content) and explain Support your discussion with evidence from the text/s.
the effect/s (content). Support
your position with evidence from the
text/s.
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Task IE7: [Insert optional question] After
7 (reading/researching) (literary text/s and/or
Procedural- informational text/s on content), write
Sequential
(product) in which you relate how (content).
Support your discussion with evidence from the text/s.
Task IE8: [Insert optional question] After
(reading/researching) (informational text/s on
8 content), developing a hypothesis, and
Hypothesis- conducting an experiment examining (content),
Experiment
write a laboratory report in which you explain your
procedures and results and confirm or reject your
hypothesis.
Task A9: [Insert optional question]
After (reading/researching)
(literary text/s and/or informational
9 text/s on content), write
Evaluation
(product) in which you discuss
10
Problem- (content) and evaluate (content).
Solution
Support your position with evidence
from the text/s.
Task A10: [Insert optional question]
After (reading/researching)
(literary text/s and/or informational
text/s on content), write
(product) in which you identify a
problem (content) and propose a
solution. Support your position with
evidence from the text/s.
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Optional Demands for Adding Rigor to Teaching Tasks
Demands may be added to a teaching task to increase its rigor. You might choose to add either a single demand
or multiple demands depending on your students’ needs, grade level standards, or content. Additional demands
can also be used as a mechanism to provide additional differentiation for individuals or groups with similar
instructional needs.
D1 Be sure to acknowledge competing views. (Use with Argumentation tasks.)
D2 Give examples from past or current (events; issues) to illustrate and clarify your position.
D3 What (lesson/s, conclusion/s, implication/s) can you draw about (content)?
D4 In your discussion, address the credibility and origin of sources.
D5 Identify any gaps or unanswered questions.
D6 Include (bibliography, citations, references, endnotes).
D7 Include (charts, tables, illustrations, and/or stylistic devices) to help convey your message to your
readers.
D8 Explain how (key detail/s, historical events, scientific ideas or concepts, or steps in a technical
procedure) and (key detail/s, historical events, scientific ideas or concepts, or steps in a technical
procedure) are (connected or related).
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Informational/Explanatory Rubric for Grades 6–12 Teaching Tasks (Literacy Design Collaborative)
Scoring Not Yet Approaches Expectations Meets Expectations 3.5 Advanced
Elements 1 1.5 2 2.5 3 4
Focus Attempts to address Addresses prompt Addresses prompt Addresses all aspects of
prompt but lacks appropriately but with appropriately and prompt appropriately and
focus or is off task. maintains a clear, steady
D: Attempts to a weak or uneven maintains a strongly
address additional focus. focus. developed focus.
demands but lacks
focus or is off task. D: Addresses D: Addresses additional D: Addresses additional
additional demands demands sufficiently. demands with thoroughness
and makes a connection to
superficially.
controlling idea.
Controlling Attempts to establish Establishes a Establishes a controlling Establishes a strong
Idea a controlling idea but controlling idea with a idea with a clear purpose controlling idea with a clear
lacks a clear purpose. maintained throughout the
general purpose. purpose maintained
response. throughout the response.
Reading/ Attempts to present Presents information Presents information from Accurately presents
Research information in from reading materials reading materials relevant information relevant to all
response to the relevant to the purpose parts of the prompt with
prompt but lacks to the prompt with effective selection of sources
connections or of the prompt with accuracy and sufficient and details from reading
relevance to the minor lapses in
purpose of the accuracy or detail. materials.
prompt. completeness.
Development Attempts to provide Presents appropriate Presents appropriate and Presents thorough and
details in response to details to support the sufficient details to detailed information to
the prompt, including focus and controlling support the focus and strongly support the focus
controlling idea. and controlling idea.
retelling but lacks idea.
sufficient
development or
relevancy.
Organization Attempts to organize Uses an appropriate Maintains an appropriate Maintains an organizational
ideas but lacks control organizational organizational structure to structure that intentionally
and effectively enhances the
of structure. structure to address the address the specific presentation of information
specific requirements requirements of the as required by the specific
of the prompt, with
prompt. prompt.
some lapses in
coherence or awkward
use of the
organizational
structure
Conventions Attempts to Demonstrates an Demonstrates a command Demonstrates and maintains
demonstrate standard uneven command of of standard English a well-developed command
English conventions
standard English conventions and cohesion of standard English
but lacks cohesion conventions and with few errors. Response conventions and cohesion
and control of cohesion. Uses with few errors. Response
language and tone with includes language and includes language and tone
grammar, usage, and some inaccurate, tone appropriate to the consistently appropriate to
mechanics. Sources inappropriate, or audience, purpose, and the audience, purpose, and
uneven features. specific requirements of specific requirements of the
are used without Inconsistently cites the prompt. Cites sources prompt. Consistently cites
citation. using an appropriate sources using an appropriate
sources. format with only minor
format.
errors.
Content Attempts to include Briefly notes Accurately presents Integrates relevant and
Understanding disciplinary content in disciplinary content disciplinary content accurate disciplinary content
relevant to the prompt; relevant to the prompt with thorough explanations
explanations but shows basic or uneven
understanding of content with sufficient that demonstrate in-depth
understanding of explanations that understanding.
is weak; content is content; minor errors
irrelevant, inappropriate, demonstrate
in explanation. understanding.
or inaccurate.
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Argumentation Rubric for Grades 6–12 Teaching Tasks (from the Literacy Design Collaborative)
Scoring Elements Not Yet Approaches Expectations Meets Expectations 3.5 Advanced
1 1.5 2 2.5 3 4
Focus Attempts to address Addresses prompt Addresses prompt Addresses all aspects of
prompt but lacks appropriately and appropriately and prompt appropriately
focus or is off task. establishes a position maintains a clear, steady
D: Attempts to but focus is uneven. focus. Provides a with a consistently strong
address additional generally convincing focus and convincing
demands but lacks D: Addresses position.
focus or is off task. additional demands position.
D: Addresses additional
superficially. D: Addresses additional demands with
demands sufficiently
thoroughness and makes
a connection to claim.
Controlling Idea Attempts to establish Establishes a claim. Establishes a credible Establishes and
a claim but lacks a claim. maintains a substantive
clear purpose. and credible claim or
proposal.
