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Activity for Teacher Educators

Describe, Draw, Describe (DDD) is an activity that teachers can use to engage children in developing spatial relations ideas. The activity, described here, creates opportunities for children to describe what they see in an image, draw what they see, and then describe what they drew. The describing allows children to work on both seeing things in relation to one another and in space, and learning how they can describe those relationships. As they represent what they see, children put into action their thinking about the relationships to show what they see. The activity can be used in pieces, for example only asking children to describe an image or only asking them to draw what they see. Or the pieces of the activity can be used together to connect describing and drawing.

What we have outlined here can be used with preservice and inservice teachers to support their learning about engaging children in the DDD task, as well as what to look for in children’s spatial relations understanding.

This activity gives participants a chance to understand the mathematical work that children are doing when they describe spatial relations. We ask them to describe a given image, as well as consider what children would say. We want participants to become familiar with a set of spatial ideas in children’s descriptions that they can listen for and support.

Preparation. Whenever possible, it is best to try the activity out with real children first! You will be able to leverage your own experience with children and what you noticed about their participation and learning. Additionally, if you ask them to draw what they see, you can collect their work samples and explore them with participants (we have also provided you some work samples in the powerpoint, downloadable below.)  Additionally, participants should have access to the Describe, Draw, Describe handout, either electronically or in printed form.

Start by sharing an image with the participants, just as we would with children.
You can show an image using a document camera or overhead projector; you can share a page from a picture book, or even draw on the white board. Ask participants to “describe what you see.” The language of what you see is purposeful. We want participants to understand that there are multiple ways to describe and recognize, and that we may not all see an image in the same way. We encourage participants to describe what makes sense to them, in the language they have to describe it. The participants together will generate a range of ideas. You will see that one person’s noticings will stimulate other responses.

Use what was generated to sort the responses (what was noticed) based on the spatial ideas addressed.
You can allow the participants to create the categories or you can provide them. The chart below provides a set of spatial relation ideas you could use for sorting.

Once participants have talked about what is the same and different about the responses you can also determine what types of responses did not occur and consider why.

Repeat with a different image.

Provide the participants with a new image that is likely to get them to notice different spatial ideas. For instance, choose an image that shows movement or items stacked.

Follow the same procedure above. Then discuss what was the same and different about what was initially noticed. Try to find examples of images that will elicit responses related to each category in the chart above.

Use artifacts of practice to deepen engagement.

After participants have discussed the mathematical ideas that emerge within this task, you can deepen their engagement with Describe, Draw, Describe (DDD) and spatial relations by exploring artifacts of practice.

Student Work Samples. Look through a variety of student work samples of what children drew. You can use what you have collected yourself or some of the examples included in the powerpoint. As always, support participants to focus on what children know and what children can do.

Classroom video. Watch and discuss classroom video examples together to support participants in generating ideas of how they might take up this activity in their own classrooms.  Some videos and reflection questions you might use are available here.

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This activity gives participants a chance to understand the mathematical work that children are doing when they describe spatial relations. We ask them to describe a given image, as well as consider what children would say. We want participants to become familiar with a set of spatial ideas in children's descriptions that they can listen for and support.

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Same and Different Images

Lots of images to support students in developing the habit of comparing and connecting representations. Thanks Brian Bushart for pulling them together!

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Great tasks you can do in 10-15 minutes every day...click the image and they're yours.

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Mathematizing the World Routine – Advancing Equity and Strengthening Teaching through Elementary Mathematical Modeling

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Mathematizing the World Routine Planning Tools

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2026 Math Collaborative Summer Book Club

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2026 Math Collaborative Summer Book Club

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This year, Math Collaborative’s Summer Book Club will do a deep dive into a re-released favorite! We will take care of our minds and spirits and journey together through Elham Kazemi and Allison Hintz’s second edition of Intentional Talk and Listening: How to Structure and Lead Productive Mathematical Discussions.

We hope you join us each week or any week you can make it! Please let me know if you plan on attending! Please feel free to pass along this info to friends!

With Joy, Appreciation and Care,

Stef

Zoom info: https://ucla.zoom.us/my/stefmath 608-038-5114

https://www.routledge.com/Intentional-Talk-and-Listening-How-to-Structure-and-Lead-Productive-Mathematical-Discussions/Kazemi-Hintz/p/book/9781625316448

Wednesdays at 6:00 PM Pacific time

Dates             Readings

June 17         Chapter 1 - Introduction

June 24         Chapter 2 – Open Strategy Sharing

July 8             Chapter 3 – Targeted Discussion: Compare and Connect

July 22           Chapter 4 – Targeted Discussion: Why? Let’s Justify

July 29           Chapter 5 - Targeted Discussion: What’s Strategic and Why?

August 5       Chapter 6 - Targeted Discussion: Define and Clarify

August 12     Chapter 7 - Targeted Discussion: Troubleshoot and Revise - Final celebration!

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Here are 5 weeks of number sense routines (5-10 minute math talks) tailored to each grade K-5. Every slide deck includes a "how to" and a week each of:

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Always, Sometimes or Never? Number | NRICH

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Are these statements always true, sometimes true or never true?

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Which One Doesn't Belong or Which One Is Unique?

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© site Copyright 2024 Talking Math with Your Kids • Saint Paul, MN, 55114 Rights to individual sets controlled by the named authors

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Dr. Chris Emdin: Why 'Ratchetdemics' Can Be Revolutionary for the Youth

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Awakening Student Agency Through A Pedagogy of Voice Introduction

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Awakening Student Agency Through A Pedagogy of Voice Introduction

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Awakening Student Agency Through a Pedagogy of Voice

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2025 Math Collaborative Summer Podcast Club

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2025 Math Collaborative Summer Podcast Club

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This year, Math Collaborative’s Summer Book Club takes a twist! We will take care of our minds and spirits in anticipation of the new book, Pedagogies of Voice by Shane Safir, Marlo Bagsik, Dr. Sawsan Jaber, and Crystal M. Watson (release date July 29, 2025, POV30 coupon code on Corwin https://www.corwin.com/books/pedagogies-of-voice-288927). Two years ago, we read Street Data (Safir & Dugan) and this year, our Summer Book Club reinvents itself as our Summer Podcast Club. Listen to the selected podcast each week and join us for a thoughtful and honest conversation about the topic. We hope that while you listen to the selected weekly podcast on Street Data Pod (Spotify), you also take care of yourself by exercising, relaxing or multi-tasking. We will come together on Wednesday evenings to chat at 6:00 PM Pacific beginning June 25, 2025 through August 27, 2025. The meetings will likely be about 45 minutes. Lenses / questions for each podcast have been posed below.

Once the book is released, we will discuss if we’d like to continue exploring through the school year or have it be our focus next summer.

We hope you join us each week or any week you can make it! Please let me know if you plan on attending! smathewson@me.com

2025 Math Collaborative Summer Podcast Club

Author intro https://youtu.be/yQ8JACNmhDo

Cult of Pedagogy intro (May 21, 2025)

https://www.youtube.com/watch?v=IuDbC42AY48

https://shanesafir.com/street-data/pod/

Zoom ID: 608 038 5114 password: mathlab

REVISED

Dates             Podcasts, lenses/questions

June 25         Street Data Pod #2

Headlines: create a headline that summarizes the impact for you.

July 2             Street Data Pod #4

Golden Line: which phrase/idea resonated most with you?

July 9             Street Data Pod #6

What are your thoughts about marginalizing standardized tests?

July 16           Street Data Pod #15

How did this change the conversation for you?

July 24***      Street Data Pod #17

How might we create learning spaces that are also “home?”

July 29           Pedagogies of Voice released

July 30           Street Data Pod #26

Immediate Impact: what will you change right away?

August 6       Street Data Pod #27

What is your “good question?”

August 13     Street Data Pod #28

What made you take pause?

August 20     No meeting

August 27     Street Data Pod #31 / future plans

What is sticky for you?

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In Episode 4, co-author Jamila Dugan is back and giving us the inside scope on equity traps and tropes. First, we dig into how this chapter came to be (spoiler alert: from a rant!) and the conversation shifts to the luminous landscape of radical dreaming, exploring, in Jamila's words: "What does it actively mean to freedom dream and who am I dreaming with?" Shane, Jamila, and Alcine think about how to live a life of big dreams and abundance, and the ways that hustle and grind culture often dims our dreams. Jamila shares some brilliant tips, like reverse calendaring and-drum roll-taking the email app off your phone!
For Further Learning:

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Hi, I'm Shane Safir ... and I'm Alcine Mumby... and this is Street Data Pod, where we dream with you about next-generation schools that affirm, love, and value every learner. Here, we have conversations about healing, hope, and listening at the margins.

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https://www.forward-ed.com/post/curriculum-as-covenant

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If you're looking for a super fun, indoor activity for the kids, try making this Easy No-Cook DIY Playdough! It so soft, squishy and takes less than 10 minutes to make!

Thank you, Sprouts Farmers Market, for sponsoring this post! As always, all thoughts and opinions are truly my own.

I don't know about you, but my kids LOVE playdough!! Every time I leave the house to run a few errands, my kids always ask if I can pick up more playdough, which can get a little pricey. Since we are spending more time at home these days, I've decided to save a little money and make my own No-Cook DIY Playdough!

I've experimented with several playdough recipes, and I must say that this recipe is my favorite. Not only is it quick and easy to make, but it's also the perfect consistency for hand strengthening, which is beneficial for preschoolers.

This Easy No-Cook Playdough only requires a few ingredients, such as flour, oil, and salt, that are located right in your pantry! During my most recent shopping trip to Sprouts Farmers Market, I stocked up on several of these items. Sprouts always has everything that we need for our dinner recipes and DIY crafts, and it's always well-stocked!
HOW TO MAKE NO-COOK DIY PLAYDOUGH
I created this recipe to make small batches so it would be easy and quick to make several colors. Each batch makes about one cup of play dough.

