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[Final Shared] A Space for All of Us_ Setting Up the Classroom Environment.pdf

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Share phrases or sentences from the position statement that seems particularly important or that stood out to you.

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As you review the set of images of classroom environments, discuss and jot down your noticings and wonderings about these images.

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I noticed in the K-2 environment that it is very hands on. A lot of manipulatives a lot of real-world engagement so that the students can learn the concepts and have a better grasp of what it is that they are learning.

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I noticed in the math environment 3-5 there are still manipulatives, but it is less physical whole-body integration. The concepts can be understood better at these grade levels, and they can understand the vocabulary in a way where the manipulatives can help in regard to the actual counting that needs to be done but the overall concepts can be understood.

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https://docs.google.com/presentation/d/1jAJRUoBKAfHVi3GKR9L2-j1OfJArFu5hd8fyjAtBO4U/edit?usp=sharing

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What did you notice about these spaces? What did you wonder?

What beliefs would a teacher need to have about mathematics to create and uphold these spaces?

What beliefs would a teacher need to have about their students?

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Discuss how these norms connect with the criteria for equitable, responsive classrooms.

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What is Math Identity and Why is it Important?

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What is Math Identity and Why is it Important?

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Which aspect of math identity had the greatest impact--either positive or negative--on your personal experience of math learning? How / why?

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Kids start learning math the moment they start exploring the world. Each skill - from identifying shapes to counting to finding patterns - builds on what they already know.

There are certain math milestones most kids hit at roughly the same age. But keep in mind that kids develop math skills at different rates. If kids don't yet have all the skills listed for their age group, that's OK.

Here's how math skills typically develop as kids get older.

Babies (ages 0-12 months)

Begin to predict the sequence of events (like running water means bath time)

Start to understand basic cause and effect (shaking a rattle makes noise)

Begin to classify things in simple ways (some toys make noise and others don't)

Start to understand relative size (baby is small, parents are big)

Begin to understand words that describe quantities (more, bigger, enough)

Toddlers (ages 1-2 years)

Understand that numbers mean "how many" (using fingers to show how many years old they are)

Begin reciting numbers, but may skip some of them

Understand words that compare or measure things (under, behind, faster)

Match basic shapes (triangle to triangle, circle to circle)

Explore measurement by filling and emptying containers

Start seeing patterns in daily routines and in things like floor tiles

Preschoolers (ages 3-4 years)

Kindergartners (age 5 years)

Move from using hands-on methods to using paper and pencil to work out math problems

Work with money

Do addition and subtraction with regrouping (also known as borrowing)

Understand place value well enough to solve problems with decimal points

Know how to do multiplication and division, with help from fact families (collections of related math facts, like 3 × 4 = 12 and 4 × 3 = 12)

Create a number sentence or equation from a word problem

Begin basic algebra with one unknown number (like 2 + x = 10)

Use coordinates to locate points on a grid, also known as graphing ordered pairs

Work with fractions, percentages, and proportions

Work with lines, angles, types of triangles, and other basic geometric shapes

Use formulas to solve complicated problems and to find the area, perimeter, and volume of shapes

Understand that numbers can be represented in many ways (fractions, decimals, bases, and variables)

Use numbers in real-life situations (like calculating a sale price or comparing student loans)

Begin to see how math ideas build on one another

Begin to understand that some math problems don't have real-world solutions

Use mathematical language to convey thoughts and solutions

Use graphs, maps, or other representations to learn and convey information

Remember that kids develop at different paces. Some may gain some math skills later than other kids or have some that are advanced for their age.

If you're concerned about progress in math, find out why some kids have trouble with math, and next steps to take.

About the author

About the author

Amanda Morin is the author of "The Everything Parent's Guide to Special Education" and the former director of thought leadership at Understood. As an expert and writer, she helped build Understood from its earliest days.

Reviewed by

Reviewed by

Brendan R. Hodnett, MAT is a special education teacher in Middletown, New Jersey, and an adjunct professor at Hunter College.

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What math skills do kids develop in different grades? Explore this list of math milestones children typically hit, from birth through high school.

