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Moving forward into the 2021–2022 school year seems like a daunting task. After all of the disruptions and challenges accompanying the pandemic, supporting mathematics students and teachers involves rethinking our priorities and resources. How do we attend to perceived potential learning and opportunity gaps? How do we foster social and emotional well-being and cultivate a safe learning environment? How can we best understand and value the varied learning each of our students has experienced during the past school year? And the list goes on.
1.
Continuing the Journey: Mathematics
Learning 2021 and Beyond
Moving forward into the 2021–2022 school year seems like curriculum, effective teaching and learning, high expectations,
a daunting task. After all of the disruptions and challenges and the support and resources needed to maximize their
accompanying the COVID-19 pandemic, supporting learning potential” (NCTM 2014, p. 5). Reaching this goal
mathematics students and teachers involves rethinking our at this unique moment in time is not easy. Even before the
priorities and resources. How do we attend to perceived pandemic, numerous programs were not meeting the needs
potential learning and opportunity gaps? How do we foster of many marginalized students. The disruptions of the past
social and emotional well-being and cultivate a safe learning 18 months, which affected students inequitably, make the goal
environment? How can we best understand and value the of meeting the needs of all students even more challenging.
varied learning each of our students has experienced during The process of continuing the journey in mathematics learning
the past school year? And the list goes on. is complex work and a long-term investment that calls for
diligence and an ongoing commitment.
With so many interruptions and variations in student
experiences during the COVID-19 pandemic, there are those Continuing the Journey: Mathematics Learning 2021 and
who assume that students will have lost a year or more Beyond is the result of a joint effort of the Association of State
of learning. But students did learn during the pandemic. Supervisors of Mathematics (ASSM), NCSM: Leadership in
They learned through virtual lessons; in-person classroom Mathematics Education, and the National Council of Teachers
instruction; and interactions with teachers, other students, and of Mathematics (NCTM); it presents considerations, questions,
family members. As in past years, students will come into the and potential solution processes to educators and school
classroom with mathematical learning experiences, and we leaders to address the challenges highlighted by the COVID-19
need to honor the knowledge they have gained from those pandemic. Continuing the Journey: Mathematics Learning
experiences. We acknowledge that not all students had the 2021 and Beyond addresses three key areas that have
same experiences, and many have learning needs that may implications for equitable access to high-quality mathematics
not have been met. As a community, mathematics educators teaching and learning: (1) a focus on grade-level content;
must commit to provide equitable learning experiences so (2) instruction through equitable, effective teaching practices;
that all students “have access to high-quality mathematics and (3) planning for advocacy.
AREAS WITH SERIOUS IMPLICATIONS FOR EQUITABLE ACCESS TO
HIGH-QUALITY MATHEMATICS TEACHING AND LEARNING
GRADE-LEVEL EQUITABLE, EFFECTIVE ADVOCACY
CONTENT TEACHING PRACTICES Engage in work that supports
On-grade-level mathematics Position all students as all students’ equitable access
content must be the focus competent, confident, and to high-quality, meaningful
of students’ work. capable learners and doers mathematics.
of mathematics.
1 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
2.
As you read through this document, we ask that you consider opportunities for students to learn and develop a deeper
your role in mathematics education and your sphere of mathematical understanding as teachers draw out essential
influence, engage in reflection, and identify actions you and features of mathematical ideas while cultivating the
those you serve can take as advocates. How can you advocate mathematical potential in a student’s thinking.
for starting the year with on-grade-level content? How can
Use tools such as concept maps to bring coherence to prior
you advocate for equitable, effective instruction with the
knowledge and on-grade-level content. They provide a visual
teachers and leaders you serve?
organizer to connect standards, learning outcomes, practices
and processes, and the instructional design of lessons.
Adhering to Grade-Level Content Such tools may serve as the Velcro to support students in
Supporting each and every student is essential in developing connecting their ideas both within and across grade levels.
a deep mathematical understanding; in understanding and
critiquing the world through mathematics; and in experiencing When mathematics lessons lack coherence, they appear
the wonder, joy, and beauty of mathematics—all of which to be stand-alone lessons or units with little attention to
contribute to a positive mathematical identity (NCTM 2018, connections and coherence to other lessons or units of
2020a, 2020b). To support students in this desired outcome, instruction. Coherence should include connected topics
we must design learning opportunities for students to make during lessons and units. For example, many students
connections within and across grade levels, emphasizing believe that decimals are completely different from fractions.
reasoning and sense making to ensure the highest-quality When they think that way, they do not generalize their
mathematics education for each and every student (NCTM understanding from one representation to the other, so
2020a, 2020b). everything seems to be a brand-new idea. We can leverage a
deeper understanding of the content by creating intentional
Regardless of what has come before, on-grade-level connections across the content.
mathematics content must be the focus of our work with
students. This includes on-grade-level mathematics content As important as the content students learn are the
for PreK-8 and on-course-level mathematics content for high opportunities students have to engage in mathematical
school. Particularly for the coming school year, mathematics processes and practices as they learn the content. Students
educators must begin with building coherence through grade- should be encouraged to make sense of mathematical
level content and shifting from remediation to supporting and problems and persevere in solving them; reason abstractly and
scaffolding to facilitate student understanding. quantitatively; construct viable arguments and critique the
reasoning of others using precise language; use mathematics
Building Coherence through Grade-Level Content to model the world; and develop generalizations (NGA Center
and CCSSO 2010, pp. 6–8). The processes and practices must
Achieving the goal of organizing grade-level content around
be viewed as coequal partners with the specific content
big ideas while attending to learning progressions will
priorities for a course or grade. One cannot be fully realized
require that district and state or provincial curriculum leaders
without the other, and both are essential in helping students
provide sufficient guidance for grade-level and vertical teams
reach the broader mathematical goals for student learning.