SUPPLIES:
* flour
* salt
* cooking oil
* food coloring
* large mixing bowl
* measuring cups and spoons
* aluminum foil, parchment paper or wax paper
* wooden spoon for stirring

Start by combining the flour and salt in a medium bowl. Slowly stir in the oil, warm water, and about 2-3 drops of food coloring until the mixture is well combined.

Transfer the dough to a flat surface and let it sit for a few minutes. It may feel a bit sticky, but whatever you do, DO NOT add more flour! As the dough sits, the salt will absorb the extra moisture.

Let the dough sit for about 5-7 minutes, before kneading. If your dough is sticky after sitting, add an additional 2 tablespoons of flour at a time until it is the perfect consistency.
HOW TO STORE YOUR DIY PLAYDOUGH:

Store in a zip-lock bag or airtight container, and place in the refrigerator. If you notice the playdough getting a bit dry, you can add a few drops of water and knead that into the dough.

The kids loved this DIY Playdough and couldn't believe that their very own mommy actually made it! Aside from this playdough recipe being so extremely affordable, it's absolutely perfect! It holds its shape beautifully, but without being crumbly, and requires zero cooking.

Go ahead and give it a try and be sure to let me know how it turns out!

Easy No-Cook DIY Playdough

Little Eats & Things

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Easy No-Cook DIY Playdough - Little Eats & Things

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If you're looking for a super fun, indoor activity for the kids, try making this Easy No-Cook DIY Playdough! It takes less than 10 minutes to make!

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What does mathematical reasoning look like?

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What might happen if we focused on the verbs rather than the nouns of mathematics?

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Partitive Division Problems (Life Science Contexts)

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Partitive Division Problems (Life Science Contexts)

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10212024 Cotsen PDiv USE DOWNLOAD Unpacking Slides

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We lead with the heart and courageously work to address racial injustice in education, embodying Black joy, confronting anti-Blackness, and building authentic partnerships with all who support creating equitable school communities across the nation.

​In September 2016, DeShawn Jackson, along with Black Men Uniting to Change The Narrative, planned a day of encouragement. The previous school year, another local elementary school had more than 200 Black men show up to greet students as they arrived at school. The men wore their uniforms, regalia, and Sunday best in hopes of disrupting the inaccurate imagery students have of Black men. Black men make up only 3% of teachers in k-12 education.​

DeShawn had taught at the school for six years and even attended the elementary school as a child. He wanted students to see Black men in a positive light, not like they often are portrayed in media. A bomb threat was called into the school on the day of the event, so school leadership officially canceled it. Yet, after the threat was investigated, nothing was found, and the school remained open. So, DeShawn personally greeted students and about fifty others joined him.

When students arrived at John Muir Elementary School, they were greeted with cheers and high-fives while drums played in the background. Black men showed up because they wanted to counter the perception students have about education and race. Inside the school, staff wore shirts that said "Black Lives Matter We Stand Together."Julie Trout, the art teacher, designed the shirts. Teachers' lessons for the day were focused on racial justice.

​Social Equality Educators (SEE), a group within the Seattle teachers union, decided October 19th would be "Black Lives Matter At School Day." They'd wear Black Lives Matter shirts to school, created a second design with "#Say Her Name," and educators picked from several options. Parent Teacher Associations made buttons and stickers for teachers and students to wear. This attracted national news, which spread to Philadelphia, where the Racial Justice & Organizing Committee of the Caucus of Working Educators was inspired to plan a similar action.

We envision a nation in which we live in authentic and conscious relationship with ourselves, each other, and the lands we occupy with the least amount of harm and violence, in which we are unapologetically Black and collectively love and care for one another as extended intergenerational families. To achieve safety, equity, and liberation for all Black lives, we attend to systems and networks of care to make the domination, erasure, and dehumanization of Black life obsolete. ​

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National coalition organizing for racial justice in education.

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Resources below are from Teaching For Racial Equity by Perry, Zemelman, Smith

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Resources below are from Teaching For Racial Equity by Perry, Zemelman, Smith

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Procured by Vanessa Heller, pages 204-210

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"Generation Mixed" refers to kids born after the 2000 Census, which for the first time allowed respondents to "Mark One or More" options to describe their racial identity. Join this conversation about the experiences of Generation Mixed at school - from the perspectives of the kids themselves and the caregivers and teachers trying to support them. Professors and social justice movers Drs. Ralina Joseph and Allison Briscoe-Smith have been in community with many mixed-race kids and the adults in their lives, personally and professionally. They've collected their observations and insights in their book, Generation Mixed Goes to School: Radically Listening to Multiracial Kids. They delve into "how the silencing and invisibility of their experiences often create a barrier for mixed-race kids to engage in nuanced conversations about race and identity in the classroom," and also talk to caregivers and teachers who have found effective ways to connect with and support their mixed-race children or students. They joined us to share what they've learned. As always, we take questions and comments from the EmbraceRace community - that's you! Check out the related resources below the transcript.
EmbraceRace: You all probably know Allison Briscoe-Smith, who has been on with us many times since 2016, ever since the beginning when we didn't have that many folks tuning in. She, along with Ralina Joseph, wrote this great book called Generation Mixed Goes to School: Radically Listening to Multiracial Kids.

And what's great about this book and about what you'll hear tonight is that these doctors, we'll call them doctors because they are doctors, have really listened to kids and they'll bring us some of that tape, the voices of multiracial kids, them speaking to their experience.

Dr. Allison Briscoe-Smith is a clinical child psychologist who specializes in trauma and issues of race. She combines her love of teaching and advocacy by serving as a professor and directing mental health programs for children experiencing trauma, homelessness, or foster care. An adjunct professor at the Wright Institute, much of Allison's work is with Bay Area schools and nonprofits as a clinician, consultant, and trainer. Most importantly, most impressively, she is a multi-time guest on Talking Race and Kids. Five times, people, more than anyone else! Really great to have you back, Allison. Welcome.

Dr. Allison Briscoe-Smith: Thanks for having me.

EmbraceRace: Introducing Dr. Ralina Joseph. Dr. Joseph is a scholar, a teacher, and a facilitator of race and communication. She's Presidential Term Professor of Communication, founding an acting Director of Center for Communication, Difference, and Equity and Associate Dean of Equity and Justice and Graduate programs at the University of Washington, Seattle. Ralina is the author of numerous articles and three books. Most recently, with Dr. Allison Briscoe-Smith, Generation Mixed Goes to School, which just came out this year. She's currently writing, Interrupting Privilege, a book of essays based on her public scholarship. Glad to have here as a first time guest. Welcome.
And let me start with the question we typically start with, which is, in this case Generation Mixed Goes to School: Radically Listening to Multiracial Kids. Professionally or personally with us, what brought you to that topic?
Dr. Allison Briscoe-Smith: I mean, I think that the simplest answer was that this was a braiding in of our experiences of our research, our service, and our personal lives. Ralina and I, in addition to being, colleagues and co-authors, are beloved friends and sister friends. We love each other's families, take care of each other's families, are godparents for each other's families. And what brought us to this book was an opportunity to actually center the experiences of our children and to think about our kids who are experiencing and living through the world as multiracial folks.

So it was, again, we used this analogy of like a braid of moving things together. And so there was a professional space with Ralina as a scholar in this kind of space around Critical Mixed Race Studies, understanding and thinking about the scholarship. There was my opportunity in thinking about things like implicit bias and clinical skills, but most importantly, we wanted an opportunity to spend more time with each other and to listen to our kids and kids like them. And these were the places that we were curious. What are kids going through? What do they want from their education? How are we armed and equipped to serve them? So that's how I would think about it, but a big, big excuse is spend more time with each other, which was also much needed.

EmbraceRace: That's awesome. Ralina, I'm sure you would agree with much of that. Ralina, what would you like to add about what drew you to the work that you do?

Dr. Ralina Joseph: No, absolutely. And before COVID, we did write much of this on writing retreats together. So it was just absolutely the best excuse to have to see each other. And yeah, my first book was on mixed race. I went to graduate school to study mixed race. My first identity is as a Critical Mixed Race Study scholar and Dr. James Banks, who is the father of multicultural education, and the one who started this wonderful Multicultural Education Series where we published in, had approached me about writing this book. And I initially said no, because I couldn't imagine doing it by myself.

And when I thought about the opportunity that it could be about our conversations, that actually we had legitimacy as scholars in this space, as well as mothers. And I think some of what makes this book really so special is that we have this really personal voice in there. The children's voice is personal. Our voice is personal. And, as Allison said, we're braiding in these literatures that don't often talk to each other.

EmbraceRace: Can you explain the title a bit, of why Generation Mixed At School, and what you mean by radically listening?

Dr. Ralina Joseph: Sure, yeah. So I mean the first part of this is we're looking at this post-"mark one or more" generation, right? This new 2000 census [in which people could first the first time choose to check multiple races]. This different moment of racialization. We can think about a post loving moment, right? And then kind of post mark one or more moments. So this felt this was really kind of a sea change in understanding how race and mixed race was articulated by a different group and then the radical listening piece. So the center that I direct, the Center for Communication, Difference and Equity, we've been engaged for a number of years in these radical listening projects, where we bring different communities in to share their stories around any number of topics. So we started off with the first time you've experienced discrimination. We have stories of Black Seattleites.

This last year, we did things around Black folks in quarantine, for example, that was called Quarantining while Black. And so it felt like this was a methodology. So it's about how we reciprocally bring people in. They get to choose who they'd like to speak with so we don't dictate who is in conversation with each other. They get to actually choose that. There's also choice around the questions, so they receive the questions in advance and they were able to script the questions as well. There's also choice then in that they receive the entire script of the interview and they're able to say, "I want this highlighted. I don't like this highlighted." And then finally, we come together as a community to listen to the pieces and the participants actually get to invite in family, friends, and other people who they really want to hear their stories. So it's a different way of really furthering listening, and engagement for people whose stories don't always get told.