Reading/ Research Attempts to Presents information Accurately presents Accurately and
reference reading from reading details from reading effectively presents
materials to develop materials relevant to the important details from
response but lacks materials relevant to purpose of the prompt to reading materials to
the purpose of the develop argument or develop argument or
connections or prompt with minor
relevance to the lapses in accuracy or claim. claim.
purpose of the
completeness.
Development Attempts to provide Presents appropriate Presents appropriate and Presents thorough and
details in response to details to support and sufficient details to detailed information to
the prompt but lacks effectively support and
develop the focus, support and develop the
sufficient controlling idea, or focus, controlling idea, develop the focus,
development or claim, with minor controlling idea, or
relevance to the or claim.
purpose of the lapses in the claim.
reasoning, examples,
prompt.
or explanations.
Organization Attempts to organize Uses an appropriate Maintains an Maintains an
ideas but lacks organizational appropriate organizational structure
structure for organizational structure that intentionally and
control of structure. development of to address specific effectively enhances the
requirements of the
reasoning and logic, prompt. Structure presentation of
with minor lapses in reveals the reasoning information as required
and logic of the by the specific prompt.
structure and/or argument.
coherence. Structure enhances
development of the
reasoning and logic of
the argument.
Conventions Attempts to Demonstrates an Demonstrates a Demonstrates and
demonstrate standard uneven command of command of standard maintains a well-
English conventions English conventions and developed command of
standard English standard English
but lacks cohesion conventions and cohesion with few conventions and
and control of errors. Response cohesion with few errors.
cohesion. includes language and Response includes
grammar, usage, tone appropriate to the language and tone
mechanics, language Uses language and audience, purpose, and consistently appropriate
and tone. Sources are tone with some specific requirements of to the audience, purpose,
used without citation. inaccurate, the prompt. Cites and specific requirements
inappropriate, or
uneven features. sources using of the prompt.
appropriate format with Consistently cites
Inconsistently cites sources using appropriate
sources. only minor errors.
format.
Content Attempts to include Briefly notes Accurately presents Integrates relevant and
Understanding disciplinary content disciplinary content disciplinary content accurate disciplinary
relevant to the prompt content with thorough
in argument but relevant to the
understanding of prompt; shows basic with sufficient explanations that
content is weak; explanations that demonstrate in-depth
content is irrelevant, or uneven
inappropriate, or understanding of demonstrate understanding.
content; minor errors understanding.
inaccurate. in explanation.
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Positive Norms to Encourage in
Math Class By Jo Boaler
1. Everyone Can Learn Math to the Highest Levels.
Encourage students to believe in themselves. There is no such thing as a
“math” person. Everyone can reach the highest levels they want to, with hard
work.
2. Mistakes are Valuable
Mistakes grow your brain! It is good to struggle
and make mistakes.
3. Questions are Really Important
Always ask questions, always answer ques-
tions. Ask yourself: why does that make sense?
4. Math is about Creativity and Making Sense
Math is a very creative subject that is, at its core, about visualizing patterns
and creating solution paths that others can see, discuss and critique.
5. Math is about Connections and Communicating
Math is a connected subject, and a form of communication. Represent math
in different forms eg words, a picture, a graph, an equation, and link them.
Color code!
6. Depth is much more Important than Speed
Top mathematicians, such as Laurent Schwartz, think slowly and deeply.
7. Math Class is about Learning not Performing
Math is a growth subject, it takes time to learn and it is all about effort.
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Recent Research on Best Practices for Teaching Math
The following research has been proven by neuroscience and is slowly taking the math world by storm!
1. Everyone can be good at math.
“The results showed that the people who worked on an exercise for a few minutes each day experienced
structural brain changes. The participants’ brains were “rewired” and grew in response to a 10-minute
mental task performed for just 15 days over three weeks. Such results should prompt educators to
abandon the traditional fixed ideas of the brain and learning that currently fill schools—ideas that
children are smart or dumb, quick or slow… If brains can change in three weeks, imagine what can
happen in a year of math class if students are given the right math materials and receive positive
messages about their potential and ability.”
Read more: https://www.youcubed.org/think-it-up/anyone-can-learn-to-high-levels/
2. Counting on your fingers is helpful for developing number sense and should be
encouraged.
“Evidence from both behavioral and neuroscience studies shows that when people receive training on
ways to perceive and represent their own fingers, they get better at doing so, which leads to higher
mathematics achievement. The tasks we have developed for use in schools and homes (see below) are
based on the training programs researchers use to improve finger-perception quality... Neuroscientists
often debate why finger knowledge predicts math achievement, but they clearly agree on one thing: That
knowledge is critical. As Brian Butterworth, a leading researcher in this area, has written, if students
aren’t learning about numbers through thinking about their fingers, numbers “will never have a normal
representation in the brain.”
Read more: http://www.theatlantic.com/education/archive/2016/04/why-kids-should-use-their-fingers-
in-math-class/478053/
3. Speed and time pressure block working memory and contribute to student aversion
to math.
“For about one-third of students, the onset of timed testing is the beginning of math anxiety…
Sian Beilock and her colleagues studied people’s brains through MRI imaging and found that math
facts are held in the working memory section of the brain… But when students are stressed, such
as when they are taking math questions under time pressure, the working memory becomes
blocked, and students cannot access math facts they know… As students realize they cannot
perform well on timed tests, they start to develop anxiety and their mathematical confidence wears
away. The blocking of the working memory and associated anxiety is particularly common among
higher-achieving students and girls... Conservative estimates suggest that at least a third of
students experience extreme stress related to timed tests, and these are not students from any
particular achievement group or economic background. When we put students through this
anxiety-provoking experience, they distance themselves from mathematics.”
Read more: https://www.youcubed.org/think-it-up/speed-time-pressure-block-working-memory/ and
http://alturl.com/95eqm
113
4. Number sense and math facts can and should be taught without the pressure of
memorization and timed tests.
Some strategies to teach deep understanding of math facts include number talks, math apps and games
and using math cards with multiple representations of numbers and facts instead of simple equations and
numerical answers.
Read more: https://www.youcubed.org/fluency-without-fear/
5. Visual math improves math performance.
“In a ground breaking new study [researchers] found that the most powerful learning occurs when
we use different areas of the brain. When students work with symbols, such as numbers, they are
using a different area of the brain than when they work with visual and spatial information, such
as an array of dots. The researchers found that mathematics learning and performance was
optimized when the two areas of the brain were communicating…(for math questions that
encourage this use of visual and symbolic representations see https://www.youcubed.org/tasks/).