to collaborate and create greater coherence using content
standards, standards for mathematical practice, priority Attending to mathematical coherence by incorporating
grade-level and course content, and individual units of study. big ideas and essential understandings and engaging in
Mathematical coherence occurs when concepts are connected mathematical processes and practices with support can
within and across grades while incorporating attention to leverage students’ current thinking to cultivate a deeper
mathematical processes and practices. That students see mathematical understanding. When a coherent and connected
how the mathematics they are currently learning builds approach is taken, content from previous grades that
on their prior knowledge is important. Making coherence students may seem to be “missing” can be taught along with
explicit becomes the Velcro™—that is, how the content grade-level content or just-in-time support. This moves the
sticks together—connecting concepts to move student conversation away from a focus on “right or wrong answers”
understanding forward. toward leveraging different student perspectives and
incorporating different strategies used by their peers into their
Mathematics includes a wide range of concepts and skills
own thinking. On the basis of the strength of both teacher and
that are intricately connected. Leveraging those connections
student inputs, these reasoning pathways help move student
and learning progressions will move student understanding
learning forward with grade-level content. See additional
forward. The analogy of using Velcro provides a window into
recommendations from in the Determining Necessary Prior
how mathematical ideas should be developed, connected,
Knowledge section of Moving Forward. The Call to Action
and organized in grade-level course content for the ultimate
section below also provides reflection questions and resources
benefit of students. This will ensure equal access and create
to guide this work.
2 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
3.
Shifting from Remediation to Supporting and Supports and scaffolds should not be framed as remediation.
Scaffolding Research shows that just-in-case remediation increases, rather
Acknowledging and building on student strengths is essential. than decreases, gaps in student achievement, whereas just-in-
“Many students who are challenged by mathematics never time scaffolding supports and values student knowledge while
‘get to’ the more engaging mathematics and therefore never being intentional to ensure success with grade-level content
experience the beauty and joy of doing mathematics” (Kobett (TNTP and ZEARN, 2021). Begin the year with grade-level
and Karp 2020, p. 12). The advancement of
student learning requires building on students’
strengths and understanding the beliefs teachers This is our time to humanize mathematics by creating
hold about what content is taught, how it is greater opportunities to learn for all students and ensure
delivered, and who receives it (Aguierre, Ingram,
that all students have access to grade-level content.
and Martin 2013). Teachers’ beliefs influence
the decisions that they make regarding how
they teach mathematics and can be productive
or unproductive (NCTM 2014). Unproductive beliefs hinder content and instruction and include opportunities for just-
the implementation of effective instructional practice and in-time scaffolds and support. Develop scope and sequences
limit student access to important mathematics content and that focus on grade-level content and include scaffolding
practices, whereas productive beliefs support each and every supports that address content from prior grade levels as
student having access to high-quality mathematics learning needed. A coherent and connected approach to content, along
opportunities that build on student strengths. Lessons and with scaffolds and support, ensures students with different
units designed with student strengths in mind include the learning needs have access to and can be successful with
following: grade-level content. See additional recommendations from
Moving Forward in the Structures That Organize Students for
• Intentional balance of time for students to acquire Instruction section and the NCTM Back to School Guides for
conceptual as well as procedural mathematics knowledge additional resources.
• Experiences for students to access prior knowledge that
can be vehicles for developing conceptual and procedural Call to Action: What Can I Do Now?
knowledge Consider the following productive ideas for content (see
• Opportunities to provide learning experiences that allow Table 1) that are true for every school year (NCSM and NCTM
each student to feel capable and safe (Aguirre, Ingram, and 2020; NCTM 2018, 2020a, 2020b) and are an important part
Martin 2013, p. 58) of success for 2021 and beyond. Then consider the questions
and resources as you and your leadership team are making
Scaffolding and supporting students are key components of decisions about intentional connections across content.
an equitable approach. So, what is scaffolding? Scaffolding is
defined by Holton and Clarke (2006) as “an act of teaching This is our time to humanize mathematics by creating greater
that (a) supports the immediate construction of knowledge opportunities to learn for all students and ensure that all
by the learner; and (b) provides the basis for the future students have access to grade-level content. Creating greater
independent learning of the individual” (p. 131). Going back coherence and providing just-in-time support for grade-level
and addressing every part of every standard is unnecessary. content provide a lever that helps students develop a deeper
In fact, scaffolding may not happen in just one lesson— mathematical understanding, increase their motivation for
scaffolding can occur throughout multiple lessons. Scaffolding learning because they see the connections, and enhance their
and supporting students make tasks accessible without identities as confident and effective doers of mathematics.
lowering the cognitive demand, the type of questioning used, Each student knows different things and will need different
or ways of promoting mathematical discourse. Additionally, types of support. We can design continual mathematical
support during a lesson can come from other students when on-ramps by listening to students’ reasoning and considering
all students are placed in visibly randomized groups (Liljedahl learning pathways that connect concepts in ways that are
2020). See additional recommendations from Moving Forward helpful and accessible (Powell and Gray 2021).
in the Determining What Students Know section.
3 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
4.
TABLE 1 PRODUCTIVE IDEAS FOR CONTENT
Ideas for Adhering to
Grade-Level Content Questions to Consider Resources
Keep joy and humanity • How do you find ways to enjoy your work? Rehumanizing Mathematics for Black, Indigenous,
in teaching and learning. • In what ways can you design lessons and tasks to facilitate and LAtinx Students
wonder, joy, and beauty in the learning of mathematics? Humanizing Disability in Mathematics Education:
• What strategies are used to explore your own Forging New Paths
mathematical identity? Mathematics for Human Flourishing
• How do you support students in their positive mathematics
identity?