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We asked our audience how they’re using the real world to teach math and compiled their most intriguing responses.

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Culturally Responsive Teaching

When students' eyes are opened to the broad community of mathematicians, they reject stereotypes and gain confidence in their own abilities.

What is mathematics? Who does mathematics?

I want my students to believe they can be mathematicians. But math is plagued by stereotypes, and many students think math isn't for people who look like them. As a math teacher, it's my responsibility to offer counternarratives that help students see themselves as mathematicians and expand what they think it means to "do mathematics."

Expanding Students' Ideas About Who Does Math

One place to start is math teacher Annie Perkins's Mathematicians Project. She has assembled and organized information on mathematicians who aren't "White dudes." I have followed in her footsteps, creating brief weekly presentations on mathematicians who aren't White men to share with my students. And Annie isn't the only person working to combat stereotypes about mathematicians. For Black History Month, Dr. Kristopher Childs shared information about a Black mathematician every day. Talithia Williams's excellent book, Power in Numbers, explores stories of women in mathematics. George Gheverghese Joseph's The Crest of the Peacock surfaces the non-European roots of modern mathematics. There are many more resources that share an inclusive vision of who does mathematics and what mathematics is, undermining stereotypical narratives.

Here's one example of a story I tell as a counternarrative. Colombian mathematician Federico Ardila-Mantilla is a geometer who researches efficient ways to move objects through confined spaces. In a 2019 article sharing an algorithm for efficient robotic movement, Ardila-Mantilla connected his research to the Black Lives Matter movement. Police departments have begun using lethal robots on people suspected of crimes. He worries, "What if it becomes trendy for mathematicians to start working on optimizing robots, but not to think about what is being optimized, or whom that optimization benefits?" Math is not only solving problems. Math is also deciding which problems are worth solving and understanding the consequences of those solutions. In math class there is often one right answer. In the world, less so. Ardila-Mantilla unpacks these tensions in his article, and I want to introduce my students to those same questions.

There are plenty more stories I can tell. Black mathematician Gladys West was a hidden figure whose pioneering behind-the-scenes work laid the foundation for modern GPS systems. Black data scientist Samuel Sinyangwe works to collect and visualize data on police violence and advocate for reforms. Mathematics is a rich place, and my students deserve to hear narratives that reflect the richness of what mathematics is and what mathematics can be.

I have two goals with my presentations. First, I want to share the work of mathematicians who are Black, Indigenous, people of color, women, queer, transgender, nonbinary, and more-people who my students might not expect to be mathematicians. Second, I want to share the richness of the discipline of mathematics and the many different ways that mathematicians imagine and contribute to a better world.

Expanding Students' Ideas About What Math Is

School mathematics often feels narrow, as if moving forward in math only means calculating longer and more complicated things. I try to find mathematicians who do work in different branches of math, fields that provide new perspectives on what it means to do and use math. Data science. Knot theory. Topology. Complex analysis. Virtual reality. Chaos theory. Hyperbolic crochet. I barely know what these things are-but I've enjoyed googling them, finding a few fun pictures to throw in a slideshow, and sparking student curiosity. The presentations are short and sweet, and they've worked well for me both in person and virtually. I share a bit about the mathematician and their work, show some pictures, and take questions. I often don't know the answers to students' questions, and that's fine by me.

Math is a human endeavor. Sharing pictures and stories from a wide range of mathematicians helps the discipline feel real, relevant, and accessible, especially to students who might not see their identities represented elsewhere. Most high school students know little about who mathematicians are and what mathematicians do. As a math teacher, I believe I have a responsibility to fill that void with a positive and inclusive vision of our discipline, dispelling the myth of a boring and irrelevant profession.

I'm a White man attempting to be an antiracist math teacher. There's a lot that I don't know. One important part of becoming antiracist is learning from others who are asking the same questions. If I try to invent everything myself, I'm likely to screw things up. Leaning on the wisdom of others can accelerate my growth as an educator.