EmbraceRace: I love that. And so we're going to listen to a piece of tape from your project, and I wonder if you want to set it up at all.

Dr. Allison Briscoe-Smith: So I mean, the thing that we want to highlight is what we did is we brought, as Ralina was talking about, but in practicality, we brought in kids with their siblings and their families to talk to each other. And so what you're going to hear here are two kids getting a chance to talk to each other. I kind of want to leave it so that you can kind of hear it, but you'll hear that these are young children who are asking each other questions about how they understand these concepts and we feel it's a beautiful window into understanding what's going on for kids. So I'll key it up in that way and we can talk a little bit about it after.

"Grace vs. Race" audio clip (whole clip followed by transcription)

Dr. Allison Briscoe-Smith: So we were asking a lot of different questions about race. Do you guys understand what race is?

Older Sibling: Yeah.

Younger Sibling: Race. Yes.

Dr. Allison Briscoe-Smith: What is it?

Younger Sibling: It's something when you do on dinner or something, and then when you are at church.

Older Sibling: No, that's grace, not race! Race is color, skin color-

Younger Sibling: Race. Oh, race! So it's a skin color that people have.

Dr. Allison Briscoe-Smith: Yeah. And what kind of skin color do you guys have?

Younger Sibling: I have Asian skin color, kind of Black. I don't know what I have.

Dr. Allison Briscoe-Smith: Yeah. And what about you? What do you have?

Older Sibling: I have a yellowish White.

Dr. Allison Briscoe-Smith: Yeah.

Older Sibling: Kind of a dark tan.

Dr. Allison Briscoe-Smith: So is that a question that you've ever gotten before? Where someone asks what's your race?

Older Sibling: Yeah.

Younger Sibling: No. Never before.

Dr. Allison Briscoe-Smith: Never before. So what do you say when people ask a question about you?

Older Sibling: I usually tell them about my parents.

Dr. Allison Briscoe-Smith: And what do you say?

Older Sibling: I tell them that my dad is mixed. My mom is Korean.

Dr. Allison Briscoe-Smith: Mhm. And your dad is mixed, what?

Older Sibling: I don't... I think... I can't remember everything.

Younger Sibling: His dad's mixed. He's White. He's American. He-

Older Sibling: I can't remember everything, so you're going to need to help me.

Younger Sibling: He's Filipino. He's Native American. And then he's Asian.

Older Sibling: Yeah, he's Asian.

Younger Sibling: And then he has a lot of relatives.

Older Sibling: Yeah. He has a really lot of them.

Dr. Allison Briscoe-Smith: Yeah. So that's a great understanding of what your parents are, but what-

Older Sibling: I think I'm just Asian.

Dr. Allison Briscoe-Smith: Just Asian? And how do you think about yourself when you think about race?

Younger Sibling: Well, I think I am Asian, American. And then different kinds because I have a long, long line of family.

Dr. Allison Briscoe-Smith: Yeah, definitely.

Audio Clip Ends

EmbraceRace: Yeah, that was great. And so much to say about that. How is this audio clip typical or not typical of sort of what you've been hearing broadly or what of themes that you've been hearing from mixed race kids?

Dr. Allison Briscoe-Smith: I mean, I love this clip so much. We had an opportunity to hear so many different kids kind of talk and think together of it. I can't help, but smile just because they were ridiculously cute kids. You can actually see something that we heard a lot, which is how siblings are co-constructing their identities together. And so that's actually something that I don't think we thought we would find or thought we would hear, but we got to see that siblings are teaching each other about it. It's also funny, right? And it also gives us a window into little kids are making sense of the world around them. Yes, in terms of race, but also it's confusing. Right? It's moving. It's these different kinds of pieces.

I love the way he talks about nationality and race and history. So I think what it gives us a window into is the kind of complexity that's there, but also how family is co-constructing. It wasn't as if these kids didn't have a sense of, as he says, "the long history, the lots of people." Also, they had a claimed sense of identity. They had multi multiracial, and there was still a claimed sense of an Asian identity, an Asian skin color. So I could go on and on, but I think the thing that this shows is, I'll speak about the research, is it's pretty consistent with the research.

Dr. Ralina Joseph: They're nine and 10 years old. I actually had just looked in the back of the book in our index. It was in December, yeah.

Dr. Allison Briscoe-Smith: So they're little in some ways, but this is consistent. The kids are trying to make sense of race, that they are looking at it, they're being impacted by it. But also the new kind of piece in terms of the literature is actually speaking about the ways that kids are co-constructing their identity. And what you see here is the opportunity for the older child to scaffold and to help. And the key part is to do this in the context of joy and connection. There's laughter, and what you can kind of hear in the background is they're coloring. They're coloring in the midst of all this. They're doing all of this kind of piece in the background. So I think it's a great window to hear that these conversations are happening in family. There is joy as much as there's been, within the research, a sense of kind of over pathologizing the experiences of multiracial kids. There are places of joy and resilience and support for each other. So that's one of the reasons why I love this piece so much.

The literature is actually speaking about the ways that kids are co-constructing their identity. And what you see here is the opportunity for the older child to scaffold and to help. And the key part is to do this in the context of joy and connection... These conversations are happening in family. There is joy as much as there's been, within the research, a sense of kind of over pathologizing the experiences of multiracial kids. There are places of joy and resilience and support for each other.

Dr. Allison Briscoe-Smith

EmbraceRace: Yeah. I can really relate to it as a multiracial person, just the question you get, "What are you?" when people have those curiosities, just going right to your parents and going to nationality and, "Well, my mom... Well, my..." And, Allison, I think that with you asking, "But what about you?" And I think so often with mixed race kids, it's really challenging, or they're not supported in the "what about you" part, right?

Dr. Ralina Joseph: I just wanted to give a quick shout out to our Research Assistant who was doing the recording there. We actually just happened to have two multiracial research assistants as well. This was Anjuli Brekke, who's now a Professor at University of Wisconsin Parkside, and the other Research Assistant, Michelle Sturgis, who did a tremendous job here with this work and really carefully clipped out, as you heard, the clip and the music, that was Anjuli. But they really did a wonderful job with this caring work.

One of the unexpected things that we came into this book having a very clear picture of, we want to write it this way. We kind of narrate that in the beginning. And of course, we end up letting our data speak to us, and it told us a very different story than what we wanted to write. And one of the stories was about siblings. And we didn't realize there was going to be so much here about siblings and how multiracial siblings were really teaching each other in a lot of ways more than what their parents were teaching them or even the schools were teaching them.

And it was particularly interesting in moments when the kids were racialized differently because there were a number of conversations that we heard where one child was talking about, "Well, you're racialized this way. I'm racialized this way. This happens to me. This doesn't happen to you. People ask you what you are. People assume that you're Black. They don't assume this." And they were figuring it out together. And so we've felt really honored to have that view into the sibling lives in ways that really aren't reflected by the literature. So we felt like this was a really interesting piece that we're bringing. And these are the littles here, but we hear it kind of more exposed in some of the teen pieces, and I think we're going to hear another piece that shows us kind of an older sibling pair as well in a little bit.

EmbraceRace: So these two, you said, Ralina, were nine and 10 years old, right? So they're pretty close in age. And you mentioned that, of course, there could be many differences across siblings, between their age, differently racialized or not.
Did you find that the younger had as much to do, had much power in the co-creation as the older child generally?
Dr. Allison Briscoe-Smith: I mean, we heard about this in kind of different ways. What I was really impacted by was actually how much weight the older had, how much of a burden the older had, often. So it wasn't in that there was a lot of kind of responsiveness to what the younger was bringing, but there was a kind of tendency that I saw and actually heard about narrated both by younger children or the younger sibling, which is, "I learned a lot faster," or, "I had had to do this much quicker," or "I had the opportunity to learn because there was someone ahead of me." The other piece to add to the complexity is that these children are racialized differently. We have, as was kind of mentioned, and this was the part that they were talking to each other about. How do you understand your race? How do identify?

So I want to be clear. We weren't asking them, "What are you?" I saw that within the chat. That's not what the question was. The question is, "How do you understand this? And what does this mean for you?" So the sibling kind of piece, and actually we also had other people and family members, "My auntie, my cousin." There was this kind of familial socialization, this familial holding. But I'd say in general, I did find that there was both a burden and expectation on the olders and also this kind of general narration that the younger ones were kind of coming up quicker in part because they seeing what was happening for their older sibling. And then again, holding onto the big variation that we have, which is that kids and their families got racialized differently according to how they looked, how they presented in their own identity.

EmbraceRace: Right. And I know from reading the book as well that not only did you not ask, "What are you," they got to choose their different prompts and you chose people to talk to based on their having parentage that was varied, as opposed to kids who themselves identified as multiracial. Is that right?
Did you find that the kids were more accepting of having different racial identities as siblings than sort of parents were or families?
Dr. Ralina Joseph: Yeah, yeah. Absolutely, absolutely. I think that our kids were really attuned to how they were racialized, and so it really went alongside, "I'm racialized as Black, and so this is a part of the way in which I go about the world." We also want to be clear that racialization shifts, so for some kids, it was a different type of racialization at school and a different type at home. And this fluidity, while it might seem confusing to adults and perhaps to monoracial adults, was not confusing to the kids. They understood how they were navigating this space. And we had some older teens who talked about, "As an elementary school kid, I identified particularly multiple minority kids, so kids who were Filipino and Black, for example. As an elementary school kid, I identified as Filipino, and then I identified as Black and now I identify as Blasian, and all of these identities make sense to me. But then sometimes I'll go into a space, and I'll say that I'm actually back what I used to identify as."

And this was kind of a singular and a continuous type of a space. But with monoracial parents, it was a bit harder. And we saw this in particular with some White mothers who in particular did not want their multiracial African American son identifying as Black. And we spent a lot of time grappling with what we were trying not to simply read as anti-Blackness, but to understand as a desire for protection of their child that was ultimately futile but that ended up feeling a lot like anti-Blackness.