Focus on and teach • How do you notice, name, and document student Strengths-Based Teaching and Learning in
to student strengths. strengths? Mathematics: 5 Teaching Turnarounds for Grades
• How can you use strengths to design lesson plans and K–6
scaffolding? Teaching Math to Multilingual Students:
• What strategies are used to build student mathematics Positioning English Learners for Success
identity and agency? Acorns to Oaks: Nurturing Growth through
Strengths-Based Practices
A Wonder-Full Task Leads to a Wonder-Full
Intervention
Connect to the major • Will this learning be needed in future grades or courses? Learning Progressions
work of the grade/course • Is this standard or concept connected to important ideas in Catalyzing Change Series
learning. previous grades or courses? (Moving Forward 2020, p. 6) Curriculum Focal Points for Prekindergarten through
• Will it elevate student reasoning and sense making with Grade 8 Mathematics: A Quest for Coherence
explicit connections to grade-level content?
Mathematics Unit Planning in a PLC at Work (PK–2,
3–5, 6–8, HS)
Essential Understandings series
Work collaboratively • How does my team work together to develop a shared The Math Pact: Achieving Instructional Coherence
in grade-level teams understanding of the essential learning? within and across Grades, (Elementary, Middle, High)
or mathematics • What process do we use to identify a variety of strategies Mathematics Assessment and Intervention in a
departments. and representations to support student learning? PLC at Work
• What prerequisite skills can be an on-ramp to the Mathematics Instruction and Tasks in a PLC at Work
essential learning?
Mathematics Homework and Grading in a PLC
• How does my team create common formative at Work
assessments that align to essential learning? How
does my team work together to build understanding Jumpstart Formative Assessment Toolkit
of connections across content? Mathematics Formative Assessment, Vol. 2
• What process do we use to determine the depth of the The Formative 5: Everyday Assessment Techniques
standards and recognize to what degree concepts and for Everyday Classroom K–8
skills need to be mastered for student understanding? Strategies for Formative Assessment
Benefits of Formative Assessment
Scaffold learning to • How do we intentionally use prerequisite knowledge Mathematics Unit Planning in a PLC at Work
grade-level standards standards or progressions to scaffold the lesson for (PK–2, 3–5, 6–8, HS)
by using prerequisite students to access grade-level content?
knowledge. • How do we intentionally scaffold during instruction
to provide access to grade-level content?
Eliminate tracking of • How do we use just-in-time supports and scaffold student Detracking School Mathematics to Ensure Equitable
students on the basis of learning so every student has access to more engaging and Empowering Programs and Opportunities
perceived abilities. mathematics during which they experience the beauty Closing the Opportunity Gap: A Call for
and joy of doing mathematics? Detracking Mathematics
• How do we use ongoing formative assessments to build NCTM Catalyzing Change Series
on student strengths?
Inform families of the • How do we communicate with families the expectations NCTM Back-to-School Guide for Families for
essential learning and for grade-level content? resources
major work of the grade. • In what ways do we partner with families and communities
to support on grade-level learning?
4 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
5.
Foundations for Equitable, Effective As we continue the learning journey into the 2021–2022
Teaching Practices school year, we must cultivate more productive mindsets
that support all students, including those who are typically
As we continue the mathematics teaching journey in 2021 marginalized. This cultivation requires educators at all levels
and beyond, focusing on equitable instructional practices within the system to disrupt the unproductive mindsets
is imperative, as is being mindful of instructional strategies and practices that inhibit learning. Productive mindsets
employed to support students in on-grade-level learning. We and practices and suggestions for disrupting unproductive
need to position all students as competent, confident, and mindsets and practices are provided in Table 2.
capable learners and doers of mathematics, while affirming
their strengths every day in ways that cultivate positive The suggestions in Table 2 are just a few of the mindsets and
mathematical identities and support students in developing practices needed to ensure each and every student is provided
agency (Aguirre, Mayfield-Ingram, and Martin 2013). We the opportunity to engage in high-quality and meaningful
want to empower our learners and cultivate equitable mathematics every year. Another important element of setting
instructional mindsets. a foundation for equitable, effective teaching practices is
building students’ positive mathematical identities.
Resist the temptation to revert to what we know are
ineffective teaching practices. One of the most equitable Building Positive Mathematical Identities
things you can do for all of your students, including those
Students and teachers have experienced many varied teaching
who have been previously marginalized, is to adhere strongly
and learning opportunities in both formal and informal
to NCTM’s eight effective mathematics teaching practices
spaces. Learning continued during the pandemic and should
(NCTM 2014). Students with different learning needs are
be recognized and valued. We also need to acknowledge that
capable learners of grade-level content when presented with
students and teachers experienced losses and challenges—
equitable, effective teaching practices that humanize students’
including personal, milestone, social, and emotional—during
mathematics experience. Supporting students with strengths-
this time. Now more than ever, teachers should learn about
based Individual Education Plans (IEP’s) that position students
who their students are, identify their strengths, and teach to
as doers of mathematics promotes individual confidence while
build on their strengths (Kobett and Karp 2020). This is a time
enhancing the learning for all (Tan et al. 2019).
to make sure students see themselves as capable thinkers and
doers of mathematics.