Additionally, they found that training students through visual representations improved students’ math
performance significantly, even on numerical math, and that the visual training helped
students more than numerical training.
Read more: https://www.youcubed.org/think-it-up/visual-math-improves-math-performance/
114
How Students Should be Taught Mathematics: Reflections from Research and Practice
by Jo Boaler, Professor of Mathematics Education, Stanford University
Mathematics classrooms should be places where students:
Develop an inquiry relationship with mathematics, approaching math with curiosity, courage, confidence
& intuition.
Talk to each other and the teachers about ideas – Why did I choose this method? Does it work with other
cases? How is the method similar or different to methods other people used?
Work on mathematics tasks that can be solved in different ways and/or with different solutions.
Work on mathematics tasks with a low entry point but a very high ceiling – so that students are
constantly challenged and working at the highest and most appropriate level for them.
Work on mathematics tasks that are complex, involve more than one method or area of mathematics, and
that often, but not always, represent real world problems and applications.
Are given growth mindset messages at all times, through the ways they are grouped together, the tasks
they work on, the messages they hear, and the assessment and grading.
Are assessed formatively – to inform learning – not summatively to give a rank with their peers. Students
should regularly receive diagnostic feedback on their work, instead of grades or scores. Summative
assessments are best used at the end of courses.
Mathematics classrooms should be places where students believe:
Everyone can do well in math.
Mathematics problems can be solved with many different insights and methods.
Mistakes are valuable, they encourage brain growth and learning.
Mathematics will help them in their lives, not because they will see the same types of problems in the real
world but because they are learning to think quantitatively and abstractly and developing in inquiry
relationship with math.
115
JMCS Best Practices and Strat
8 Mathematical What It Means JMCS Strateg
Practices
Students understand Give students toug
Make sense of the problem, find a and let them work t
problems and way to attack it, and them. Allow wait tim
persevere in work until it is done. yourself and your
solving them. They continually ask students. Work for
themselves “Does this progress and “aha”
make sense?” moments. The mat
Basically, you will find becomes about the
practice standard #1 in process and not ab
every math problem, one right answer. L
every day. The with questions, but
hardest part is pushing pick up a pencil. Ha
students to solve students make hea
tough problems by in the task themsel
applying what they
already know and to
monitor themselves
when problem-solving.
11
tegies for Teaching CCSS Math
gies Practices Reflection Questions
gh tasks ● Write out math proofs. •Do the activities and
through Students show math discussions in my
sentences on the left classroom focus on
me for and written students’ thinking, how
justification on the and why a particular
” ● right. strategy was used instead
th Students check their of just getting the right
e answers. answer?
bout the
Lead •Do I use rich-tasks that
t don’t allow multiple entry points
ave and solutions or do I
adway provide more simple and
lves. direct problems?
•Do I focus on 1-2
problems for students to
solve and discuss
thoroughly or provide a
worksheet of problems to
solve?
•Do I focus on tricks, such
as key words, that focus
on memorization and can
often be unreliable or
conceptual
understanding?
16
Reason abstractly Get ready for the Have students draw
and quantitatively. words contextualize representations of
and decontextualize. If problems. Break ou
students have a manipulatives. Let
problem, they should students figure out
be able to break it to do with data
apart and show it themselves instead
symbolically, with boxing them into on
pictures, or in any way of organization. As
other than the questions that lead
standard algorithm. students to
Conversely, if students understanding. Hav
are working a students draw their
problem, they should thinking, with and w
be able to apply the traditional number
“math work” to the sentences.
situation.
Construct viable Be able to talk about Post mathematical
arguments and math, using vocabulary and ma
mathematical your students use i
critique the language, to support just in math class, e
reasoning of or oppose the work of Use "talk moves" to
others.
11
w Identify key information in •Do the activities and
word problems and discussions in my
ut the develop mathematical classroom focus on
expressions to solve. students’ thinking, how
what and why a particular
strategy was used instead
d of of just getting the right
ne type answer?
sk
d •Do I use rich-tasks that
allow multiple entry points
ve and solutions or do I
r provide more simple and
without direct problems?
•Do I focus on 1-2
problems for students to
solve and discuss
thoroughly or provide a
worksheet of problems to
solve?
•Do I focus on tricks, such
as key words, that focus
on memorization and can
often be unreliable or
conceptual
understanding?
l List/Group/Label, Gallery •Do I encourage students
ake Walk to explain and prove their
it — not solutions/answers or
either! accept memorized
o
17
others. encourage discours
Work on your class
environment from d
so that it is a safe p
discuss ideas.
Model with Use math to solve Math limited to mat
mathematics. real-world problems, is worthless. Have
organize data, and students use math
understand the world science, art, music
around you. even reading. Use
graphics, articles, a
data from the news
or other sources to
math relevant and
Have students crea
real-world problem
their mathematical
11
se. processes or formulas?
sroom
day one •Do I expect justifications
place to (why or why not?) or
simple yes/no answers?
th class
in •Do I model effective
c, and arguments and critiquing
real the reasoning of others?
and
spaper •Do I provide activities for
o make students to defend their
real. arguments using concrete
ate objects, diagrams,
ms using examples, definitions,
and/or data?
18
•Do I provide students
opportunities to share
their solutions with the
class?
•Do I give my students
ample opportunities to
model their mathematical
thinking?
•Do I discuss with my
students how the models
being used helped them
find solutions?
•Do I encourage my
students to use multiple
models to represent the
knowledge.
Use appropriate Students can select Don’t tell students w
tools strategically. the appropriate math tool to use. Try to l
tool to use and use it the decision open e
correctly to solve and then discuss w
problems. In the real worked best and w
world, no one tells you
that it is time to use
the meter stick instead
of the protractor.
11
what math?
leave
ended •Do I help students
what identify more efficient
why. models by discussing
when a specific model
19 might be more
appropriate?
•Do I model the use of
appropriate tools when
solving problems and
discuss why a particular
tool was selected with my
students?
•Do I provide a variety of
tools in which students
can explore and practice
using?
•Do I ask students which
tools might be helpful to
solve specific problems or
tasks?
•Do I engage my class in
discussions about
advantages and
limitations of specific
tools?
•Do I ensure my students
know how to use tools
correctly?