EmbraceRace: Mm-hmm (affirmative). So that's a generous interpretation. Yeah. I mean, right?

Dr. Ralina Joseph: We try. We try.

Dr. Allison Briscoe-Smith: We worked hard on that, but it did really come up. And you asked a question about what the radical listening is really about, and that was it, which is both that the radical listening, as Ralina mentioned, was that we had a plan for the book. We had an outline. Then we really listened, and then the kids blew up our book, right? We had to reorganize and reshift. That was also the kind of piece too that what we're encouraging here is the radical listening of our children and to multiracial children, which is that many times the children were voicing things that were distant to what the parents wanted including their identity, and that a radical listening means slowing down, listening deeply to what is being said, paying attention to the power, and staying uncomfortable.

And so in this context of the framed kind of generous interpretation, it's in part because those were folks that, for whatever reasons, weren't able to lean into the discomfort. The discomfort meant that their children had a proximity to Blackness, were identified and saw themselves as Black, and were also therefore subject to oppression and racism. And there was a desire, implicitly or explicitly, that their Whiteness would protect them. So that's what we mean, this is uncomfortable and can be challenging, but if we really need to center the voices of what kids are saying, they're telling us, "I felt like I was this in third grade and this in fifth grade, and I'm this now." Okay.

What we're encouraging here is the radical listening of our children and to multiracial children, which is that many times the children were voicing things that were distant to what the parents wanted including their identity, and that a radical listening means slowing down, listening deeply to what is being said, paying attention to the power, and staying uncomfortable.

Dr. Allison Briscoe-Smith

EmbraceRace: You have another great clip that we'd like to get to. And of course, people have lots of questions, so I'm not asking you to respond to this now, but that last point about moms of multiracial children with a Black parent raises this question of the differences, right? The many differences within the category of multiracial identity. So often it shows up as want and as if, and we know that all racially defined groups have lots of variation within them. Sometimes it feels like that is lost with respect to multiracial and that the experiences can be quite different. So I just want to put a pin in that and make sure we get back to least a couple pieces within it.

Dr. Allison Briscoe-Smith: We are about to hear a clip. This is a teenage participant. She is, I think, 16 here. And she's sharing a story about what happened at school with her 10 year old sister and with us.

"Singling Out Mixed Kids" Audio Clip (from 3:00 to 6:47, followed by transcription)

Older Sister: I remember in eighth grade there was a teacher. It was at lunch, and I was the only Black kid in the classroom that I was in with a class full of White kids. And of course, out of all the people, he asked me a question, saying, "Do you know how Richard Sherman's dreadlocks work?" And I said, "Well, they're braids that are left in for a long time." And he was like, "Oh-"

Younger Sister: They don't really come out.

Older Sister: Yeah. And he was like, "Oh, cool." And just basically saying a bunch of stuff about dreads and Richard Sherman and how one of his dreads fell out. And I was just like, "Okay." And I just remember feeling very pointed out in that moment because everyone looked at me when he asked me that question. And I just felt really on the spot, and I was a little bit embarrassed to be... Not embarrassed, but the fact that he pointed me out in front of everybody just wasn't the best experience.

Dr. Allison Briscoe-Smith: How do you understand why he did what he did?

Older Sister: Well, because I'm Black, and so I must know what dreads are. None of the kids in the class know what it is even though Richard Sherman is a famous football player. Everyone knows who he is, but let me just ask the one Black person in this room who would know.

Dr. Allison Briscoe-Smith: That also makes me think about this kind of question about how is it that your teachers see you. So in that moment, do you think he saw you as Black, as mixed?

Older Sister: I definitely think he saw me as Black in that moment and that basically I know everything about Black hair and all Black people. I must know what Richard Sherman's hair is. I must know what dreadlocks are. Because of the way my skin color is, I have to know. He didn't even know my name. So he was just like, "You..."

Dr. Allison Briscoe-Smith: What do you want to communicate with teachers and schools about what Black kids, what mixed kids need?

Older Sister: Not pointing out kids, not putting them on the spot, not judging them, not-

Younger Sister: Not like, "Hey, you, tell me what this person does. Teach the class about this person because I don't know, and I'm supposed to teach you about this because you're Black."

Older Sister: What she said.

Dr. Allison Briscoe-Smith: That's it?

Older Sister: I mean, people get taught to teach. They should also be taught how to teach certain lessons and say certain things. And not everybody knows that, even teachers. Teachers make mistakes. Teachers point kids out. Teachers-

Younger Sister: Can I say something?

Older Sister: ... say things they aren't supposed to. And I think that's a boundary that they shouldn't be passing and that they should learn about.

Younger Sister: And maybe sometimes teachers will put people on the spot because maybe they'll pick someone over you even if you're smart. They'll probably think, "Oh, Black people aren't smart because they're dumb, and White people are smart, so I should call on this person." And then they never get your opinions about something that you might actually have a really good opinion on something.

Dr. Allison Briscoe-Smith: Has that happened to you in class before?

Younger Sister: No. I usually get picked a lot because I just raise my hand all the time. So yeah, I've been in games a lot of times because I raise my hands all the time, and they'll be like, "Oh, you've already been it." I'm like, "Oh, sorry. Yeah, I have."

Dr. Allison Briscoe-Smith: For you to put that hand up, sounds like you make your voice heard.

Audio Clip Ends

EmbraceRace: Wow. That was great. Ralina, I'm sure that multiracial kids who are identified, racialized as Black, and a lot of Black kids couldn't relate to parts of that. What's special about the experience of the multiracial kid in this conversation?

Dr. Ralina Joseph: Yeah. I think that you really see the sibling dynamic that Alison was talking about earlier there, that the younger sibling who's six years younger is really learning how to protect herself against racism by hearing her older siblings' stories. Right. We love this so much that we use this in the book. She just always raises her hand. We're like, that's what we want.

Yes, absolutely. And that this prediction of understanding racism and even this prediction moment is not going to be a negative, but rather it's going to help stoke her confidence. And this is also a family, and we talked a bit about this, that's very clear in talking about race. They've talked about how the older daughter casually used the N word and dad was like, "Stop. We're going to do some reading together. We're going to have a big conversation." This was their weekend spent on what does it mean for our family to use the N word? When the whole family is engaged in conversations around race and racism, how it can really be this protective factor for children.

EmbraceRace: I know you want to add, Allison, so I'm giving you space.

Dr. Allison Briscoe-Smith: You know me very well. There's so much that stands out. Again, the ways that siblings are doing this together. Again, I think just to lean in a bit to the pain in the moment. We had a whole framing around a chapter about what it means to be singled out. Right. What it means to be called in that kind of way. And then, what it means to be a spotlight and the basic thing we need to listen to is, it doesn't feel good. This young woman, in addition to a number of the kids that we talked with said things like, "I wish our teachers were trained better. I wish they were supported better." And so that's a very explicit call, right? For supporting teachers, what kids are asking for is, "Hey, you're supposed to be able to take this on and help me with this. And what's ending up happening is that I, as a kid, I'm getting tasked with handling all of this." There's that piece.

And the other part is, again, that we highlight throughout the book, but I think that the younger sibling so wonderfully illustrates is the agency that kids have, the resistance that the kids have, but also to just hold on for a moment and think about, and this needs to be marshaled every day, right. It's to show up and to keep on raising the hand. Right. I think there's just a lot that shows up within these kinds of pieces. This is why actually listening was so rich. And so you can hear us. We're laughing. There are times we're crying. We have the graduate assistants who were helping us also had their own lived experience. Want to say that this was not some sort of controlled, scientific. This was scientific in terms of the centering of voices and really being able to listen. I could go on, but I always feel really impacted by hearing about these kids' experiences and what they're doing about it.

EmbraceRace: One question I have is about surprises, right? Clearly there were some already for you, right, as two mixed race people who have mixed race children and have done a lot of work in the space, but there were surprises. What were some of those surprises as you were researching and writing this book? And what are some of the big takeaways for the folks who haven't picked up your book yet?

Dr. Ralina Joseph: I think that for me, one of the huge, not huge surprises. I don't know if this is a surprise, but I was dismayed by it. But every one of our kids who was part Black, talked about implicit bias. Without a doubt. And talked about being punished in schools, talked about differential... Recesses taken away, being given detentions, not given explanations for math problems and watching this happen in various ways. And so to see this with all of our kids who are racialized as Black was... We knew this, but to hear it over and over again, and to hear eerily similar stories was disturbing.

EmbraceRace: There was another interview you all did with two, I think, brothers who are half Asian and half White who talked about being singled out as well. And then said, "Well, we're luckier than most kids." They were very aware of a racial hierarchy "because we mostly pass for White, but kids from other groups," again, who get racialized differently, "they get singled out a lot more." That was really interesting. This is happening and all the kids are seeing it. And a lot of times adults don't think the kids are seeing it or want to believe that, or don't think the kids saying, "My teachers need training in this. It's pretty clear that this is happening at school." That this is important, but not in the training. Right. Really interesting stuff.

Dr. Ralina Joseph: No, I was just going to say that actually speaks the true dangers. I know this is probably not the audience to say this in front of, but I am not a proponent of a multiracial category. I think that it actually is a truly flattening device. I think that there's a way for us to do multiple levels of data collection, so that people can be multiracial and. And we've known this from plenty of other scholars, but the multi-racial Asian, and particularly Asian American and White experience, is fundamentally different, right, in terms of voting patterns, in terms of life chances across a variety of things.

And so we talk about with those brothers, for example, this experience of White adjacency, right. And how they're very conscious of it and are not trying to use it in a way that that is evil, but they understand how power is operating. Right. And then how to insert themselves in a way in which the kids who are part African-American are not provided with that luxury. Right. And so for us thinking about an upper middle class kid who is Japanese American and White versus a working class kid who might be Samoan and Black, and for those two children to be categorized by a school system as the same, is troubling to me. Is actually troubling to me. And when we hear the stories telling our experiences are incredibly different here.