We should recognize, believe in, and build The media uses terms like learning loss and learning gaps and
on the strengths of our students. makes other deficit-based statements. These terms devalue
the incredible work that students, educators, and families have
done this year. Using such terms as learning loss and gaps
It is important that we cultivate equitable instructional
can lead to a developed sense of inadequacy and damage.
mindsets and practices, build students’ positive mathematical
Terms such as unfinished learning, opportunity to learn, and
identities, create an equitable classroom culture, support
accelerated learning are strength-based terms that should be
making connections through effective instructional practices,
used to affirm what students bring to the learning experience
and engage teachers and teams to create a culture of
and to value existing knowledge. One role of mathematics
collaboration. Discussion of each area follows.
teachers and leaders is to ensure continual affirmation of
Cultivating Productive Instructional Mindsets and students’ positive mathematical identities through their
Practices learning experiences, feedback, and building on their strengths
that will support them in developing strong and resilient
Each and every student needs and deserves access to high- positive identities.
quality, meaningful mathematics. We should recognize,
believe in, and build on the strengths of our students. In An essential step to promoting positive mathematical identity
too many schools, unproductive mindsets and practices is to start with knowing the learners (Brown and Seda 2021).
have harmed many students and particularly marginalized Each and every student is unique, with different experiences.
students such as Black, Brown, and Indigenous students; As teachers learn about who students are, they can also learn
students who are economically disadvantaged; students who about their strengths, interests, cultures, and hobbies. As
are learning the English language; and students with special teachers, we should get to know our students as individuals
needs and disabilities. In the pandemic-disrupted school so we can plan intentional instructional experiences that
years, marginalized students may have taken the brunt of create access to grade-level content. As leaders, we need to
the impact of disrupted learning when schools went virtual. know those whom we serve as individuals so we can plan
Because of the challenges of mitigation strategies, IEPs may intentional learning experiences to support access to grade-
have been temporarily disregarded, some students may have level content.
had inequitable access to technology at home, and language
acquisition supports may have evaporated in many places.
5 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
6.
TABLE 2 DISRUPTING UNPRODUCTIVE MINDSETS AND PRACTICES THAT INHIBIT LEARNING
Unproductive Mindsets and
Productive Mindsets and Practices
Practices to Disrupt
Create a mindset that views mathematics as an interconnected web of • Disrupt a linear view of learning that creates barriers
concepts, knowledge, and skills. between content by suggesting a particular skill or
concept must be “mastered” before students can
access other content.
Believe that all students are mathematically brilliant. Assessing, activating, • Disrupt deficit language and views that negatively
and building on prior knowledge are a normal part of the teaching process. affect students’ mathematical identity, agency, and
Students become confident in their mathematics ability when teachers believe in authority in learning mathematics.
their ability. • Disrupt a view that perceived learning gaps are
problematic, treating students as less capable, and
lowering students’ sense of efficacy.
Deeply understand standard(s) and learning goals in a unit. Focus • Disrupt learning organized around performance
instruction around learning goals that consider the concepts, language, and goals that suggests students learn concepts in
representations students are developing in the unit. Support connections isolation and cannot move forward in their learning
between concepts across the grade levels so that students can strengthen their until they have mastered a particular set of skills
understanding of grade level content as well as prior and future content. and procedures.
Elicit and examine your knowledge of students. Value the knowledge and • Disrupt language around “learning loss” that could
skill students bring to each topic and signal this value through asset-based lead to remediation practices.
language. Examine student work (e.g., diagnostic, formative) to consider which • Disrupt making assumptions about what students
concepts, language, and representations students already show evidence of can and cannot do based on limited evidence that
using on previous tasks and which concepts, language, and representations could lead to remediation.
might need additional just-in-time support during the unit.
Invite students into the mathematics of the unit. Use instructional routines • Disrupt instruction that values only one solution
that support student access to low-floor, high-ceiling tasks and ensure students pathway and only some types of knowledge.
can bring their breadth of knowledge and experience in understanding and
engaging in the task.
Make connections explicit. When planning a unit, consider the concepts, • Disrupt instruction that initially teaches concepts in
language, and representations within the unit and how they connect to one ways that send the message that mathematics is a
another. How will these connections be developed? Which connections will be list of isolated procedures.
important to name during instruction? What connections to mathematical ideas
and real-world contexts will strengthen and support student learning?
Value reflection and revision. Have students grapple with cognitively • Disrupt grading all assignments. This practice sends
demanding tasks without the fear of everything being graded. Model reflection the message that mathematics is procedural and
as a teacher so students see learning as a journey. Allow students to revisit or having the right answer means the destination has
redo assignments so they develop a deep understanding of concepts (Safir and been reached or the learning has stopped.
Dugan 2021).
Establish a culture of feedback. Create a culture in which you provide • Disrupt using feedback that is a single score,
students both written and verbal feedback to help them grow as learners. Create percentages, or grades that are given to
a protocol for students to provide each other with feedback. Conference with students afterward and only represent a token
students to better understand how they are thinking about a concept. Co-create system of points.
rubrics with students so they are clear on the expectations that are rooted in
language of behaviors we are trying to see (Safir and Dugan 2021).
Practice responsive teaching. Use formative assessment throughout a lesson • Disrupt the use of single assessments or summative
to respond to the needs of learners, knowing when to scaffold or when to allow assessment as the sole assessment practice.
students to productively struggle. Provide opportunities for students help other • Disrupt the use of traditional methods that
students, share mathematical ideas, or help to work through a misconception emphasize evaluation of student achievement and
(Acceleration Workshop, CT Center for School Change 2021). school accountability rather than student learning
and understanding.
• Disrupt the view of assessment as limited to
testing students.
6 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
7.