Attend to Students speak and Push students to u
precision. solve mathematics precise and exact
with exactness and language in math.
meticulousness. Measurements sho
exact, numbers sho
precise, and explan
must be detailed. O
change I’ve made i
allowing the phrase
don’t get it.” Studen
have to explain exa
what they do and d
understand and wh
their understanding
apart.
Look for and make Find patterns and Help students iden
use of structure. repeated reasoning multiple strategies
that can help solve then select the bes
more complex Repeatedly break
problems. For young numbers and prob
students this might be into different parts
recognizing fact what you know is tr
families, inverses, or solve a new proble
the distributive Prove solutions wit
property. As students relying on the algor
get older, they can For example, my st
break apart problems are changing mixed
and numbers into numbers into impro
familiar relationships. fractions. They hav
12
use Right is right (TLAC), no •Do I use appropriate
opt out (TLAC) grade-level math
vocabulary during
ould be instruction?
ould be
nations •Can my student use
One precise language when
is not communicating with
e, “I others?
nts
actly •How can I support my
do not students in understanding
here math vocabulary?
g falls
•Do I provide students
with opportunities to be
precise?
ntify Students write steps on •Do I ask my students
and left side of paper and regularly “What do you
st one. perform calculations on notice?”
right side.
apart •Do I provide lessons in
blems which patterns can
s. Use emerge?
rue to
em. •Do I provide
thout opportunities for my
rithm. students to work together
tudents and share their thinking?
d
oper
ve to
20
prove to me that th
have the right answ
without using the “s
Look for and Keep an eye on the I heard Greg Tang
express regularity big picture while a couple of years a
working out the details he gave some advi
in repeated of the problem. You think fits this standa
reasoning. don’t want kids that perfectly. He said t
can solve the one students how the p
problem you’ve given works. As soon as
them; you want “get it,” start makin
students who can generalize to a vari
generalize their problems. Don’t wo
thinking. of the same proble
your mathematical
reasoning and app
other situations.
**A Guide to the 8 Mathematical Standards:
http://www.scholastic.com/teachers/top-teaching/2013/03/guide-8-m
12
hey •Do I pose problems or
wer tasks that draw attention
steps.” to repetition?
speak •Do I encourage
ago and discoveries?
ice I
dard •Do I have students
to show explain shortcuts when
problem discovered in class?
they
ng them
iety of
ork fifty
em; take
ply it to
mathematical-practice-standards
21
SFUSD Signature Strategy #1: Three Read Protocol
What is this strategy?
The Three Read Protocol is a strategy that models how to do a close read of a complex math word problem or
task. This strategy includes three separate readings of a math scenario with specific goals. The 1st read is for
understanding the context. The 2nd read is for understanding the mathematics. The 3rd read is to elicit
inquiry questions based on the scenario.
Why would I use this strategy?
The Three Read Protocol is designed to engage students in sense-making of language-rich math problems or
tasks. It deepens student understanding by surfacing linguistic as well as mathematical clues. It focuses
attention on the importance of understanding problems rather than blindly trying to solve them. It is an
alternative to just simplifying the language. It also allows for natural differentiation within a class of diverse
learners.
When do I use this strategy?
This strategy can be used for many math tasks that include complex language structures and/or lend
themselves to a variety of interpretations. While this is a particularly useful strategy for EL students, all
students can benefit from the deeper understanding of word problem structures and open-ended
questioning.
How do I use this strategy?
The Three Read Protocol uses the “problem stem” of a word problem. This is essentially the word problem
without the question at the end. The purpose of only presenting the problem stem is to have student focus
on the contextual and mathematical information before dealing with any question that is involved. It also
allows for students to create their own questions for a given scenario, which is an excellent skill to develop
both in math and in reading in general. It is important that the teacher choose the problem carefully and
anticipates potential linguistic and mathematical roadblocks the students may encounter.
1. 1st Read: Teacher reads the problem stem orally.
The teacher may have visuals to accompany the oral read of the problem stem. Students are asked to
listen to the “story” with the goal of turning to a partner and retelling it. They do not have to memorize
the information. After the Turn-and-Talk, the teacher asks students to volunteer information they
remember from the story. Teachers and students ask clarifying questions about the vocabulary as
needed.
2. 2nd Read: Class does choral read or partner read of the problem stem.
The teacher projects the problem stem so the whole class can see it. The teacher leads the class either in
a choral read of the problem or has partners read the problem orally to each other. Choral read is
preferable because it allows all students to participate without excessive pressure, but partner reads
can work fine if that is a better fit to the classroom culture or age of students. The teacher explains that
math stories usually have information about quantities, which are numbers and the units that are being
counted. An example is 25 cats, where “25” is the quantity and “cat” is the unit. Sometimes the quantities
are implied. For example, “some cats” implies a quantity but we do not know what it is. There can also be
implied units. An example is “I have one at home.” The implied unit in this case depends entirely on the
context of the story. Bottom line: The discussion of quantities and units can be important for focusing
student attention, but whether the teacher delves deeply into the explicit and implicit information
depends entirely on the math problem and the needs of the students.
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3. 3rd Read: Class or partner read the problem stem orally one more time.
The teacher asks students to do one more read of the “story” and ask them to think “What is missing to
make this a good math problem?” Students volunteer their answers to that question. Responses will likely
vary because many students assume there is a question without actually reading one. Without correcting
student responses, the teacher probes until the class decides that a question is missing. The teacher asks
“Is there only one question that we can ask of this story?” Students responses may vary, but there are
usually many different questions that can be asked of almost any scenario. The teacher asks partners to
determine at least two questions that can be asked of the problem stem. After a few minutes, a couple of
students volunteer their questions. The teacher writes a couple of the questions and clarifies language
as appropriate. After each question, the teacher asks the class “Can this question be answered with the
information from this story?” and the class discusses why or why not.
4. Students work in collaborative groups on the problem.
The teacher can choose to either have students work in groups on a question they choose or a question
another student volunteered, or the teacher can pose his/her own question for the class to work on. If
groups are asked to choose their own questions, it is important that the teacher circulate and clarify
expectations around the work. Another strategy is to use one of the questions volunteered by a student
and have the whole class work on it. Finally, this can be an opportunity to differentiate the math work
because the range of possible questions to a problem stem is broad.
