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Equitable A Pathway to Math Instruction STRIDE 2 (Math Equity Toolkit)

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1
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
A Pathway to
Equitable
Math Instruction
Fostering
Deep
Understanding
Methods for deepening student conceptual
understanding through orchestrated math
discussions that build on and connect
multiple strategies.
2
STRIDE
1
Fostering
Deep
Understanding
The purpose of this tool is to highlight the diversity of
student thinking, misconceptions, alternate solutions,
and connections so any student, regardless of level, can
contribute in meaningful discussion and gain agency
and deep conceptual understanding.
Teachers also build pedagogical content knowledge,
cultivate the exibility to work with diverse students,
and practice continuous improvement.
This tool is designed for middle school math teachers, but
can be adapted for any grade level. It is used for planning
the lesson, data collection during the lesson, as well as
reection after the lesson and for future planning. It is a
cyclical, living practice of continuous improvement.
To use this tool, follow the Planning
Checklist, which breaks down the
process into four distinct parts: Un-
derstand the Activity, Plan the Ac-
tivity, Collect Student Samples and
Thinking During the Lesson, and
Reect on Content Understanding
After the Lesson.
First, Understand the Activity by
reading 5 Practices for Orches-
trating Productive Mathematics
Discussions, how to choose cogni-
tively demanding tasks, and how to
facilitate equitable student discus-
sions and honor student responses.
Next, Plan the Activity by identify-
ing a specic learning goal, task,
content, and math practice. Antic-
ipate student strategies that may
be used, practice sequencing stu-
dent thinking, and consider virtual
adaptations and necessary adjust-
ments.
Then, During the Lesson, collect
student samples and data, and
guide student discussion.
And nally, After the Lesson, exam-
ine what student data says about
your students’ current level of un-
derstanding, identify student re-
sponses you did not anticipate or
understand, and identify potential
areas of development for your own
content understanding.
HOW TO USE THIS TOOL
CONTENT DEVELOPERS
Rolanda Baldwin
Director or Mathematics
UnboundEd
Mindy Shacklett, EdD
Mathematics Coordinator
San Diego County Office
of Education
Jean Siqi Yang
Teacher Leader
TeachPlus
FEEDBACK ADVISORS
Dr. Joanne Rossi Becker
Professor Emerita, Mathematics
San José State University
Dr. Mirna Miranda-Welsh, EdD
Title III Coordinator
Los Angeles County Office
of Education
Dr. Susie W. Håkansson
Mathematics Education Consultant
and Past President
TODOS: Mathematics for ALL
Dr. Judit Moschkovich
Professor, Mathematics Education
University of California, Santa Cruz
THEMES
Content + Conceptual Understanding
GUIDING PRINCIPLES
Equitable access to grade-level
priority math standards.
Learning opportunities for students
to engage with the standards for
mathematical practice.
Assets-based formative assessments
to inform instruction.
STRIDE 2
2
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
3
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
INTRODUCTION 4
GETTING STARTED 5
USING THE PLANNING CHECKLIST 6
PLANNING CHECKLIST 7
EQUITABLE MATH DISCUSSIONS ACTIVITY TEMPLATE 8
RESOURCES 11
EXAMPLE OF ACTIVITY TEMPLATE 12
Table of Contents
4
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
Equitable Math Discussions
through Student Discourse
Introduction
Have you ever taught a lesson and then encountered students solving the presented problems in a different way?
Do opposing student thoughts scare you or excite you?
Is your rst reaction to differing thoughts “Tell me more!” or “Let me show you”?
Purpose
The purpose of this tool is to highlight the diversity of students’ thinking, misconceptions, alternative solutions, and
connections so any student, regardless of level, can contribute and gain equitable access to grade-level content. Stu-
dents gain agency, voice, and deep content knowledge through carefully planned discussions. Teachers also build deep
content knowledge, cultivate the exibility to work with diverse students, and practice continuous improvement.