Another benefit of knowing students is so teachers and and interaction and implement tasks that support and
teams can make intentional connections between students promote a community of learning. Have students help
and high-quality, relevant instructional tasks. Engage craft the norms and identify behaviors and actions that are
students in meaningful grade-level mathematics and observe examples of the norms after working together as a group.
how they interact, listen to the questions they have, the Throughout the year, continuously revise and reinforce norms
reasoning they use, how they navigate challenges, and how as a group. In addition, having students work and gather as
they communicate their thinking. This level of engagement a group helps to create classrooms where all viewpoints are
will open doors, invite students into the mathematics, and seen and heard and form a community feel for students and
help them develop a sense of belonging. As students see the teacher alike in the space (Safir and Dugan 2021).
themselves as doers of mathematics, they build confidence
Ensure that you are being a warm demander (Hammond
and develop mathematical agency and authority. As
2015) in your learning spaces. A warm demander continuously
teachers and students continue to develop mathematical
makes time to build a classroom culture in which teachers
understanding, ensure that teachers and students are
honor their students as individuals, build supportive
developing productive beliefs regarding the teaching and
relationships, and cultivate safe spaces where students can be
learning of mathematics.
themselves so they bring all of their mathematical brilliance
Creating an Equitable Classroom Culture into the learning space (Berry 2021). All the while, support
students’ engagement in rigorous mathematics that ensures
A positive, supportive classroom community is necessary for they are reasoning, sharing ideas, and struggling productively.
advancing student learning. Classroom communities bond Equitable classrooms are places where all students feel
through trust, relationship, and experience. Teachers have welcome and get the support they need while striving to meet
worked very hard during the pandemic in various ways to high expectations.
engage students to create a virtual equitable classroom
culture by using breakout rooms, screen sharing, the chat box, Making Connections through Instruction
and other technology tools this past year. One positive that
NCTM defines mathematical connections as the ability to
has come out of the pandemic is that many virtual lessons
“recognize and use connections among mathematical ideas;
have started with more concerted efforts to strengthen
understand how mathematical ideas interconnect and build
community as students have worked in a somewhat isolated
on one another to produce a coherent whole; recognize and
fashion. Many lessons that both students and teachers learned
apply mathematics in contexts outside of mathematics”
during pandemic-era instruction should be carried forward
(2000, p. 64). To ensure that students truly understand and
into subsequent school years. A focus on building an equitable
make sense of mathematics, we must plan instructional
classroom community is one of those lessons.
experiences that provide students with regular opportunities
We must continue to cultivate and support an equitable to make connections.
classroom culture that strengthens and supports positive
Copley, Karp, and Dougherty (2017) remind us that “tasks
mathematics identities. Aguirre, Mayfield-Ingram, and Martin
that focus only on procedural aspects may not help students
(2013) identify five equity-based instructional practices
learn the mathematical idea deeply” (p. 7) nor make
that support this goal: (1) going deep with mathematics,
important connections between and within mathematical
(2) leveraging multiple mathematical competencies,
representations that can assist in deepening conceptual
(3) affirming mathematics learners’ identities, (4) challenging
understanding. Thus, it is important that leaders and teachers
spaces of marginality, and (5) drawing on multiple resources
collaborate to create a coherent plan that includes a well-
of knowledge (p. vi). Together with the eight mathematical
organized curriculum as well as instructional tasks that
teaching practices described in Principles to Actions: Ensuring
support students as they make sense of mathematics and
Mathematical Success for All (2014), these practices provide a
make connections among important mathematical ideas. A
strong foundation for strengthening the teaching and learning
coherent plan, along with in-depth teacher content knowledge
of mathematics for each and every student, providing an
and implementation of research-informed equitable teaching
equitable classroom culture.
practices (NCTM 2014), influences the manner in which the
As we begin this school year, we must be intentional with content is taught, the ideas that are shown as important, and
building a positive, collaborative learning community with the type of tasks selected.
our students. Those who have been to school likely have not
Planning authentic classroom activities that allow students
had opportunities to engage with the full complement of
to explore a mathematical concept once it is introduced,
their classmates. Working to rebuild classroom communities
extending that knowledge, and connecting it to other concepts
is a critical step for mathematics teachers and teams to
within and across grades will deepen students’ understanding
take at the beginning of the year, as is working to maintain
of mathematics concepts. Connections that draw together
and strengthen them throughout the year. Take time during
key ideas and topics within a domain help students develop
mathematics class to co-create class norms for engagement
7 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
8.
a coherent understanding of the concept or process they are activation of prior knowledge strategies, more focus on the
learning. Engaging students in real-world problems in context essential content of each grade, more focus on the academic
around areas of interest that affect them individually and, language and strategies used to develop mathematical rigor,
in their communities, gives mathematics meaning. The goals and less repetition of content from year to year. Vertical
are to teach on-grade-level content and provide support collaboration will promote identifying content connections
and scaffolding as needed to foster students’ sense making. between grade levels and between concepts, which in turn
Table 3 shows three practices with areas for consideration that will help teacher teams guide their students to understand
every teacher can use to achieve these goals. the same connections.
For additional suggestions related to making connections Teacher teams will need additional time and support to ensure
through instruction, see the Teaching Practices Supporting that students are provided with access to grade-level content
Mathematics Teaching and Learning section in Moving Forward. through horizontal and vertical collaboration. Collaboration
will offer opportunities for teams to develop effective
Creating a Culture of Collaboration strategies to support diverse learning needs, specifically
For mathematics educators to collaborate with others is developing robust and intentional scaffolds and questions
vital to meeting the needs of each and every learner. The to support or extend learning. Maximizing access to grade-
collective wisdom of a mathematics team is much greater level content by knowing student strengths, interests, and
than a single individual member of a team. Mathematics experiences is crucial.
leaders and teachers must ensure that structures are created To avoid undercutting students’ interest or enthusiasm for
for intentional collaboration and must identify policies and learning, math teacher teams should focus on strengths and
other structures that might impede collaboration and the avoid over assessing students to identify gaps during the
work of teachers. first days and weeks of school (Moving Forward). Teacher
As noted in Moving Forward, teachers need dedicated teams can focus on developing an understanding of student
time during their daily schedule to collaborate with their strengths to implement strategies that tap into prerequisite
peers who teach the same grade or same course, including knowledge during a new unit of instruction and provide
special education and English language learner specialists. opportunities for students to engage in tasks to access their
To create coherence across grade levels and mathematics funds of knowledge and highlight their ideas, strategies,
course matriculation, teacher teams should collaborate and representations. For more ideas, see the Most Effective
vertically. Working with the teacher teams in grades before Teaching Practices, Determining What Students Already Know,
or after a specific grade level promotes more effective and Introducing New Learning sections of Moving Forward.