As I look for new ways to help every student succeed in math, weekly mathematician presentations are a concrete place to start. Researching and preparing to share about a mathematician each week is an opportunity to reflect and hold myself accountable. What else am I doing to develop my skills as an antiracist educator? Where else can I work to undermine inequities in my classroom and in my school?

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How to Help Students See Themselves as Mathematicians

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When students' eyes are opened to the broad community of mathematicians, they reject stereotypes and gain confidence in their own abilities.

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How might these various elements contribute to a student’s sense of math identity?

How might an environment be designed with these elements (development, culture, real-world math connections) in mind?

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How is the environment designed to encourage extended exploration and access to mathematical supports?

How is the environment designed to promote peer collaboration and/or flexible groupings?

What might be done to make the environment even more supportive of collaboration, exploration and access?

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How is the environment designed to encourage extended exploration and access to mathematical supports?

How is the environment designed to promote peer collaboration and/or flexible groupings?

What might be done to make the environment even more supportive of collaboration, exploration and access?

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Boosting Confidence With Student-Driven Math

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How is the environment designed to encourage extended exploration and access to mathematical supports?

How is the environment designed to promote peer collaboration and/or flexible groupings?

What might be done to make the environment even more supportive of collaboration, exploration and access?

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Teaching Students to Prove Their Mathematical Thinking through Questions, Charts, and Discourse

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How is the environment designed to encourage extended exploration and access to mathematical supports?

How is the environment designed to promote peer collaboration and/or flexible groupings?

What might be done to make the environment even more supportive of collaboration, exploration and access?

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How is the environment designed to encourage extended exploration and access to mathematical supports?

How is the environment designed to promote peer collaboration and/or flexible groupings?

What might be done to make the environment even more supportive of collaboration, exploration and access?

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A Student-Centered Model of Blended Learning

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[Final Shared] Walkthrough Environments Tool

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"Acknowledging and addressing factors that contribute to differential outcomes among groups of students are critical to ensuring that all students routinely have opportunities to experience high-quality mathematics instruction," I believe this to be particularly important because in order for all students to be successful we must acknowledge that all students will not be on the same level and then it must be addressed if it isn't addressed then we have been of disservice to our students. Differentiation is key because in the end it is not about equality but about equity, not all students need the same thing to be successful.

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I know within this grade K-2 everything is well-label and visuals are used so if the students can't read they have the pictures to help them put things back in their correct spots. It is also hands on.

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"Creating, supporting, and sustaining a culture of access and equity require being responsive to students’ backgrounds, experiences, cultural perspectives, traditions, and knowledge when designing and implementing a mathematics program and assessing its effectiveness." I believe this statement stands out because it is essential to incorporate your student's background when lesson planning not just for math, but any subject. Children need to be acknowledged especially the fact that they're not all the same, they're not all on the same level and that's ok.

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I love how the K-2 is full of visuals and labels because unfortunately, some children can't read still when in certain grades so the visuals and labels are crucial for classrooms.

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Creating, supporting, and sustaining a culture of access and equity require being responsive to students’ backgrounds, experiences, cultural perspectives, traditions, and knowledge when designing and implementing a mathematics program and assessing its effectiveness.

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I notice how for grades 6-12 there are word walls, desks are in groups, a lot of peer interactions, hands on activities, etc.

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“No student should be asked to leave any part of who they are at the door. This includes race, languages, and ways of knowing, as well as the family and community knowledge and skills students bring with them.” This quote stood out to me because it is important for teachers to ensure that a piece from each child is incorporated into their classroom. This allows them to feel welcomed and a part of their classroom community. More importantly, children will feel welcomed and more at home when their interests and a part of their background are incorporated into their classroom. Teachers need to view math as an opportunity to make it interactive and fun because when it is engaging children will be engaged rather than unfocused/ uninterested. Integrating manipulatives and interacting activities can make math more welcoming and engaging for children. I love the idea of putting up their work around the classroom because this is what children love to see. They feel appreciated and proud when they see their work around the classroom.