This tool is meant to be used with all grade levels and content. It is used for planning the lesson, data collection during
the lesson, as well as reection after the lesson and for future planning. It is a cyclical, living practice of continuous
improvement. For a detailed demonstration on how to use this tool, see the short video here.
WHO BENEFITS HOW
STUDENT
TEACHER
Gains equitable access to meaningful and
engaging grade-level content.
Builds deep content knowledge.
Gains agency and uses their voice.
Gains flexibility when confronted with
diverse or unexpected thinking.
Builds deep content knowledge.
Practices continuous improvement.
Represent
all
levels of student thought:
initial thoughts, misconceptions,
alternative solutions, connections.
Intentional anticipating and sequencing
potential student thinking to reach learning
goals.
Center content understanding around
student
questions, diverse thinking, misconceptions,
and alternative strategies.
Explore diverse student thinking through
class discussion and questioning.
Facilitate active sharing, discussion, and
connections between different students'
thinking.
Reflect before, during, and after the lesson
to grapple with your own development needed
for the content.
5
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
Getting Started
Understand the 5 Practices for Orchestrating Productive Mathematics Discussions
5 Practices for Orchestrating Productive Mathematics Discussions, by Margaret Schwan Smith and Mary Kay Stein
1
, is
the foundational reference for this tool. It involves ve stages for highlighting and sequencing student thinking:
1. Anticipate likely student emerging ideas and alternative solutions to mathematical tasks.
2. Monitor students’ actual responses to the tasks.
3. Select student responses to feature during the discussion.
4. Sequence student responses in a purposeful order to build a coherent math story.
5. Connect different students' responses through math discussion.
Understanding this practice is the major step to begin implementation. If you are not familiar with the 5 Practices, it is
recommended that professional development, coaching, and independent research be coupled with this resource. Level
1 and 2 readings are must reads, and level 3 is an optional deep dive.
LEVEL 1
INTRODUCTION
LEVEL 2
IMPLEMENTATION
OPTIONAL LEVEL 3
DEEP DIVES
5 Practices for Orchestrating Productive
Mathematics Discussions (SFUSD)
Chapter 1:
Introducing the Five Practices
The 5 Practices in Practice: Successfully
Orchestrating Mathematical Discussion
1
Margaret Schwan Smith and Mary Kay Stein,
5 Practices for Orchestrating
Productive Mathematics Discussions
(Reston, VA: National Council of Teachers of
Mathematics, 2011).
6
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
In order to use this tool, follow the Planning Checklist, which breaks down the
process into four distinct parts: Understand the Activity, Plan the Activity, Col-
lect Thinking During the Lesson, and Reect After the Lesson.
Understand the Activity by reading 5 Practices for Orchestrating Productive
Math discussions, how to choose cognitively demanding tasks, and how to fa-
cilitate equitable student discussions and honor student responses.
Then, Plan the Activity by identifying a specic learning goal, task, content,
and math practice. Anticipate student strategies that may be used, practice
sequencing student thinking, and consider virtual adaptations and necessary
adjustments.
Then, During the Lesson collect student samples and data and guide student
discussion. And nally, After the Lesson, examine what student data says about
your students' current level of understanding, identify student responses you
did not anticipate or understand, and identify potential areas of development for
your own content understanding.
Using the Planning Checklist
7
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
UNDERSTAND THE ACTIVITY (Before the Lesson)
1. Understand the 5 Practices for Orchestrating Productive Mathematics Discussions.
“Chapter 1: Introducing the Five Practices”
2. Understand how to choose cognitively demanding tasks.
A. What are cognitively demanding tasks?
B. Libraries for finding tasks: Math Assessment Project or Illustrative Mathematics
3. Understand how to facilitate equitable student discussion and honor student responses.
A. Facilitating Effective Discussions
B. Equitable Discussion in a Multicultural Classroom
4. Choose tasks.
Consider Priority Standards by Grade Level.
Consider 2020-21 Priority Instructional Content in Mathematics.
Planning Checklist
COLLECT STUDENT SAMPLES AND THINKING (During the Lesson)
1.
Collect student samples and data.