TABLE 3 SUPPORT AND SCAFFOLDING TO TEACH ON-GRADE-LEVEL CONTENT
Instructional Practice Considerations
Integrate activities that allow • Intentionally plan activities that allow students to relate new topics to concepts or skills previously
students to make connections learned and connect mathematical ideas to one another.
between mathematics topics • Encourage students to use multiple strategies to show and explain their thinking.
and concepts within and across • Have students articulate connections by asking how they use previous learning in the new task or
grade levels. to describe how the new topic relates to a previous topic.
• Engage in collaborative planning with colleagues about the ways specific concepts are taught and
connections between the concept both within grade levels and across grade levels.
Use and connect multiple • Engage students in the use of multiple
representations. representations. VISUAL
• Encourage students to use and select
representations purposefully to foster sense PHYSICAL SYMBOLIC
making and explore relationships and
connections between various representations.
CONTEXTUAL VERBAL
VERBAL
Huinker’s (2015) diagram (in NCTM 2020b, p. 51) indicates
important connections among various representation forms.
Develop students’ abilities • Select tasks and facilitate class discussions that provide opportunities for students to explore and
to recognize and identify focus on mathematical structure.
mathematical structures from • Looking for and making use of structure can support students in making connections.
previous experiences and apply • Students should recognize and identify structures from prior learning experiences and apply this
understanding in new situations. understanding in a new situation (NCTM 2014).
8 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
9.
Educators and leaders must continue to focus on creating resources as you and your leadership team create a culture
access to grade-level content and refrain from the use of of learning for all, implement equitable effective instructional
tracking, ability grouping, or other ineffective models to try strategies, plan grade-level learning experiences, and cultivate
to meet the needs of each and every learner. See Moving productive mindsets and practices.
Forward for more ideas on structures in the Structures That
As we continue to move forward, educators need to position
Organize Students for Instruction section or the Structures
all students as competent, confident, and capable learners and
That Support Teachers section.
doers of mathematics, while affirming their strengths every
Call to Action: What Can I Do Now? day. As educators focus on grade-level content and equitable
instruction, we are now called upon to advocate for a system
Consider the following productive ideas for equitable effective
that confronts problems and engages in advocacy strategies.
instruction (see Table 4). Then consider the questions and
TABLE 4 PRODUCTIVE IDEAS, QUESTIONS, AND RESOURCES FOR EFFECTIVE EQUITABLE INSTRUCTION
Productive Ideas
for Equitable Questions to Consider Resources
Effective Instruction
Invite students into • What intentional actions do teachers and teams take to Creating Classroom Communities –
the mathematics. understand their learners? Mathematics Teacher Focus Issue
• What reflection opportunities are created for teachers and teams Strengths-Based Teaching and Learning
to self-assess any potential deficit views or language that is in Mathematics: 5 Teaching Turnarounds
used to hinder developing positive mathematical identities? for Grades K–6
• How do teachers and teams create an effective classroom culture?
Make connections • How do the teacher and teams analyze current and previous The Math Pact Series: Achieving
explicit. grade-level content to identify connections across grade levels? Instructional Coherence within and
• What protocols or processes are employed to support across Grades
vertical coherence? Taking Action Series
• How do teachers and teams support vertical connections Reasoning and Sense Making in the
with the process standards? Mathematics Classroom Series
• How do we intentionally plan for the use of physical 5 Practices for Orchestrating Productive
(manipulatives), visual (drawings, charts, etc.), contextual Mathematics Discussions, 2nd edition
(provide a context/real-world scenario/application), symbolic
(use of numbers, expressions, and equations), and verbal
(have student dialogue/explain thinking)?
Ensure grade level • What opportunities are included in lessons for students to Principles to Actions: Ensuring
instructional experiences show responses in more than one way or using more than Mathematical Success for All
reflect the rigor of one representation? Taking Action Series
the standards. • How do students reflect on different strategies in each lesson?
Connect mathematical • How are teacher teams intentionally planning for students’ S3D: Fostering and Improving Small-Group,
content with engagement in the process standards? Student-to-Student Discourse
mathematical • What learning experiences are included to develop problem solving? 5 Practices for Orchestrating Productive
processes. • What tasks are used to create access to the content Mathematics Discussions, 2nd edition
(i.e. low-floor, high-ceiling tasks)? Good Questions
• What opportunities are intentionally created in a lesson for Mathematical Thinking
student-to-student discourse?
Cultivate instructional • What resources are useful in planning for supporting productive The Living Tree of Mathematics
mindset and practices. mindsets and practices? The Culturally Relevant Cognitively
• How are teacher teams addressing ways to disrupt potential Demanding (CRCD) Mathematical
biases and deficit thinking and language? Task Framework
• What opportunities are intentionally planned to create lessons Principles to Actions
and tasks that build on student strengths?