In summary:
What the teacher does What the students do
1st • Identifies appropriate problem stem • Sit with a partner
Read • Anticipateslinguisticand/or mathematical • Listen to the “story”
• Turn to partner to discuss the “story” in his/her
challenges own words
• Creates visuals to support understanding • Say what they remember of the story
• Orally reads the “story” (problem stem)
2nd • Has “story” available on overhead, projector, or poster • Read chorally with the class or with a partner
Read • Leads class in choral read or partner read • Volunteer quantities and units he/she identifies
• Leads discussion of quantities and units
3rd • Has partners read with specific goal in mind • Read one more time with partner
Read “What is missing to make this a math problem?” • Brainstorm with partner several questions that
• Leads discussion of potential questions could be asked of this scenario
• Clarifies language of the questions, as needed • Volunteer question for the problem stem
Considerations for use of the Three Read Protocol
1. Is the problem stem sufficiently interesting as a story?
○ It does not need to be long, but it should have a narrative purpose.
2. Does the problem stem have quantities, both explicit and implicit?
○ Ideally it has easily identifiable explicit quantities, but implicit ones as well for a richer
discussion and potentially more interesting math investigations.
3. Does the problem stem have extraneous explicit quantities or a variety of implicit quantities?
○ This strategy can model how to comprehend math problems with this characteristic and teach
students how to discern salient information.
4. Is the language of the problem stem likely to create obstacles for EL students or is the context of
the problem likely to be unfamiliar to students from diverse backgrounds?
○ Surfacing the language structure and contextual clues within the problem stem allows students
to focus on the mathematical structures and evaluate the reasonableness of their work.
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Sample Problem Stems
Animal Shelter Problem
Tasha wants a pet. She goes to the animal shelter to ask how much it will cost to adopt and care for a dog. The
vet at the shelter tells her that big dogs have an adoption fee of $200; a vaccination fee of $300; and they eat
about 35 pounds of food per month. Small dogs have an adoption fee of $300; a vaccination fee of $450; and
they eat about 18 pounds of food per month. The vet says that dog food costs about $3 per pound.
Judy’s Berries
Judy loves to eat berries for breakfast, lunch, and dinner. She sees that Clear Lake School is having a
fundraiser to raise money for a new playground. The students are selling fruit baskets to raise the money.
Strawberries sell for $3 per basket. Blueberries sell for $4 per basket. Raspberries sell for $5 per basket.
Judy has $20 to spend on berries.
Joining a Gym
Carlos wants to join a gym. The gym offers three membership options. The first one is called "Pay as you go"
and costs $6 each time you work out. The second one is called "Regular deal” and costs $50 per month and $2
each time you work out. The third one is called "All-in-one price!” and costs $100 per month for unlimited use
of the gym.
Squirrels and Their Acorns
Austin likes to watch squirrels find and store acorns for the winter. Brown Squirrels can carry two acorns at
a time. Gray Squirrels can carry three acorns at a time, and Black Squirrels can carry five acorns at a time.
There is a pile of 24 acorns.
Further Resources
SFUSD Mathematics Department website: www.sfusdmath.org/3-read-protocol.html
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Gallery Walk Activity Protocol
A “Gallery Walk” is an activity that allows students opportunities to discuss and display their work around a
room much like artists would display their artistic pieces in an exhibit. It is a non-threatening way for
students to receive feedback on their work, check their understanding, and see multiple solution paths for a
given math task. It is an ideal opportunity for students to see and discuss multiple ways of approaching and
representing math thinking (as described in the “Rule of Four”).
Here is one way of conducting a Gallery Walk in your classroom.
1. An assigned task/activity is worked on by a team of 3–6 students that form Home Base Teams.
2. When the teams finish their collaborative work poster, they each display their work around the
classroom. This can take the form of a poster, or it can be a simple as laying their work out on the tables.
3. Inform the teams that they will visit each one of the posters to generate a discussion about the
work/poster that each group did. There are (at least) two options to direct students’ observations and
thinking during the Gallery Walk itself.
a. Students may post comments and/or clarifying questions on their peers’ work with sticky notes.
Provide, or have students generate, clear guidelines to help groups write useful and appropriate
comments. These might include:
● Address the math, not the person.
● Disagree respectfully.
● Keep this a safe math space.
Give the groups specific prompts to respond to as they comment on and ask questions of each other’s
work. For example,
● What part of the math do you agree with? Why?
● What part of the math work do you disagree with? Why?
● What looks similar to what you did?
● What looks different from what you did?
● What part of the work really helps you understand what the people who made the poster
were thinking?
● What part of the work do you find confusing?
● What questions do you have for the authors of the poster that will help you understand their
thinking.
Or generate prompts that relate specifically to the math content.
b. Students may fill out a Gallery Walk Response Sheet instead of posting their thinking directly on
their classmates’ work. In this case, you may choose to direct students to pick one or more posters to
focus their thinking on. Questions might include all of those above plus:
● What are some mathematical characteristics of the work that fit...? (Insert an aspect of the
math content that you are focusing on.)
● What do you notice that is similar among all or most of the posters?
● What do you notice that is different on all or most of the posters?
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4. Once each team has visited each of the posters, they return to their poster and review the comments left
by the other groups or discuss their Gallery Walk Response Sheets. If there is time, and after considering the
work that they saw on others’ posters and/or the comments and questions that were left, they may modify
their work on the poster.
5. Collect any clarifying questions that were posted on the posters with sticky notes or on the Response
Sheets and begin a whole group discussion.
6. Each clarifying question is directed to the collaborative group that generated and/or created the poster.
To facilitate the discussion, you read each clarifying question and then write them on a white board/smart
board/chart paper, identifying each question by team name/number (i.e., Group 1, 2 or by the Group’s
invented name, if appropriate).
7. Then a discussion ensues. You give each team the floor when their question is brought up. The team is
given the opportunity to answer the clarifying question, which can be done by one representative from the
group or by all the group members chiming in when appropriate and/or when there is a pause after a team
member adds a final comment.
As your class develops a comfortable routine around Gallery Walks, you can extend the activity by asking
students to generate the questions that they will respond to as they observe the posters, either on sticky
notes or on Response Sheets. In some cases, you can have students generate prompts or questions that they
would like observers to respond to about their specific poster.
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SFUSD Signature Strategy #2: Math Talks
*see "Number Talks Basics" after the SFUSD Strategies for more information
What is this strategy?