REFLECT ON STUDENT AND TEACHER CONTENT UNDERSTANDING (After the Lesson)
1.
Examine what student data says about your students’ current level of understanding.
2. Identify student responses you did not anticipate or understand.
3. Identify potential areas of development for your content understanding.
PLAN THE ACTIVITY (Before the Lesson)
1. Identify a specic learning goal, task, content standards, and mathematical practice.
2. Anticipate student strategies that may be used to complete the task.
4. Consider virtual adaptations or other necessary adjustments.
(See “Resources” at the end of the Toolkit.)
3. Practice sequencing student thinking.
A. Complete the task with various colleagues and sequence solutions.
B. Review Chapter 1 or the 5 Practices
USE TEMPLATE BELOW
8
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
Equitable Math Discussions Activity Template
LEARNING GOAL
LEARNING TASK
CONTENT STANDARDS
8 STANDARDS
FOR MATH PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for regularity in repeated reasoning.
1. CONTENT PREPARATION (Use Priority Standards by Grade Level.)
For a detailed demonstration on how to use this tool, see the short video here.
9
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
2. ANTICIPATED STUDENT STRATEGIES SEQUENCING (Use Progression Documents by Topic.)
ORDER
Sequence strategies
for learning target
WHO / OBSERVATIONS
Which students are
using the strategy?
ANTICIPATED STRATEGY
Anticipate the various strategies/methods your students
will apply to arrive at a solution. One strategy per row.
10
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
3. REFLECT UPON YOUR LESSON AND PLAN NEXT STEPS
TASK NEXT STEPSREFLECTION NOTES
Examine what student
data says about your
students’ current level
of understanding.
Identify student
responses you did
not anticipate or
understand.
Identify potential
areas of development
for your content
understanding.
11
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
2
Keith Nabb, Erick B. Hofacker, Kathryn T. Ernie, and Susan Ahrendt, “Using the 5
Practices in Mathematics Teaching,”
Mathematics Teacher
Vol. 111, No. 5 (March
2018): 366-373.
1. Understand the Activity
5 Practices for Orchestrating Productive Mathematics Discussions
a. Level 1: 5 Practices for Productive Math Discussions Overview (SFUSD)
b. Level 2: Using the 5 Practices in Mathematics Teaching (NCTM 2018)
2
c. Level 3: 5 Practices for Orchestrating Productive Mathematics Discussions
(Smith & Stein Book)
2. Understand the Content
Priority Standards by Grade Level
a. Achieve the Core Priority Standards by Grade Level Website
b. Achieve the Core Priority Standards by Grade Level 2020-2021 PDF
Content Standards
Content Standards CCSS
Standards for Mathematical Practice CCSS
3. Understand the Sequence
Progression Documents by Topic
a. Achieve the Core Progressions by Topic
b. UnboundEd Content Guides
4. Cognitively Demanding Tasks
a. What are cognitively demanding tasks?
b. Math Assessment Project
c. Illustrative Mathematics
5. Facilitate Small Group Discussion
a. Complex Instruction
b. Facilitating Effective Discussions
c. Equitable Discussion in a Multicultural Classroom
6. Virtual Adaptations
a. Select and sequence asynchronously.
Give students tasks asynchronously and submit to the teacher. The teacher
selects, sequences, and prepares an asynchronous or synchronous lesson around
students' responses.
b. Select and sequence using Google Slides deck.
Provide a shared Google Slides deck for students to populate with their thinking
by typing on their assigned slides or uploading a picture of their work. The teacher
can then select and sequence the slides, and share with students synchronously
or asynchronously.
c. Select and sequence with interactive whiteboards.
Use interactive whiteboards to select, sequence, and share solutions.
Resources
12
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
EXAMPLE
Equitable Math Discussions Activity Template
LEARNING GOAL
LEARNING TASK
CONTENT STANDARDS
8 STANDARDS
FOR MATH PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for regularity in repeated reasoning.
1. CONTENT PREPARATION (Use Priority Standards by Grade Level.)
For a detailed demonstration on how to use this tool, see the short video here.