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To create and uphold equitable spaces in my mathematics classroom, I firmly believe that mathematics is accessible to all students, regardless of their background or prior experiences. I understand that every student has unique strengths and challenges, and I am committed to recognizing and valuing their diverse perspectives and ways of thinking. I believe in the power of growth mindset, understanding that mistakes are opportunities for learning, and I actively promote a classroom culture where students feel safe to take risks and ask questions. I also believe in providing multiple entry points to mathematical concepts, using real-world contexts to make math relevant, and continuously reflecting on and adapting my teaching practices to meet the individual needs of my students. Above all, I hold a deep belief in the potential of each student to succeed in mathematics, and I am dedicated to fostering their confidence, curiosity, and a sense of belonging in my classroom.

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These norms not only promote a positive mindset towards mathematics but also align closely with the criteria for equitable, responsive classrooms. They create an atmosphere where students feel valued, supported, and motivated to engage with math, irrespective of their background or prior experiences, thus contributing to a more inclusive and effective learning environment.

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These norms connect because it covers the importance for an equitable and responsive classroom, its important for the educator and student to understand that understanding is more important than speeding through the content. Visuals are essentials because it allows students to make the connection to the concept being discussed and helps strengthen their brain comprehension.

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The aspect of math identity that had the most significant impact on my personal experience of math learning was "beliefs about one's self as a mathematics learner." These beliefs played a pivotal role in shaping my journey through mathematics. When I held positive beliefs about myself as a mathematics learner, my experience was much more positive. I felt confident, motivated to learn, and capable of overcoming challenges.

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The various elements of math identity contribute significantly to a student's sense of self in mathematics. Beliefs about oneself as a math learner shape confidence and willingness to engage, perceptions of how others view them influence their comfort level with participation, beliefs about the nature of math impact motivation, engagement in math activities fosters a sense of belonging, and seeing oneself as a potential math participant cultivates aspiration. Designing a supportive environment involves acknowledging developmental differences, embracing diverse cultural perspectives in curriculum and instruction, and making real-world math connections. By nurturing positive self-beliefs, providing inclusive opportunities for engagement, and highlighting the relevance of math in the world, students' math identities can be nurtured, leading to more confident, motivated, and participatory learners.

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NYCPS Math Collaborative site for additional resources

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NYCPS Math Collaborative site for additional resources

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New York City Public Schools Mathematics Collaborative

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To encourage extended exploration and access to mathematical supports in block play, the environment should include a variety of block types, sizes, and materials, as well as labeled containers or shelves for easy access.
To promote peer collaboration and flexible groupings, the block play area should have ample space for multiple children to work together comfortably. In addition, introducing collaborative challenges or projects that require teamwork and problem-solving can naturally encourage peer interaction.
To make the environment even more supportive of collaboration, exploration, and access, educators can incorporate dedicated areas for group projects with larger blocks or collaborative building materials. Providing opportunities for children to document and display their creations, such as a photo wall or written descriptions, can also foster discussions and exploration.

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To boost students' confidence and encourage extended mathematical exploration, the environment should provide easy access to a range of math-related materials, including manipulatives and visual aids, and incorporate clearly labeled areas or bins to promote independent inquiry.
To promote peer collaboration and flexible groupings, the environment should have versatile seating arrangements, collaborative workstations, and designated collaborative project areas, encouraging students to work together on math tasks and share ideas.
To make the environment even more supportive, educators can introduce structured group challenges that require cooperation, provide diverse math resources reflecting different learning styles, and regularly rotate materials to sustain students' interest and curiosity in mathematical exploration and teamwork.

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To encourage extended exploration and access to mathematical supports, the environment should include a variety of resources such as textbooks, digital tools, and reference materials readily available, with clear organization for easy student access.
To promote peer collaboration and flexible groupings, the environment can incorporate flexible seating arrangements that facilitate group discussions, provide interactive whiteboards or shared digital platforms for collaborative problem-solving, and encourage students to work in pairs or small groups on mathematical tasks.
To make the environment even more supportive of collaboration, exploration, and access, educators can establish a "mathematical discourse corner" with discussion prompts and open-ended problems, create a math library with resources for different skill levels, and schedule regular peer-led discussions or presentations to foster deeper engagement and collaborative learning.