Math Talks are teacher-led, student-centered techniques for building math thinking and academic discourse.
They are intended to last for 10–15 minutes. Math Talks can be centered on any math topic. However, they
are not used to introduce math content. Rather, Math Talks are best when the content is generally familiar to
students up to their Zone of Proximal Development.
Why would I use this strategy?
Math Talks serve to further understanding of math content while addressing Math Practice Standard #3:
Construct viable arguments and critique the reasoning of others. They give students the opportunity to
develop flexibility and fluency with mental visualization and computation. They offer opportunities to revisit
math topics as well as deepen understanding by sharing multiple ways of thinking about a concept or skill.
When do I use this strategy?
This strategy can be used at any time, but is often done at the beginning of a math class. Because it does not
need to be focused on the lesson’s content, the content of the Math Talk can vary according to the needs of
the students. Math Talks generally happen 2 or 3 times a week for 10–15 minutes each.
How do I use this strategy?
Teachers deliberately set up a safe environment where each child’s thinking is valued.
Students practice making their thinking explicit.
Everyone practices understanding each other’s thinking.
1. Teacher presents the problem.
A problem is presented to the whole class or a small group. Computation problems are always
presented horizontally (e.g., 43 + 35 =?), so as to encourage mental strategies rather than reliance on
algorithms.
2. Students think about the problem.
Students are given time (1–2 minutes) to silently, mentally think about the problem and try to find an
answer. They signal quietly to the teacher (e.g., with a thumb up against their chest) when they have an
answer.
3. Students share their answers.
A few students volunteer to share their answers and the teacher records them on the board. Without
judgment, the teacher records answers where all students can see. The teacher continues to take
answers until all students’ answers have been shared. Teacher can also ask the students to
Turn-and-Talk with a partner before sharing answers.
4. Students share their thinking.
Students share how they got their answers with a partner or with the larger group. Any student can
provide an explanation to any answer on the board. Equity sticks can be used to ensure every student has
an equal opportunity to share. The teacher records the students’ name and thinking. The teacher asks
questions that help students express themselves, understand each other, and clarify their thinking to
make sense of the problem and its solution(s). Multiple ways of solving problems are emphasized. The
student’s name is attached to the solution.
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Sample Math Talks By Level
K-2 3-5 6-8 High School
How many dots do you see? Which is greater
86 x 38 or 88 x 36?
Three corn dogs cost $5.25.
How much does one corn dog
cost?
Always, Sometimes, Never: How many tiles in figure 10?
15 x 18
Number Strings Place ½ and 2 ½ on this How many triangles do you see?
number line Estimate what 32% of 647 is.
12 + 12
12 + 13
13 + 13
13 + 14
Math Talk Layers
As you begin to implement Math Talks in your classroom, you will want to keep them simple. Your goal might
be to have 2 or 3 students share their thinking, which you capture and record without much comment or
questioning.
Initial Implementation:
❖ Provide a safe environment.
❖ Start with easier problems so that students can learn the routine.
❖ Present problems horizontally.
❖ Provide quiet think time and a silent signal.
❖ Capture student thinking as faithfully as you can.
❖ Accept, respect, and consider all answers.
❖ Develop your poker face. Respond neutrally to students’ comments.
As you and your students’ familiarity with Math Talks grows, you will find that you can begin to ask questions
and probe their thinking.
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Here are some clarifying and probing questions to try:
❖ Where did you get this number?
❖ How did you get this?
❖ Why did you do this operation?
❖ Do you mean this?
❖ Is this how you thought of it?
❖ So you are saying that…?
These questions help students in breaking down their thinking and explaining the steps they went through.
Share the Why with Students
Give the students the rationale behind the Math Talk. Let them know that they have such great thinking going
on that we can't see and this gives them a chance to share what's going on in their brains. It also gives
everyone a chance to learn from each other and informs the teacher about what they know and what we
might need to work on.
Adding Layers
❖ Ask students if they thought of the problem in the same or a different way. (This can be done verbally
or with a signal, e.g., pat your head if you thought of it the same way.)
❖ Have students Turn-and-Talk. Use this strategy when many students want to talk and may not have a
turn individually; when you want to generate more answers and/or discussion from students; or
when students need time and practice articulating their math ideas and strategies before sharing
with the whole group.
❖ Begin to ask questions that connect students thinking to each other:
➢ Who has a question for ?
➢ Who can paraphrase what is saying?
➢ Who can explain what is thinking?
➢ Do you agree or disagree with what they said? Can you explain why?
❖ Point out similarities and differences between different strategies.
❖ Ask students to point out similarities and differences between different strategies.
Responding to Math Talks
As you become increasingly comfortable with using Math Talks, you will find yourself adjusting them and
incorporating them into your pedagogical repertoire.
❖ Design new Math Talks based on issues that arise during math instruction.
❖ Design math instruction based on confusions that arise during Math Talks.
❖ Create Class Strategy posters that summarize different strategies that your class is using in Math
Talks.
❖ Simplifying Math Talks when students have difficulty. Using smaller numbers can help students
access a strategy that they can then apply to larger numbers.
❖ You can offer more than one problem during a Math Talk and allow students to choose the one they
want to solve. For example, 13 X 12 and 15 X 17 both get at multi-digit multiplication, but one uses
numbers that may be easier for students to keep in their heads as they solve the problem mentally.
Further Resources 24
SFUSD Mathematics Department website: www.sfusdmath.org/math-talks.html
San Diego Unified School District website: www.sandi.net/Page/33501
Number Talks - Helping Children Build Mental Math and Computation Strategies (K-5) Sherry Parrish
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SFUSD Mathematics Department, June 2014, sfusdmath.org
Effective Questioning2
You can promote discourse and stimulate student thinking through effective questioning. This, in turn,
develops the habits of mind suggested by the Standards for Mathematical Practice. Here is a list of questions
from the Professional Standards in Teaching Mathematics, grouped into categories that reflect the
mathematical practices.
❖ Helping students work together to make sense of mathematics: it?”
➢ “What do others think about what Janine said?”
➢ “Do you agree? Disagree?”
➢ “Does anyone have the same answer but a different way to explain
➢ “Do you understand what they are saying?”
❖ Helping students to rely more on themselves to determine whether something is mathematically
correct:
➢ “Why do you think that?”
➢ “Why is that true?”
➢ “How did you reach that conclusion?”