I can decide whether or not two situations are happening at the same rate.
Diego paid $47 for 3 tickets to a concert. Andre paid $141 for 9 tickets to a concert.
Did they pay at the same rate? Explain your reasoning.
Students will work individually. Their solutions will be shared anonymously.
Students will compare their strategies at the end of the lesson.
6.RP.A.2 Ratios and Proportional Relationships
Understand the concept of a unit rate a/b associated with a ratio a:b with b0, and use rate language in the
context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions.
13
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
2
Since 9 is 3 x 3, multiply 47 by 3, 47 x 3 = 141 . Diego would
have paid $141 for 9 tickets if he paid at the same rate he did for 3
tickets. Since this is what Andre paid for 9 tickets, they paid at the
same rate.
Student B
Student D
Student K
Student M
Student N
3
47 ÷ 3 = $15.67 , 141 ÷ 9 = $15.67
Each ticket costs $15.67, they paid the same rate
Student E
Student F
Student H
Student Q
1
Student A
Student C
Student J
Student L
Student P
Student Q
amount paid
in dollars
0 47 94 141 188
0 3 6 9 12
1. START
HERE
2. GO TO 9
TICKETS
3. SEE IF AMOUNT
PAID MATCHES
THE OTHER RATIO
number of
tickets
2. ANTICIPATED STUDENT STRATEGIES SEQUENCING (Use Progression Documents by Topic.)
ORDER
Sequence strategies
for learning target
WHO / OBSERVATIONS
Which students are
using the strategy?
ANTICIPATED STRATEGY
Anticipate the various strategies/methods your students
will apply to arrive at a solution. One strategy per row.
14
STRIDE 2 : Fostering Deep UnderstandingA Pathway to Equitable Math Instruction
Identify student
responses you did
not anticipate or
understand.
Identify potential
areas of development
for your content
understanding.
The students are in different places along the continuum. There were
4 students who leaned on additive reasoning rather than multiplica-
tive reasoning. I was able to connect their reasoning to the double
number line and model the multiplicative reasoning.
Once we discussed the double number line, I connected it to the
method of multiplying by a factor of 3 and compared it to the divi-
sion model.
We concluded with the unit rate strategy.
This student data tells me that students are in various places. I inten-
tionally started with the misconception, (i.e. using additive reason-
ing) and moved from representational to abstract, so that students
can make connections between their method and the more abstract
methods.
The goal for this unit is to build conceptual understanding, so I will
continue to make connections between the various approaches.
I need to push myself to continue to approach ratios in a conceptual
way, rather than rushing to teach a procedure. I want to continue to
understand the connections between the various strategies used to
solve these types of problems. I need to study how ratios will show
up in 7th grade, and make sure that I am setting students up for the
7th-grade approach.
Discussion for Professional Learning Community (PLC):
• How can I “honor” a wrong strategy?
• Is there an equity-oriented way to approach wrong strategies?
• When does the additive strategy apply or make sense?
• Can I think of a time or example when that strategy
does work or apply or make sense?
• Why do we as math teachers see this an absolute correct approach?
Misconception:
Using additive reasoning
3 + 6 = 9 and 47 + 94 = 141
(got stuck)
(4 students: Students G, I, R, S)
47 goes into 141, 3 times and 3 goes into 9, 3 times,
therefore they paid the same price.
(3 students: Students T, U, V)
The next lesson introduces
ratio tables. I need to be sure
that I show the relationship
between each of the methods
shown today and ratio tables.
I also can help the students
who relied on additive
reasoning to see how they
can translate to multiplicative
reasoning using a ratio table.
I will make the connection
for students who relied on
additive reasoning to see
how they can translate to
multiplicative reasoning using
a ratio table.
I will show the students who
divided that they are using
multiplicative reasoning.
3. REFLECT UPON YOUR LESSON AND PLAN NEXT STEPS
TASK NEXT STEPSREFLECTION NOTES
Examine what student
data says about your
students’ current level
of understanding.
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Completed 22 February 2021

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