Taking Action Series
• How do teacher teams work together to rethink and reimagine
Reimagining the Mathematics Classroom
mathematics classrooms that engage all students in meaningful
mathematics, particularly those traditionally marginalized in their Mathematics Instruction and Tasks in a
school, district, state, or province? PLC at Work
9 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
10.
Planning for Advocacy create understanding, so too might advocates benefit from
appreciating those with opposing views as having common
As we continue our journey following a year of disruption,
aims built on their own realities that can emerge as new
we must rethink PK–12 mathematics education. Mathematics
and shared understandings about how the system works
educators and leaders have a unique opportunity to critically
(or should work). As you read table 5, review the comparison
examine historic and entrenched practices and advocate
between the Mathematics Teaching Practices and the
for policies, structures, and approaches to teaching that
Mathematics Advocacy Practices. Consider your current
support and promote high-quality, equitable teaching and
efforts and identify additional action steps to become a more
learning of mathematics for each and every student. Current
effective advocate.
systems marginalize too many students.
Therefore, dismantling inequitable structures
and racism must be foundational to our At the core of the Mathematics Advocacy Practices is the
advocacy efforts. In Moving Forward (NCTM
acknowledgment that advocacy is akin to teaching and
and NCSM 2020), mathematics teachers and
leaders are called on to advocate for teachers, learning more so than political theatrics and persuasion.
students, and “humanizing the development
and implementation of education policies and
practices” (p. 1). For further recommendations on advocacy from Moving
Forward, see the section labeled Teachers, Students, and
We must ensure that voices from all stakeholders are included
Humanizing the Development and Implementation of Education
in decision-making for a renewed approach to convey real
Policies and Practices. Call to Action: What Can I Do Now?
challenges associated with our systems, build understanding,
and generate solutions to meet the needs of marginalized Consider the following productive ideas for advocacy (see
students. Too often, decisions are made by homogeneous table 6). Then consider the questions and resources as you and
groups of adults who are not representative of students in your leadership team are planning for the upcoming school
terms of racial, ethnic, or gender identity. Our voices are year. In addition to these resources, be sure to investigate
important, individually and collectively. Building alliances NCTM’s Principles to Actions and books in the Catalyzing
among different groups of stakeholders effectively allows Change series or NCSM’s Framework for Leadership in
us to engage in advocacy work that supports all students’ Mathematics Education for guidance about key ideas for
equitable access to high-quality, meaningful mathematics. which you might need to advocate in your context.
Advocacy Strategies and Practices Federal and state or provincial programs support much of
the work described in this section. In light of the COVID-19
As you engage in advocacy in specific situations, reflecting on pandemic, additional funding sources may be available to
questions and considering responses to them are crucial actions: support schools to help students continue learning on grade
• What is the purpose of advocacy in this situation? What do level while supplementing the foundational skills necessary
you want to accomplish? for that on-grade-level learning. This should include funding to
• What communication is needed to achieve your purpose? support teachers in their learning and planning.
• Which audiences need to hear your messages? It is time to build allies and empower all voices to be heard.
• What are the key messages that you need to convey to Seek opportunities to advocate, such as talking with the
your audiences? media, testifying at the state level, writing letters to policy
• What are the most appropriate methods to communicate makers, presenting to local school boards, or communicating
your messages? with parents and families about mathematics education.
Resources such as the NCTM Advocacy Toolkit are available to
• What resources and professional networks can you draw assist you and your colleagues in this work.
on to support your efforts?
Consider the following productive ideas (see Table 6) as
Responses to these questions help you frame your direction you plan to advocate. Reflect on the questions and identify
for advocacy and next steps. We propose Mathematics possible resources to support your journey.
Advocacy Practices (see Table 5) to complement the
Mathematics Teaching Practices from Principles to Actions As educators and leaders, we can strengthen our advocacy
(NCTM 2014), positioning advocacy parallel to one’s role as an efforts to address shared concerns and understandings about
educator. At the core of the Mathematics Advocacy Practices grade-level content and equitable teaching practices. Our
is the acknowledgment that advocacy is akin to teaching and voices are important, individually and collectively. If we do
learning more so than political theatrics and persuasion. not engage now in the decision making and take action, then
others who may not have a clear understanding of mathematics
In the same way that a teacher’s aim to build confidence and education will determine for us what is valued, supported, and
capability in students can benefit from positioning learners as funded. The stakes are high and have implications for teachers,
sense makers who bring their own experiences to bear as they administrators, students, and families in every community.
10 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
11.
TABLE 5 MATHEMATICS TEACHING AND ADVOCACY PRACTICES
Mathematics Teaching Practices Mathematics Advocacy Practices
Establish mathematics goals to focus learning. Effective teaching Establish advocacy goals to focus efforts. Effective advocacy
of mathematics establishes clear goals for the mathematics that establishes clear goals to promote mathematics education, situates
students are learning, situates goals within learning progressions, and goals within equity frameworks and evidence, and uses the goals to
uses the goals to guide instructional decisions. guide approaches and accomplish purposes.
Implement tasks that promote reasoning and problem solving. Implement communication and approaches that promote
Effective teaching of mathematics engages students in solving and mathematics. Effective advocacy engages stakeholders in
discussing tasks that promote mathematical reasoning and problem understanding problems that need
solving and allow multiple entry points and varied solution strategies. to be solved. It also requires inciting inspiration to pursue action.
Use and connect mathematical representations. Effective Use and connect mathematical representations. Effective
teaching of mathematics engages students advocacy engages stakeholders in considering different perspectives,
in making connections among mathematical representations to making connections, finding common ground, and co-creating
deepen understanding of mathematics concepts and procedures and solutions. Using mathematical representations is useful to illustrate
as tools for problem solving. problems, make arguments, and deepen understanding.