➢ “Can you make a model to show that?”
❖ Helping student learn to reason mathematically
➢ “Does that always work?”
➢ “Can you think of a counterexample?”
➢ “How can you prove that?”
➢ “What assumptions are you making?”
❖ Helping students learn to conjecture, invent, and solve problems:
➢ “What would happen if…? What if not?”
➢ “Do you see a pattern?”
➢ “What is alike and what is different about your method and her method to solve the problem?”
➢ “Can you predict the next one? What about the last one?”
❖ Helping students to connect mathematics, its ideas, and its applications:
➢ “How is this process like others that you have used?”
➢ “How does this relate to ?”
➢ “Have you ever solved a problem like this before?”
➢ “Can you give me an example of ?”
2 Adapted from NCSM Great Tasks for Mathematics, 6-12 by Schrock, Norris, Pugalee, Seitz, and Hollingshead, 2013 26
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SFUSD Mathematics Department, June 2014, sfusdmath.org
SFUSD Signature Strategy #3: Participation Quiz/Group Feedback
What is this strategy?
3
A Participation Quiz /Group Feedback is a strategy to help establish or reinforce norms for group work in a
cooperative environment. While students work together in their group on a math task, the teacher takes
public notes—on a document camera, white board, chart paper, or overhead projector—about the quality of
their group work (social moves) and the quality of their mathematical discussions (math moves). The teacher
can take notes on how students work together, their use of classroom norms, or the specific language they
use to communicate their mathematical ideas.
Why would I use this strategy?
Publically taking notes on students’ interactions allows the teacher to communicate the behaviors they wish
to encourage and value, as well as mitigating perceived status differences between students—that is,
highlighting strengths of students who are not perceived to be strong in math. Some teachers assign each
group a grade at the end of a Participation Quiz/Group Feedback. Other teachers prefer to focus on the
feedback rather than giving it a score. This protocol might be named differently, for example, “Group Work
Feedback,” to reflect the teacher’s objective.
When do I use this strategy?
This strategy can be used whenever students are working in collaborative groups.
How do I use this strategy?
1. Choose a worthy task.
The teacher chooses a task that is accessible, challenging, important, and requires students to read and talk
together. If a task is too hard, the teacher may spend more time answering group questions than observing,
and if a task is too routine students will naturally do these individually since little collaboration will be
required.
2. Decide on a focus.
The teacher decides which group norms or Standards of Mathematical Practice he/she wants students to
focus on. This decision depends on the context of the classroom. Early in the year, the teacher may focus
on establishing norms, such as getting a quick start (reading problem promptly and making sure group
understands), working together (heads leaning in and working in the middle of the group), and asking the
group questions before asking the teacher. Later in the year, the teacher may focus on refining a norm
that a particular class is struggling with, such as making statements with reasons, or the teacher may
choose to highlight strengths of specific students that have low status (students who are not generally
seen as strong in math).
3. Communicate the focus to students.
The teacher lets his/her class know that the lesson will be structured as a Participation Quiz/Group
Feedback. The teacher is clear about what he/she is looking for and uses language that students
understand. The teacher explains that he/she will publically record a snapshot of the students working
together in their groups.
3 This description is based on Smarter Together—Collaboration and Equity in the Elementary Math Classroom by Featherstone, Crespo, Jilk, et.
al. and the Instructional Toolkit for Mathematics produced by Oakland Unified School District.
131 28
SFUSD Mathematics Department, June 2014, sfusdmath.org
For example, the teacher can say that he/she is looking for:
❖ “because” statements (addresses “making statements with reasons”).
❖ students leaning in (addresses “working together on the same problem”).
❖ group questions only.
As students work, the teacher publically records statements about how groups are working together.
This can be done on a document camera, white board, chart paper, or overhead projector. The recording
sheet is split into as many spaces as there are groups (see example diagram below). Sometimes groups
do not notice this public documentation, while other times they pay attention and change their behaviors
to meet the norms.
4. Debrief the notes taken.
The teacher takes time before the end of class or in the middle of the task to debrief. Time is given for
students to read comments. The teacher highlights key evidence that supports the group work norms.
The focus on group work norms to start the class and then end the lesson can be a powerful way to
reinforce the kinds of cooperative behaviors that teachers want to establish.
When students are used to seeing this structure, teachers can use these public notes as a “quiz” to assess
students and groups on their group work skills. Generally, the focus should be on positive behaviors,
although over time honest critiques of behavior may be included as well.
Example Participation Quiz / Group Feedback Diagram
Group 1 Group 2
QS (Quick Start) QS
Miguel, “Can you re-read the problem?” So, I think this means…
I said this was 3x + 2 because… Heads leaning in
“What did you mean by…”
Group 3 “So, x means... do you get it?”
QS
“I’m not sure what to do. Can you…” Group 4
Using a table of points Reading problem
I think the pattern is +3; see, look at… “Melissa, explain to me how you got the equation.”
Using “because” statements
Group 6
Group 5 QS
QS “I don’t understand the where the 2 shows up on
“Can you repeat your idea?” the graph.”
Let’s make a graph. “Oh…I see the adding 3 in the table and graph;
“Something’s not right. What do you think?” now…that makes more sense.”
Explaining graph and equation. Students leaning in on work in the middle
Further Resources
SFUSD Mathematics Department web site: www.sfusdmath.org/participation-quiz--group-feedback.html
Smarter Together-Collaboration and Equity in the Elementary Math Classroom Featherstone, Crespo, Jilk, et. al.
132 29
SFUSD Mathematics Department, June 2014, sfusdmath.org
Sample Math Class Norms
Classroom norms can be generated by the students themselves with facilitation by you as their teacher and
fellow community member. They can also be discussed and innovated upon using a pre-determined list that
you may have. Here are some sample math class norms you can use to generate discussion about class norms
with your students.
133 31
SFUSD Mathematics Department, June 2014, sfusdmath.org
Number Talk Basics by Erika Barrish, JMCS
Before the Number Talk
1. Choose a problem in students’ zone of proximal development (number talks as formative
assessments).
2. Anticipate the kinds of strategies that students might use (see quadranthandout).
During the Number Talk
1. Students put paper and pencils away and put their fists unobtrusively on their chests to show
they are ready.