Dr. Allison Briscoe-Smith: It's troubling to the kids as well. It's troubling to the kids. They're actually saying that their experiences are different and different is okay. This flattens categorization. It's also embedded in this idea as if multi-raciality is some sort of way out of experiencing bias. And so that's in part why what we learned from listening and what we also found within the research is that it's not. It's not some sort of way out. The increasing proximity to Whiteness is not some sort of way out of this, especially when kids, and this is what we also heard, so many kids make agentic choices about when they would claim and use Whiteness. And also when they wouldn't and how many times we heard kids... We have another story of a multiracial Asian kid who chooses an Asian identity, despite everybody else saying, "No, maybe he should..." chooses and reaffirms and sticks to that. There are these opportunities, but he's actually very clear that that choice comes with particular baggage. It's not as if kids don't know, but this flattening categorization is not some sort of way out.

EmbraceRace: Allison, here's one more question I want to stay with you. It's another very blunt question. What's clear, you've mentioned from the very beginning of this conversation that the kids of course know that they're being racialized. They know their racialized very early. At a very early age, they know that. They can talk about that. Now, obviously they don't use that language, but they recognize the nuances of their racialization and that there are a lot of sources, right? A lot of sources of input for that.
And I'm wondering about if we distinguish between, and I'm thinking about the distinction between their peers and how their peers racialize, what that input looks like and feels like versus the adults in their minds, whether at home, at school. Did you come away with a sense of the distinction that the kids themselves see and how they are racialized by the others, especially by generation?
Dr. Allison Briscoe-Smith: Yeah. I think if I were to jump in on the shorthanded answer and, Ralina, definitely help me out with here, one of which is that grown folks got to catch up. There wasn't any space that I heard a ton of folks saying, "Wow, I was incredibly helped by folks who had a different generational piece." In fact, when we asked kids to tell us about the grownups on campus that were really helpful, we got nothing. We really had to go back, right, and to ask again. I think that's one. The second piece that we found is that the parents and the adults who really, really did connect were uncomfortable and tried really hard. And so that was another lesson. We also heard that kids were really relying upon other peers. And in fact, asked pretty strategically for affinity spaces to be with others.

There's a number of places where we hear kids say something about, "Oh yeah. As I think about it, it turns out all my friends are mixed kids. All my friends." There was this sense that what they were being drawn to and what they were really being supported by were children who were having similar experiences. What we asked in particular is, "What schools could do?" What kids said is train your teachers as to how to have conversations and how to see me. Give us spaces to be together. And allow us to be who we are. I think one of the quotes that stands out to me and Melissa, I remember, you and I talked about this a while back is kids saying, "It's not as if I don't know who I am. It's that other people won't let me be." That I would say that those are the pieces that kept coming up again and again.

EmbraceRace: There are a lot of questions like this, "How does one respond to a teen classifying herself as White when she's half Latino and half White? Do kids unconsciously know from observing society that identifying as White brings certain privileges and higher social status?" I'm going to say yes to the last part, but the question of how to take it? I think a lot of parents struggle with that for different reasons. Some struggle if White isn't identified. Others struggle if you're not identifying as Black and for example, instead of Black/White, if that's your racial identity because of the politics of it. Right.
How do you all think about the best response to mixed race kids who are choosing their identities. Ralina, what do you think?
Dr. Ralina Joseph: Yeah. And we've gotten this question quite a bit from troubled parents. And it's interesting, it's often the parent who does not want the child to identify as White. Right. And is really upset about their child identifying as White. And I think that for one is that we're talking about listening to kids. You want to hear about what that experience is like. One of the things that Allison and I are trying to talk about is the importance, and that you all are doing here so well, is the importance of talking about race early and often and all the time, right? So that you're not in the position of what... I was just talking to my hairdresser today about awkward conversations about race that happened with White folks, which she called "White people-ing." You're not in the moments where it's so awkward to talk about race that you're stumbling over your language, right. Sociologist Eduardo Bonilla-Silva talks about this is you don't go into this space a rhetorical incoherence.

And I don't know if this is his mom, but if you're not having race as a part of a conversation all the time... For Alison and myself, we're godmothers to each other's kids, race is something that we are able to joke about. Race is something that we're able to make transparent in all the different ways. Race is never off the table. Race is never something bad to identify. And so I think that to have conversations then about Whiteness as well, and Whiteness as being erased. And what does that mean? And what does it mean to choose Whiteness in this space, is to listen and to let the kid work that out and to see where that goes. And then I think for the parent also to talk about, "This is my racialized identity, and this is how I came here. And was I always here? I don't know." And that's one of the things that we do actually in the book is that we provide different exercises for people to go through and to practice. And this is something that you could do with your kids as well.

It's often the parent [of a multiracial child] who does not want the child to identify as White. Right. And is really upset about their child identifying as White. And I think that, for one, is that we're talking about listening to kids. You want to hear about what that experience is like... And what does it mean to choose Whiteness in this space, is to listen and to let the kid work that out and to see where that goes.

Dr. Ralina Joseph

EmbraceRace: Thank you for that. We have, not surprisingly, a bunch of questions too about how to support mixed race kids in the school context. And we have questions essentially both around, "What do I do?" There's one person who is a teacher who says that she manages the affinity groups for the school and wants to create a mixed race affinity group, both for students and for parents, wondering what to do. And then someone else, also a teacher, saying, "What do I not do? What do I make sure not to do that teachers who mean well often do with respect to mixed race kids?" Allison, any thoughts on either side of that?

Dr. Allison Briscoe-Smith: Yeah. I think we hope and encourage folks will get a chance to connect up with the clips in the book, in part because this is the focus. And part of it is we encourage these exercises, these practices that are very consistent with practices that teachers are really good at doing, which is I think teachers are uniquely equipped to help kids feel like they belong. And they need practice at talking about race and thinking about race in complex manners. And so I think it's about getting a chance to practice that. I'm really glad to hear that someone is saying that they're supporting affinity groups. That was very consistent about what kids wanted. Not only that, it was also very, very consistent that schools were saying no. Like, "No, we can't have affinity group. No, that would make White people feel bad." So I actually think sharing the resources about how to create affinity groups. And we bring some of those resources to bear by listening to teachers who do it. What kids are hungry for is a sense of belonging and a sense of connection.

In terms of what not to do, the kids actually speak to that as well. Don't just be silent about it. And don't presume that your silence is actually helping out. The kids are very, very specific that we need teachers who feel supported and equipped to talk about race and race difference and difference, period. This idea that if I don't talk about it, it'll go away is actually very painful.

I think last, a growth mindset around making mistakes and staying in connection is very important. We hear from so many folks, "I'm so scared to do the wrong thing. I'm so scared to say something." So then people end up silent. Or, "I'm so scared that my children are going to be impacted by race, so my hope is that Whiteness will protect them." The kids are very, very loud that it's not helping them, it's not protecting them. So that was a whole lot, but I think it's in part because the conversations we had with kids were really nuanced and complex, and their requests were very clear. We asked every kid, "What do you want teachers to know? And what do you want schools to do?"

EmbraceRace: Right. So there are some teachers in your book that were really exceptional in creating that belonging for kids.
How do you curate a classroom with mixed children to explore their identity safely?
Sort of along the same lines there. And it does make me think about... maybe was there someone named Mr. Daley or Mr. Day, Mr. Daley? Yeah. And he reminded me of, we had some people who were book folks, like reader folks talking about race in picture books and how to talk to kids. And always when we have anything about books, everyone wants a book list. Book list about these kids, these kids. And the folks who were on, and Megan Dowd Lambert was one of them, and Hannah Gomez said, "We hate book lists. If you really want to get something that your kid attaches to, go to the librarian and talk about your kid's interest, this, this, this." And I think that that's kind of a nice analogy.

When you're curating your classroom, is it that you get out the book list that everyone's going to have in their classroom and put those on the shelves, or you really talk to the kids about who they are? And that was something. I mean, what was special about Mr. Daley? Maybe one of you can talk about him a little bit.

Dr. Ralina Joseph: Yeah, he was really fantastic. If it's okay actually, I pulled up... because I wanted to share some of his words. And this is how we start this chapter. I would say the chapter four is really on these really special teachers. And who, as Allison points out, we ended our interview saying, well, who are the teachers that are really impactful and what we wanted to ask this question of belonging? "Who makes you feel like you belong at school?" And we got this, "Nothing, nobody." And so we had to return back, and Mr. Day was one of the ones who came back and the students really identified with. And so he teaches at a large, racially diverse public high school.

And so he started off for us, he said that for him, relationship is key. He quotes Maya Angelou and says, "People will forget what you say, but they'll always remember how you made them feel." He notes, "All the content that I do is important and it'll be beneficial for students in their educational career, and maybe their professional career." But what makes them actually, "consider myself the most successful is if my students come to my space and they feel like they feel seen, they feel validated and they feel like they were treated as more than just a person who was supposed to be graded." He says that, "Especially for a lot of students of color, if you have that established relationship, you can be talking about how paint dries and students are more likely to roll with you. They may not roll with the content, naturally because they're not interested in it. But if you're talking to them about it, they're going to follow you and they're going to work with you because they appreciate you."

So just in terms of context, when I was interviewing him, our interview kept on getting disrupted because students kept on opening up the door and being like, "Hi, Mr. Day. Hi, Mr. Day." They wanted to see him. They wanted to get acknowledged by him. He was a mixed race, African American teacher. And he started off by talking about his own journey. And that meant such a tremendous amount to all of the students who really formed kind of an unofficial affinity group around him, including a couple of our participants.

EmbraceRace: Did gender or other sort of personal group identities shape racial socialization and kids' attitudes towards all of this?