Facilitate meaningful mathematical discourse. Effective teaching Facilitate meaningful discourse. Effective advocacy facilitates
of mathematics facilitates discourse among students to build shared discourse among stakeholders to build shared understanding of
understanding of mathematical ideas by analyzing and comparing problems that need action by analyzing and comparing data and
student approaches and arguments. arguments.
Pose purposeful questions. Effective teaching of mathematics uses Pose purposeful questions. Effective advocacy uses purposeful
purposeful questions to assess and advance students’ reasoning and questions to assess situations, highlight inequities, bring awareness to
sense making about important mathematical ideas and relationships. issues, challenge perspectives, and advance action.
Build procedural fluency from conceptual understanding. Build fluency from conceptual understanding. Effective advocacy
Effective teaching of mathematics builds fluency with procedures on builds fluency with creating conditions to identify problems, analyze
a foundation of conceptual understanding so that students, over time, data, generate representations, build understanding, and pursue
become skillful in using procedures flexibly as they solve contextual action.
and mathematical problems.
Support productive struggle in learning mathematics. Support productive struggle in advocacy efforts. Effective
Effective teaching of mathematics consistently provides students, advocacy provides stakeholders, individually and collectively, with
individually and collectively, with opportunities and supports to opportunities and support to engage in productive struggle as
engage in productive struggle as they grapple with mathematical they grapple with problems, challenges, and constraints related to
ideas and relationships. promoting mathematics education.
Elicit and use evidence of student thinking. Effective teaching Elicit and use evidence of student thinking. Effective advocacy
of mathematics uses evidence of student thinking to assess progress uses evidence of student thinking to assess access and opportunity
toward mathematical understanding and to adjust instruction to learn and identify systems and structures that serve as barriers to
continually in ways that support and extend learning. mathematical learning and understanding.
11 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
12.
TABLE 6 PLANNING TO ADVOCATE
Productive Ideas
Questions to Consider Resources
for Advocacy
Identify possible • What are you passionate about? NCTM Position Statement: High Expectations
challenges or barriers. • What are obstacles that hinder mathematics teaching, NCTM Catalyzing Change Series
learning, and leading? NCTM Social Justice and Equity Resources
• What problems, challenges, or inequities need action
in your setting?
Build alliances. • Who are colleagues, both locally and nationally, who NCTM Advocacy Toolkit
can join or support your efforts? NCTM Position Statement: Access and Equity
• How do you build relationships with district data/ Catalyzing Change Series
assessment and strategic communication personnel?
• Who are the key teachers in your state who have
access and are influencing state policy?
Use the Mathematics • What do you want to accomplish in your context? Moving Forward
Advocacy Practices to reflect • What communication is needed to achieve your NCTM Position Statement: Closing the
and begin conversations in purpose? Opportunity Gap
your communities. • Which audiences need to hear your messages? NCTM Position Statement: Mathematics in
• What are the key messages that you need to convey Early Childhood
to your audiences? NCSM Position Statement: Formative Assessment
• What are the most appropriate methods to
communicate your messages?
• What resources and professional networks can
you draw on to support your efforts?
Pursue professional • What book studies can you lead or participate in Back to School resources
development opportunities related to advocacy in mathematics education? Catalyzing Change and Principles to Action Book
to continue to develop • What resources and professional development Studies
advocacy competencies. offered by NCTM, NCSM, and ASSM can you pursue? NCTM Position Statement: Curricular Coherence
and Open Education Resources
NCSM Position Statement: Tier 1 Instruction
NCSM Position Statement: RtI and Tier 2
Instruction
NCTM Webinars for PD
Videos and Tasks for PD
Identify resources that need • What are talking points that would be useful? NCTM President’s Message: Advocacy as a
to be compiled or created • What representations, images or student work Mathematics Education Community—The
to support your particular would illustrate the issues? Time Is Now
advocacy needs. NCTM Advocacy Resources
Use your voice and talent. It • Who needs to hear your message? NCTM Advocacy Toolkit
matters. • How might you invite them to the table? NCSM and TODOS Joint Position Statement: Social
Justice and Equity in Mathematics Education
12 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
13.
Your Role in Continuing to Move Forward As we continue the journey emerging from the pandemic,
these types of situations will require educators and leaders
The current system of mathematics education is not working
to build strong alliances as we work together to dismantle
for all students. The global pandemic in 2020 and 2021
inequitable structures and racism in mathematics education
highlighted the persistent inequities within the system.
that have hindered too many students for far too long. Strong
Mathematics educators and leaders continue to face
pressures, challenges, changes, and opportunities
that influence the teaching and learning
of mathematics (e.g., the adoption of new For mathematics educators to collaborate with others is
mathematics curricula, technology tools, state vital to meeting the needs of each and every learner.
or provincial standards and assessments, or
district policies).
Continuing the Journey: Mathematics Learning 2021 and advocates are equipped with the knowledge and the resources
Beyond focuses on three key ideas that mathematics to engage in important conversations. It is imperative that we
educators and leaders must address. As we adhere to all work together in advocating for high-quality mathematics
on-grade-level content through equitable, effective teaching for each and every student and support for classroom
practices and engage in effective advocacy, we can ensure teachers, families, and communities in this work.
equitable outcomes for each and every student.
13 | Continuing the Journey: Mathematics Learning 2021 and Beyond • July 2021
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Copley, Juanita, Karen Karp, and Barbara Dougherty. 2017. Putting Essential Understanding of Geometry and Measurement into
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Hammond, Zaretta. 2015. Culturally Responsive Teaching and the Brain. Thousand Oaks, CA: Corwin.
Holton, Derek, and David Clarke. 2006. “Scaffolding and Metacognition.” International Journal of Mathematical Education in
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