2. Teacher writes a problem on the board.
3. Teacher watches while students solve problem mentally and put thumbs up to indicate they
have identified one way to solve the problem.
a. Students raise an additional finger for additionalmethods.
b. Don’t rush to end thinking time! Aim for a minimum of 10seconds.
4. When most thumbs are up, teacher asks students to share answers.
5. Teacher records JUST THE ANSWER(S) without indicating success.
6. When all answers have been recorded, teacher asks if anyone can explain how he or she figured
the problem out.
7. Teacher begins by asking student which answer he or she is defending.
8. Teacher records the thinking of each student for class to see.
9. After student shares a strategy, teacher asks questions to work with that student’sthinking.
a. Does anyone have a question for ?
b. Can you say more about ?
c. Can someone explain ?
d. There are no right or wrong questions to ask, so long as you are prompting students to
explain their reasoning or the reasoning of someone else.
In general, Number Talks should last about 15 minutes.
After the Number Talk
Reflect (see quadrant handout):
1. Did students use the strategies you anticipated? What does that tellyou?
2. Did you choose an appropriate problem for students’ mathlevel?
3. Should you do a similar problem next time to build on the strategies students used, or move to a
different type of problem?
134
Number Talk Pro Tips
Ask questions
to help
students
articulate their
thinking
precisely!
Subtraction Multiplication Addition Division
Minuend – Subtrahend = Difference Factor X Factor = Product Addend + Addend = Sum Dividend ÷ Divisor = Quotient
Be precise!
“Three and eighteen hundredths” vs. “three point one eight.”
1. Celebrate mistakes and explore reasoning!
2. Allow students to come up with strategies that make sense to them.
3. Interchange symbols to encourage flexibility.
4. Students may use strategies you didn’t anticipate. That’s okay!
5. In general, Number Talks should last about 15 minutes.
135
Trouble Shooting for Number Talks
Scenario Suggestion(s)
What if I don’t 1. “What I think I heard you saying was . Is that what you are saying?”
understand what 2. “I want to make sure I understand what you mean. Could you please
a student is repeat that last part?”
saying? 3. “Who can explain what said in your own words?”
How can I get my 4. Be careful not to put words in a student’s mouth! Ask, don’t tell.
students to move beyond 1. The first time the traditional algorithm is offered, briefly explain that an
the traditional algorithm
when solving a problem? algorithm is steps that will lead to a complete solution. For subsequent
use of the algorithm strategy, simply write “traditional algorithm” on the
How can I get more board.
students involved? 2. Try to find problems that are unwieldy with the traditional algorithm but
easier using a different method.
What if I get confused or 3. Begin by offering a better strategy: “I saw someone do it like this. [Map
make a mistake? the process.] See if you can understand what they did.” Then choose a
different problem that lends itself to this new strategy so students can try
it. If no one volunteers the new strategy, ask, “Did anyone try the method I
saw someone else do?” If no one did, ask, “How do you think they might
have used their strategy to solve this problem?”
4. Ask, “how else?”
5. Don’t dwell on asking “why” the traditional algorithm works. Early in a
student’s experience with number talks, asking why the algorithm works
can be frustrating and discouraging, since it often relies on rote memory.
1. “I would like to hear from someone who hasn’t had a chance to share.”
Wait….then wait some more. If no one volunteers after a LONG wait time,
try a different approach but revisit this strategy in future Number Talks.
2. Bring a small group of students together in a private Number Talk.
3. Use Think-Pair-Share (TPS). Caution: setting up a situation where all
students are expected to share can be frightening for students who are
tentative in their thinking. Also, this scenario will likely result only in
correct answers, which can deprive the class of exploring the reasoning
behind incorrect answers.
4. Pose a problem and ask students to mentally solve it two different ways,
and record their strategies on note cards. When they are finished, place
students in small groups to share their strategies. After the small group
share, collect the cards to inform your decision on what type of number
talk to conduct next.
1. This is an opportunity to illustrate that mistakes can facilitate growth and
learning!
2. “I think there is something wrong here. Can anyone help me figure it out?”
3. “I just confused myself. Can anyone help me think about this?”
4. “I think I just made a mistake. My synapses are firing!”
5. “I’m going to think about this for a while and I hope you help me think
about it. We’ll talk about it tomorrow and try to figure it out.”
136
I know we’re supposed to 1. Establish a classroom norm that any answer, right or wrong, must be
make mistakes as sites justified. Always start by having students identify the answer they are
for learning, but what defending.
should we do when a
student’s answer or 2. Usually it becomes apparent early on which answer is “right.” If you
method is wrong? decide it’s valuable to discuss the other answers:
a. “Is the person who answered willing to tell us how you
What do I do if I don’t thought about this?” If no one answers, ask, “How might
know how to record a someone arrive at this answer?” or let it go depending on your
preference.
student’s thinking? b. If there is a lingering question about which answer is right, you
might want to say “Let’s try this strategy with smaller numbers
How do I get buy-in from that we know the answer to and see if it works.”
my high school students?
3. Resist acknowledging, verbally or nonverbally, that an answer is right or
How can I get students to wrong, especially before your students have had a chance to examine and
listen and talk to one defend the various answers.
another?
4. Don’t put individuals on the spot!
1. Write just enough so the class can clearly “see” how a student’s strategy
works.
2. Listen to a student for a while before you start to write so you can get the
gist of the strategy or where the mathematics is going.
3. Sometimes you might erase what you have written as a student talks and
start all over again.
4. Be honest if you’re really stuck. Consider inviting the student to come
record the strategy for the class.
1. Realize that for their entire school experience, high school students have
interpreted success in math as knowing WHAT to do. It can be frustrating
and discouraging to suddenly be required to explain why math makes
sense.
2. Start with Dot Talks, which present patterns to the class and ask students
to figure out the number of items in the pattern WITHOUT COUNTING.
This explores “ways of seeing.”
3. Present problems in context (word problems).
4. Be patient and don’t give up!
5. Keep talks short – less than 15 minutes.
1. “What questions do you have for ?”
2. “Do you think ’s method will work every time? Talk to about what
you think.”
3. When hands are raised, don’t call on a student yourself. Say, “ , there are
some students who want to talk to you.” Then, let the student call on
someone.
4. Help students learn to ask “does anyone have a question?” after they have
shared a strategy.
5. If a student is willing to share something he or she is not sure about,
celebrate!
6. Wait to respond to students to give the class a chance to beat you to
responding.
137
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