Dr. Allison Briscoe-Smith: Yep, yep. So yeah, gender did, skin color did, hair color did, neighborhood, school composition. There were all of these roads into identity. We, in the beginning, in part paid attention, lifted up the stories around Black boys, in part because of the ways in which their risk is particular, right? And it's not as if other kids don't have risk too. There's a risk for Black girls as well. But it is an intersectional experience. So gender definitely played a lot, as did all these other factors, right? So many parents talked about where are we going to choose to send our kids to school? Where will they be seen? Where could they possibly be seen? So many parents talked about the issues of feeling the need that they had to choose between either diversity or a "good school." Kids knew that these were the decisions that were being made, too. So yeah, it's all of those factors, gender, neighborhood, school, class were very impactful for their experiences.

EmbraceRace: We're getting, by the way, in the questions, and we did earlier as well, sort of live and earlier, a lot of people saying, "Oh my gosh." A mixed race adult saying, "I've been waiting for this conversation forever, and it's great to hear experiences like mine." So there's really a lack of research around this, right? There's more and more conversation, but it does feel like, yeah, people are hungry for more. So we're glad that you created this book.
What are the question that racially ambiguous kids get? And I was fascinated in the book that you brought some of the science behind people's responses to mixed race kids, or particularly those that they see as racially ambiguous. And I wonder if you could talk about that. And Ralina, could you tell the story about that class evaluation you got as well?
Dr. Ralina Joseph: Yeah. This is kind of a typical experience for me. So one of the stories that I tell in the book, as we're talking about, in particular, racial ambiguity and this what are you question, is that when I was an Assistant Professor, so many years ago, and I was at... probably this was maybe about 14 years ago, teaching a large lecture class. And I got the, we call them the dreaded yellow sheets. So that's when people could do their long hand comments. And I got one that said, "If you just told me what you were, I could focus better in class." Which I thought was just really, really interesting that this student, we're in the quarter system, over the course of 10 weeks, 20 lectures was just somehow so focused on trying to figure out my racial background, as opposed to all of the brilliance that I was speaking on those days. And we know that there is actual science. I did not know this at all until I started working with Dr. Briscoe-Smith here. So Allison, I don't know if you want to just share some of the science behind this?

Dr. Allison Briscoe-Smith: Yeah. So what we were able to do is to bring it together. That's a painful and salient quote that was actually really a moment of truth telling. There is brain science that actually shows that people become kind of tapped out if they can't figure out the category of the person in front of them. Now I say brain science gets tapped out, but I also want to say we can overcome that. It's not necessary to place another person. It's really about a person's discomfort at not being able to tell who the person is across from them. And I think anyone that is multiracial or is read as multiracial or has the experience has felt that, that desire of the person across from them to try to categorize, to test, to figure it out, and to place them.

So again, I'm operating as much as I can with generosity about what that is, because it manifests very painfully and it manifests, as we detail within the book, with some pretty significant consequences. Things like, "You should probably pay more attention to basketball, because clearly, that's who you are." Or the research that indicates that when you place multiracial folks who are multiracial Black and White in particular dress, that there's an association of class and intelligence because, "Oh, now I can finally read you." So these things do show up, but it's people's inability to kind of manage their brain a bit that moves them to the place of needing to categorize. And what we're actually really hoping for is that people pause a moment, listen a moment, and pay attention to the other things that might help you figure out whether or not you can connect, as opposed to... If that person missed out on the brilliance that Dr. Joseph has to offer, right? Because it's uncomfortable. Be uncomfortable and figure it out. These kids are figuring it out, we can do this as well.

EmbraceRace: I mean, the flip side of many people's unease with the racial ambiguity, or not being able to figure it out is that as you think about... we also have lots of research on this, how readily, how quickly we assess racial identity, gender identity, and age with a split second of meeting someone, no matter how fleeting, right? So when I think of, sometimes I check myself or I ask others, that person that you checked out when you checked out, you bought the candy bar or whatever, you encountered someone you don't know, it was fleeting. People can virtually always say, "Well, this is what I thought." Right? "I thought it was a man, I thought he was roughly this age, and I thought he was that racial category." And if I don't know, yes, that sense of wait, I wasn't able to place. I mean, if that doesn't speak to the investment we have in those categories, I don't know what does.

Right. And imagine as you have, and as you've listened, kids experiencing that when they know who they are, but they still get this confusion and this pain that you see.

Thank you. Yeah, this has been a great conversation. Amazing work. Yeah, really lovely to be with you both and with everyone here.

Generation Mixed Goes to School: Radically Listening to Multiracial Kids by Ralina Joseph and Allison Briscoe-Smith

Growing Up Mixed Race audio interviews - check out these audio interviews from the Generation Mixed Goes to School project, featured on the Center for Communication, Difference and Equity's Interrupting Privilege, Radical Listening website:

Things to Know if You Love a Multiracial Child - A collection of EmbraceRace resources about raising multiracial kids

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Generation Mixed Goes to School: Radically Listening to Multiracial...

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"Generation Mixed" refers to kids born after the 2000 Census, which for the first time allowed respondents to "Mark One or More" options to describe their racial identity. Join this conversation about the experiences of Generation Mixed at school - from the perspectives of the kids themselves and the caregivers and teachers trying to support them.

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Mikva Challenge - Mikva Challenge develops youth to be empowered, informed, and active citizens who will promote a just and equitable society.

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At Mikva Challenge democracy is a verb. Our curricula and programming develops youth to be empowered, informed, and active citizens who will promote a just and equitable society.

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Racial Equity Tools offers tools, research, tips, curricula, and ideas for people who want to increase their own understanding and to help those working for racial justice at every level –in systems, organizations, communities, and the culture at large.

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National Museum of African American History

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National Museum of African American History

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National Museum of African American History and Culture

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Talking about race, although hard, is necessary. We are here to provide tools and guidance to empower your journey and inspire conversation.

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Becoming a Culturally Responsive Teacher (2012)
~ YouTube

How do you reach students who are culturally and racially different from you? This course provides thought-provoking background and practical suggestions for teachers seeking the answer to this question. Participants explore their own assumptions about race, class, and culture; and learn strategies for creating classrooms that are culturally inviting to all. Classroom footage illustrates principles of Culturally Responsive Teaching, including affirming students' cultural connections, stressing collectivity as well as individuality, and managing the classroom with firm, consistent, caring control.

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Culturally Responsive Teaching - The Network

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Becoming a Culturally Responsive Teacher (2012)~ YouTube How do you reach students who are culturally and racially different from you? This course provides thought-provoking background and practical suggestions for teachers seeking the answer to this question. Participants explore their own assumptions about race, class, and culture; and learn strategies for creating classrooms that are culturally inviting to all.

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Videos

Featured Interviews and Speeches

Dr. Yolanda Sealey-Ruiz recites Maya Angelou's, Still I Rise.

Dr. Yolanda Sealey-Ruiz spoke in conversation with Dr. Bettina Love and Genevieve DeBose Akinnagbe, about Dr. Love's book, "We Want To Do More Than Survive: Abolitionist Teaching and the Pursuit of Educational Freedom".

The Arts and Humanities Department at Teachers College were highlighting remarkable faculty and students. Dr. Yolanda Sealey-Ruiz was one the faculty featured for #WomensHistoryMonth, and discussed her book, Love from the Vortex (Kaleidoscope Vibrations, LLC) - which charts her journey of finding and losing love over the span of three decades with six different men who came into her life at various times, but also offers a universal take on what can happen when one seeks love and connection with others, and the lessons that follow when that connection and love is lost.

Featured Podcasts

Dr. Yolanda Sealey-Ruiz was featured on the podcast, 3 Educated Brothas . In conversation with Dr. Edmund Adjapong, P. Gladston Williamson and Marc Pierre, Yolanda discussed her first, full-length poetry book, Love from the Vortex & Other Poems. The podcast is all about things education through the experiences of 3 Black men.

Click here to listen to the entire episode. Click here for more information on the podcast.

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The first episode of the NYC Healing Collective podcast featured Dr. Yolanda Sealey-Ruiz. In conversation with Angel Acosta, co-founder of the NYC Healing Collective, both discuss what it means to heal through the Archeology of the Self.

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The NYC Healing Collective amplifies people and insights at the intersection of healing, wellness and societal transformation.

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Click here to listen to the entire episode. Click here for more information on the NYC Healing Collective.

On Thursday, July 2nd, I joined Dr. Shamaine Bertrand and Dr. Kisha Porcher of Black Gaze Podcast to discuss how Love from the Vortex and Other Poems is the manifestation of my radical Black Self Love and Healing.

Throughout this live event, we had a critical conversation about how we must love ourselves deeply and become critically conscious about the trauma we have experienced in learning to love ourselves and others. To listen to the episode, click here.

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Videos Featured Interviews and Speeches Dr. Yolanda Sealey-Ruiz recites Maya Angelou's, Still I Rise. Dr. Yolanda Sealey-Ruiz spoke in conversation with Dr. Bettina Love and Genevieve DeBose Akinnagbe, about Dr. Love's book, "We Want To Do More Than Survive: Abolitionist Teaching and the Pursuit of Educational Freedom".

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In this four-part podcast series, the authors of Teaching for Racial Equity engage in critical conversations about race, equity, and education. They explore ...

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2024 Math Collaborative Summer Book Club

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2024 Math Collaborative Summer Book Club

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https://us02web.zoom.us/j/4219107593

Meeting ID: 421 910 7593

Wednesday, July 3, 2024             Forward, Introduction, Time Out To Talk (TOTT), pages xv-xxvi and pages 1-22

Wednesday, July 17, 2024           Chapters 1 & 2 and TOTT section, pages 23 - 70

Wednesday, July 24, 2024           Chapters 3 & 4 and TOTT sections, pages 71 - 121

Wednesday, July 31, 2024           Chapters 5 & 6 and TOTT sections, pages 123 - 172

Wednesday, August 7, 2024       Epilogue, Appendices 1, 2, 3, 4, pages 173 - 219

Contact Stef at smathewson@me.com with questions.