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To encourage extended exploration and access to mathematical supports, the environment provides a diverse range of math resources.
To promote peer collaboration and flexible groupings, the environment incorporates collaborative workspaces, encourages regular partner or small group discussions, and employs flexible seating arrangements to facilitate dynamic interaction and idea-sharing among students.

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The environment is designed to encourage extended exploration and access to mathematical supports by providing a digital platform or learning management system with a wide array of resources, including tutorials, simulations, and practice problems, available for students to explore at their own pace and convenience.
To promote peer collaboration and flexible groupings, the environment integrates online discussion forums, collaborative project spaces, and virtual breakout rooms, allowing students to interact, share ideas, and work together on mathematical concepts regardless of physical location or time constraints.

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The environment is designed to align because it provides a wide range of digital resources accommodating different learning styles and needs.

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"Acknowledging and addressing factors that contribute to differential outcomes among groups of students are critical to ensuring that all students routinely have opportunities to experience high quality mathematics instruction, learn challenging mathematics content, and receive the support necessary to be successful." It's important to monitor student progress to see where they are struggling. We want to continue to build on their strengths while supporting what they struggle with.

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k-2 classrooms should have visuals and labels, and the classroom should be filled with warm colors and tools for students to access such as a calm corner.

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I wonder how the teacher planned to organize the classroom and what advice the teacher would give to new teachers entering their own classroom. Teachers need to believe in every single child that they are capable of mathematics and that when they are doing math, they are mathematicians.

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These norms connect by having empowering statements that encourage children and boost their self confidence in mathematics.

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Children's math environment should have some sort of hands on activities. This can be implemented using manipulatives, fake money, cash register, blocks, cubes, etc. I have seen classrooms that allow children to explore these things during their morning free choice before the pledge, so they're not only using the tools during math time.

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"A firm commitment to this work requires that all educators operate on the belief that all students can learn". I am a strong believer in the fact that students learn and absorb knowledge in different ways, so if a student is struggling with a topic or lesson, perhaps they just need a different method to learn it, whether that be a more visual or more tactile approach.

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The K-2 rooms seem very supportive of the different learning styles and places a lot of importance on the students doing the activities to learn. I see an active and more hands-on approaches to learning.

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These norms promote a growth mindset and lets the class know that everyone single student has the potential to reach the same levels of success, they might just need different supports to get there.

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The aspect of math identity that had the greatest impact on me was "beliefs about one's self as a mathematics learner. I struggled with math growing up and one of my teachers told me it was common sense, which ruined my confidence in my ability to learn math. I later learned that I am just a more visual/tactile learner; I need to see it and do it myself, have manipulatives, etc.

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One of the biggest things a teacher can do for a student is give them the confidence to know they can accomplish what they are working on; they can overcome the challenges they are facing. This also transfers to their confidence as a mathematician that faces challenges in understanding content.

An environment that is designed with culture and real-world connections in mind gives students a stronger understanding of how the content relates to them. Students are more likely to be interested in content that they can see actually impacts them, they can directly interact with, and that has some relationship to them and who they are.

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This environment gives students time to think on their own and then collaborate ideas with their peers. This type of setup invites collaboration and flexible grouping as students discuss their ideas through group discourse after independently considering the math problem at hand.

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This classroom environment gives students space to really think about and defend their answers to a mathematical problem. Students have time to brainstorm their ideas, hear their peers' views, and can collaborate with those ideas or use what they're hearing to strengthen their own. It is a good environment to argue, gain knowledge, and collaborate to share that knowledge.

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This environment allows for students to work at their own pace. Students who are ahead can move on and work on an exit-ticket, while students who need more time can continue the lesson and independent work. This setup promotes peer flexible grouping because, as shown in the video, the students working on the board were working together on the math problem, presumably because they were on the same section of the lesson.

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"Providing all students with access is not enough; educators must have the knowledge, skills, and disposition necessary to support effective, equitable mathematics teaching and learning."