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Dr. Yolie Sealy-Ruiz Archaeology of Self

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Dr. Yolie Sealy-Ruiz Archaeology of Self

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Truth, Love, & Racial Literacy | Yolanda Sealey-Ruiz | TEDxPenn

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Our Culture Our Schools Culturally Responsive Education in New York City

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Slow Reveal Graph is an instructional routine that promotes sensemaking about data. This highly engaging routine uses scaffolded visuals and discourse to help students (in K-12 and beyond) make sense of data. As more and more of the graph is revealed, students refine their interpretation and construct meaning, often in surprising ways.

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Recently, I have been learning a lot about problem posing from my friend Jinfa Cai, in particular how to infuse these opportunities within the use of curriculum materials. Because, while there are rich problem solving experiences in a good curriculum, we do not often see explicit use of problem posing, especially in K-5. The Notice/Wonder routine is probably as close as it gets.

Since I am in classrooms this year, I get to try some problem posing around curriculum activities and follow up with Jinfa. As with all learning, the more things I try, the longer my list of questions for him grows! After last week's lessons, I was left with two questions that I can't wait to discuss:

1. How do we honor all of the posed problems within the timing of a lesson?

2. How do we infuse opportunities to reason about the problems that can and cannot be answered with the information in the situation?

The curriculum task

The lesson focused on addition of fractions with like denominators. As an adaptation to the curriculum materials, I only showed students the bolded part of the task and asked them to share things they noticed and wondered. Because their ideas were all falling into the 'wonder' column, I quickly pivoted to the problem posing work.

Problem-posing launch

In their journals, I asked students to take a couple of minutes to write mathematical problems they could pose about this situation. They shared their problems as a whole class and I recorded.

It was so interesting, yet not surprising, that they asked the exact same problem the curriculum task was posing! I had them work as a group to solve that problem and told them that if they finished before we came back together as a group, to re-read the other problems on the board to see if they could answer them as well.

Problem solutions

Students represented their solutions in many different ways as they leveraged their understanding of fractions, addition, and multiplication. It was particularly interesting that you could see in their work how they used one expression to derive subsequent ones. Here are a couple of examples:

Additional problems

As a whole group, we compared and connected student work like in the 2 examples above. Discussing questions such as:

Where are the ¼ cups in the expression?

Where are the ¾ cups in the expression?

Where is one expression in the other?

How did knowing this expression help you with another?

At that point, our time for the lesson was over so I quickly went around and snapped pics of the problems they posed independently at the launch.

I saw so many interesting problems that didn't get shared during the whole group time, which made me wonder how I could have done that better next time and led to my two wonderings:

1. How do we honor all of the posed problems within the timing of a lesson?

2. How do we infuse opportunities to reason about the problems that can and cannot be answered with the information in the situation?

Embedding problem-posing experiences in the curriculum and answering the teaching questions that arise is such powerful learning for both students and teachers. When students have the opportunity to pose their own mathematical problems based on a situation, they must make sense of the constraints and conditions from the given information to build connections between their existing understanding and a new understanding of related mathematical ideas (Cai, 2022). And when teachers have the opportunity to listen to connections students make, understand the math students see in situations, and make teaching decisions on how to elicit, share, and move forward with student thinking, it shares the mathematical authority in the classroom and leads to deeper, more powerful learning for all.

The other exciting part, that I wish happened more often in general, is through these experiences, Jinfa and I continually learn how research can inform practice and how practice can inform future research.

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Student work is just the best. It is the one thing that will always motivate me to write! So, let's kick this post off with a great work example from 3rd grade. In this task, students are asked to locate 1 on a number line labeled with 0 and 1/3.

When I look at student work, I typically think about 3 things:

1. I look for evidence of what the student understands. In this case, it also involves making some assumptions because we can only know so much based on written work.

2. I think about question(s) I would want to ask the student. Again, because we can only know so much from written work, there can be so many hidden understandings that can only be found by talking with the student.

3. I think about next steps, in particular, how I might create a connected, grade-level onramp experience that might be helpful for the student.

Before reading on, what do you see in the work? Take a minute to jot down your thoughts and I would love to hear them in the comments at the end!

Here are some ideas I think the student understands based on the work:

Since the denominators increase from 1 to 5, there is evidence of an understanding that whole numbers increase as they move to the right on the number line.

Since the unit fractions are all between 0 and 1, there is evidence the student knows that unit fractions are less than 1 and can be in many denominations, especially since 5ths are not typically in the work of earlier grades.

Since the fractions are spaced out fairly equally on the number line, there is evidence the student understands there should be equal intervals between numbers on a number line.

These are great understandings on which to build for a problem like this. So, what is up with those denominators increasing to the right?

At first, I thought the student was counting the denominator like whole numbers and thinking fractions are always written with a 1 above the fraction line, so it was just 'put there' but had no meaning. But then, in the same set of student work I saw this cool-down.

This student did a similar thing with the increasing denominators, but it was the sketch in the corner that caught my eye. This student is showing the number of equal parts in a whole for halves and fourths. This made me wonder if the denominator did have a fractional meaning to them and not just a counting number in a sequence.

If the student understands the denominator as a number that represents the number of equal parts in a whole, then it makes total sense that they would label them in increasing order. In that way of thinking, the denominator is like a whole number in their mind and whole numbers increase to the right on the number line. For example halves are 2 parts, thirds are 3 parts, fourths are 4 parts,...etc. So the more parts, the larger the fraction in their eyes. They see the fraction as a numerator and denominator, 2 numbers with different meanings, rather than 1 number located on the number line.

As I looked back at the progression, I realized how intricate this move is from K-2 understanding of fractions as a quantity to grade 3 understanding of a fractions as numbers.

Using the Standards as a reference, in grade 2 students use fraction language to describe quantities such as "half of a rectangle" or "a fourth of a circle." That half and fourth are quantities that can be a different size each time depending on the whole because it is a fraction of a shape. At the same grade level, within their measurement work, students are locating whole numbers on a number line.

In grade 3, students learn fraction notation to represent those quantities and later merge that understanding with the early number line work to understand a fraction as a number on the number line. That number has a specific location and does not vary in the way the quantities did in earlier grades. It is a super intricate move in fraction understanding which makes it easy to understand why students have brilliant moments like the one in the cool-downs above.

So, let's revisit the student work.

To learn more about their thinking, I would love to ask the following questions:

1. How did you use the ⅓ to locate the other fractions?
In this question, I am curious if the student split the distance from 0-⅓ into 3 parts (bc of the 3 in thirds) and then did the same on the other side of ⅓ and that is how they determined to stop at ⅕ and then put 1 - three intervals on each side of the ⅓.

2. What did you pay attention to when you put the other fractions on the number line?
Want to hear about the equal spacing of the fractions.

3. Why did you stop at ⅕? Where would you put ⅙?
This could be answered in question 1, but if not, I am so curious about ⅙.

4. [draw a number line labeled with 0-1] Where is ½?
Do they know ½ is in the middle of 0-1?

5. [draw an identical number line below the first one] Where is ⅓?
Do they know it is 3 equal intervals between 0-1? And, if the number lines are the same length, I want to see if they can reason about ½ being larger than ⅓.

Now, the fun part of thinking about next steps. How can we connect what students understand about equal-sized parts to understanding a fraction as a number on the number line? It seems like activities that connect the number of parts in a whole to the size of one part would be helpful here. Right now, I think the students understand 1/4 as 4 parts and maybe 3/4 as 3 out of 4 parts. But, when moving to understanding a fraction as a number we want them to understand 1/4 as the size of 1 part of a whole partitioned into 4 equal parts and 3/4 as 3 pieces the size of 1/4.

There are many ways to think about this but below is a sequence of learning that I think would be nice.

To connect to the 2.G.A.3 work, we could:
Have several same-sized rectangles drawn one above another. Ask students to divide the top one into halves the middle one into thirds, and the bottom one into fourths. (grade 2 work)

Ask them how they would name each piece and what they notice about the size of the pieces. (grade 2 work)

Ask them how they would record the size of each piece. If they use fraction notation that is grade 3 work.

Transition to creating fraction strips for halves, thirds, and fourths. Label the fractional pieces of the strips and discuss the size of the pieces and the number of pieces in the whole. (The fraction strips from Marilyn's Do the Math kits made the WORLD of difference in my 5th graders understanding of fractions!)

Place the fraction strips above a number line labeled with 0 and 1.
Ask students where 1/2, 1/3, and 1/4 would be located on the number line based on what they know about the length of the fraction strips.

Ask where they think non-unit fractions might go? Where is 2/3? Where is 2/4? Where is 3/4?

Ask for different ways they could label 1.

Then, I would have them revisit the cool-down and see what they do! There are also more of my favorite number line tasks from IM here. Try some out and dig into the student thinking. It is always such an amazing learning experience!

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Over the past months, like many people, I have been thinking a lot about what it looks like to reimagine a curriculum that was intended for use in an in-person setting for digital implementation. Alongside an amazing working group of teachers and coaches, we have gone piece by piece through the IM K-5 curriculum components and discussed the purposes, opportunities, and challenges of each when transitioning to a digital setting, whether it be synchronous or asynchronous. Discussions around the student experience and student engagement were challenging, but really pushed us to think deeply about the activity purposes, questions, and ways in which to design opportunities to learn more about each student.

Together, we are in the process of creating ' storyboards ' for warm-up routines and generalizing processes for planning activities, but the piece that I really cannot get off my mind is assessment. While I think every activity is an assessment opportunity, I am thinking in particular about the cool-downs and checkpoints in the curriculum. These pieces were designed as in-person formative check-ins on student understanding after a lesson and section. Now, in some cases, they will be completed asynchronously. While it seems like one of the easier pieces to recreate digitally with the platform being the biggest decision, there are nuanced constraints that limit what a teacher might learn about student understanding when completed asynchronously. In the classroom, a teacher can observe students as they work and is available to answer questions. The sheer nature of sending home an assessment digitally can raise a lot of different, yet related questions.