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I noticed that the children were given the time and space to move about the room and use manipulatives to enhance their learning.

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"This collaboration is essential to ensure that all students have the necessary support to maximize their success in the mathematics classroom."

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I notice that each image is trying to initiate mathematical thinking.

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In my Math Childhood Education Program, we did a lot of work surrounding Building Thinking Classrooms, I wonder how many teachers are aware of the impact of where students work in the classroom in association with performance. It is amazing just seeing the students stand up and how much more engaged they are in the learning.
A teacher would need to believe that the classroom space they have is enough and their students can handle movement around the classroom, even if it can be chaotic. Teachers need to believe in their mathematics curriculum and the potential it has to be tweaked to meet student's needs.

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You can see that these values respect the needs of an inclusive classroom. The learning is not limited to the material provided but rather visuals and making mistakes throughout the problem-solving process.

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Achieving access and equity requires that all stakeholders— ensure that all students have access to a challenging mathematics curriculum, taught by skilled and effective teachers who differentiate instruction as needed; monitor student progress and make needed accommodations; and offer remediation or additional challenges when appropriate.

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I noticed that in grade 3-5 rooms, some students are sitting in small groups with an adult working on an assignment. There are a few students who are working on their own on the rug. All the math materials are labeled and organized in storage containers.

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Students are in small groups, and they are sitting together close enough that they can see and hear each other speak during discussions. Some students are given their own space to complete independent work on their own. A teacher must have the belief that mathematics is accessible to everyone and every student is capable of success in math.

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These math norms are very motivating and inspirational. They are meant to develop a positive mind set in the process in learning math content, and not the result.

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The aspect of math identity that has the greatest impact on my experience with math is my beliefs about myself as a math learner. Math has always been one of my strengths in school, especially when I received better grades in math than ELA. It boosted my confidence, and I was willing to be more engaged during math. It became my favorite subject because of my belief that I was good at it.

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As students advance to higher grade levels, they start to see real world word problems that involves critical thinking. If they see the math problem in action (cutting a pie and labeling the slices as fractions), they can relate more to the problem in text. This will increase their understanding for fraction, boosting their confidence in math.

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The teacher in the video uses cognitive guided instruction to encourage extended exploration of the problem by having the students work out the math problems using their own thinking. Answers are not given to them. Through questions and discourse student plan out a strategy own their understanding of the problem.

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"A firm commitment to this work requires that all educators operate on the belief that all students can learn." This stood out to me because as educators it is our job to ensure a level playing field for all of our students. All students are capable learning in their own way. We need to make the effort to really know our students to ensure we can support their success in the way that best suits each individual child in our classroom.

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Looking at the K-2 environments, I notice that there are manipulatives and materials clearly labeled and easily accessible to students. I also see that there are collaborative spaces for students to explore and learn together. There are multiple different variations of math being introduced and made available to students outside of simply using a book. Students are encouraged to move around and explore while still learning.

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I notice that the teacher in the article left space for her students to have creative freedom and say over their classroom. This creates a welcoming environment for students and allows them to really feel like the space is their own, not that they are in their teacher's space. A teacher would need to believe that students are capable of learning through exploration in order to create and uphold these spaces. Teachers need to be able to trust in the process and establish set boundaries and expectations so students are given the freedom to explore and learn in their own ways.

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These norms encourage a different viewpoint of math in highlighting that in can be fun and achievable by anyone. These norms closely relate to having an equitable responsive classroom in that they encourage everyone that they are capable of learning and accomplishing things that might seem challenging at first. It creates a safe space for everyone to learn and achieve.

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The aspect of math identity that I think has the biggest impact is the "beliefs about one's self as a math learner." I previously worked as a math teacher in a charter school, and have experienced first hand that many students come into math with the mindset of "I am just not good at math." To many people, math can be intimidating to learn. Those with a predetermined mindset might not open themselves up or allow themselves to really try to achieve great things in the subject. It is our job as educators to take those students with that mindset and show them that learning math can be FUN and that they are capable of it.