Family members or caregivers might wonder:

Is this graded?

What if they cannot do it independently?

How much should I help and if I don't help, will they get a bad grade?

If I do help them, how do I know if the work is aligned to how they are learning?

Teachers might wonder:

Should I grade this?

Did the student do it independently?

If the student had help, how much?

If the student had help, what did they know and where did they struggle?

I think these questions stem from two very caring places: family members want their child to learn and be successful and teachers want to know what students understand so they are able to best support them as learning continues throughout the unit.
I have been thinking about these questions alongside the results from Learning Heroes most recent family survey ( / webinar ), in particular, the areas outlined in yellow. slides

I wonder if this year is an opportunity to reframe how we approach assessment, reframe how we engage families in discussions around assessment, and break the typical lens families might have of assessment being the 'final grade.' H ow could reframing assessment as 'finding out what students understand and are able to do' change the way in which: families approach assessments at home, teachers respond to what they receive from home, and shape ongoing communication between families and teachers throughout the year?
Here are few ideas I keep tossing around to support this communication:

At the beginning of the year, when meeting with families, let them know that they are not expected to teach their child the math content but instead encouraged to continually ask students questions (provided for them) to find out what they know, such as:

Retell the problem to me in your own words.

What do you notice? What questions do you have?

How do you know?

Where do you see the problem in your picture or diagram?

Explain how you solved the problem?

Reinforce learning as an ongoing journey that, in order to best support students, teachers need to know what they understand and are able to do. Explain that the activities and assessments are ways in which to do this, so every conversation during the year will focus first on what students know.

Create a schedule to meet with family members or caregivers to check in with some pre-arranged topics to discuss that focus on what they are seeing or hearing:

What do you see your child understanding and able to do?

When, if ever, do they get stuck?

What can I do to help?

(Consider communication types families view as most effective on page 24 .)

On each assessment, put 2 checkboxes at the end for students to choose from (no repercussions for whichever is chosen - also established in the opening family meeting):

___ I did this all by myself

___ I did this with some help.

[text box for the student or whoever helped] What was the sticky point?

Provide opportunities for students to retake problems they didn't get correct as the unit progresses. For example, if a student gets 2/3 problems correct on the Section A checkpoint, put the one they missed on the Section B checkpoint after the next week's instruction. This communicates to families that learning happens across time.

All of this is so challenging with no easy solutions, but I would love to engage in discussions that push us to reimagine learning at a distance that includes strengthening teacher<>family relationships and moves beyond creating digital worksheets of in-person materials.

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(Originally written for IM: https://illustrativemathematics.blog/2019/05/07/designing-coherent-learning-experiences-k-12/)
One challenge in curriculum design is considering all we know and believe to be true about math teaching and learning and translating that into realistic and actionable pieces for teachers and students. Our recent post about the K-5 curriculum focused around our belief that each and every student should be seen as a unique person with unique knowledge and needs. And while that post centered on elementary materials, to truly design around this belief we must look past K-5 to consider each student's unique K-12 mathematical journey. A journey that, for most students, looks very different as they move from elementary to middle to high school.

While it is not surprising that the math content is different across grades, there is something really important and powerful in curriculum design that can make a student's mathematical journey much more connected and accessible than it currently is. As we write the final component of our complete K-12 OER math curriculum, we continually focus on the consistent structures that support the focused, coherent mathematics within, and across, grade levels.

Why consistent structures for students?

Consistent problem-based curriculum structures can help remove some of the guesswork and anxiety about how math class is going to look and feel each year. Imagine a student entering any grade level knowing they will be given the opportunity to grapple with mathematical ideas, both independently and with their peers, before being expected to apply the understandings on their own. The problem-based atmosphere supports students in understanding their important role as a mathematical thinker and over time students get used to generating the mathematical knowledge as a community of learners.

Within our problem-based design is a consistent lesson structure:

This means every day, no matter what grade they are in, students know they have an accessible invitation to the mathematics during the warm-up, are given opportunities to build ways of thinking and doing the mathematics, discuss the ideas of the lesson as a whole group during the synthesis, and show what they know on the cool-down. This predictable flow allows students to focus on the math of the lesson because they know there will be opportunities to discuss ideas, ask questions, and consolidate their learning.

Another important consistent structure within the curriculum are the instructional routines. Instructional routines provide an opportunity for all students to engage in, and contribute to, mathematical conversations. The predictable routine for the interactions between teachers and students allows each student in the classroom to focus on the mathematics rather than the actions that need to be taken. And the more experience students have with the routine gives them greater ability to focus on the mathematical content.

Why consistent structures for teachers?

Within the classroom, consistent structures give teachers time to focus on what's most important-the students. As a teacher, there is something really nice about not having to worry about "what comes next" in a lesson or "how the activity runs," just as it is for students. When all of those things are kept consistent, it offers teachers the time to listen for the mathematical understandings students have and think more deeply about the classroom experience: What questions are being asked? How are they are asked? With a consistent structure, there are more opportunities to think about whose ideas are being shared, and the ways in which these ideas are represented.

Outside of the classrooms, it would be amazing to have the opportunity to collaborate K-12 in professional learning. I would love to have a world where a district did not always split into K-5, 6-8, and 9-12 math professional development sessions, but instead had a room of K-12 teachers discussing the coherent progression of mathematical ideas, representations, or routines in a curriculum.

For example

Below is a series of warm-up routines from the IM math curriculum by grade. While there are some really cool math connections that show up in this series, I would like you to imagine a student who experienced the structure of these routines since kindergarten.

What might they know about the structure of the routine by the time they are in Algebra 1?

How might what they know about the structure help them attend to the mathematics more by the time they reach Algebra 1?

How might what they know about the structure help them be seen as a unique person with unique knowledge by the time they reach Algebra 1?

Kindergarten

Grade 2

Grade 3

Grade 6

Algebra 1

Next Steps

For more info on the upcoming K-5 curriculum, you can visit the IM site here. We would love to read your reflections to the questions above! Please respond in the comments.

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I cannot count how many times classroom norms has been a topic of my conversations in the past month. From creating and facilitating professional learning to thinking about how a curriculum can offer support in this area, I find myself obsessively thinking about ways in which norms might support both students and adults in their learning.

If you asked me a year ago about the norms in my classroom, I would have felt pretty good about how the list hung proudly on my classroom wall, was collaboratively established by students, and appeared to be in place during their math activities.

However, like the majority of my teaching life, the more I learn, the more I realize how much there is still left to learn. In this particular case, it is norms in a classroom.

I think most people would agree that establishing norms is important. Norms can encourage students to work collaboratively and productively in a classroom, elicit use of the Mathematical Practices and help students see learning mathematics as more than just doing problems on a piece of paper. But, how often do we create norms in our classroom only to complain a month or two later that students aren't thinking about any of them when working together and we struggle with how to refocus students to keep in mind those things they said were important at the beginning of the year? I know I have been there and looking back, wonder how I could have done that better.

While I think good curriculum tasks, lesson structures, and relationships I had with students helped me a lot in encouraging students to be mindful of the norms in the classroom, I don't think I put an equal amount of effort into maintaining norms as I did establishing them. With that, I wonder what it even looks and sounds like to maintain them?

To me, maintaining norms is about moving from a poster on a wall to a living and breathing culture in the classroom. But, what things can a teacher do to make the norms not only a list, but a part of their classroom math community?

Of course, as the journey begins on writing the IM K-5 Math curriculum, I am also wondering how a curriculum can support teachers in establishing and maintaining classroom norms in a meaningful way. Even more specifically, what could this look like in Kindergarten when we have the opportunity to influence the way students view learning mathematics?

As I think through these questions, I would love to hear how you think about norms in your math classroom. What things can we do as teachers to support students in thinking more about what it means to learn and do mathematics? How could a curriculum, especially in Kindergarten, help teachers in this process?

This entry was posted in Collaboration, Illustrative Mathematics, Mathematical Practices, Mindset, Systems and Structures, Uncategorized on by .

As a teacher, curiosity around students' mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn't Belong?, I was quick to go back to the classroom and try them with my students. It didn't matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading

This entry was posted in 3rd Grade, 4th Grade, 5th Grade, 6-8 Grade, Illustrative Mathematics, Notice/Wonder, Number Talks, Uncategorized, Which One Doesn't Belong? on by .

The other day, Jamie tweeted about this example problem from a 4th grade program:

After looking at the Standards, Learning Progressions, and discussing the 3rd grade fraction work, all of us on the thread agreed it was an appropriate question for 4th grade but then started to question whether we would give this problem as is, or adapt it. I really appreciate these conversations because they move us beyond 'this problem sucks, don't do it' or 'do this problem instead, it's more fun' to thinking about a realistic thought process teachers can use when working with a text that may not be aligned to the standards or lessons that may not best meet the needs of their students.

While all of the adaptations we discussed sounded similar, I couldn't help but wonder how each impacted so many classroom things in different ways. What may seem like a small change can easily impact the amount of time it takes in class, students' approaches to the problem, what we learn about student thinking, and the follow-up question we could potentially ask.

There is no right or wrong adaptation here, but I wanted to sketch out how each change to the problem impacts teaching and learning.
Change 1: Remove the context and only give the expression 1 - ²⁄6.
This would be the quickest way to do this problem in class. With this change, I would be curious to see how students think about 1 in the expression.

Do they record it as ⁶⁄ 6? Do they draw a diagram? If so, what does it look like? How is it partitioned? Are the pieces removed in the diagram or is there work off to the side? A potential follow-up could be to ask students to write a context to match the expression. I think in their contexts I would have the opportunity to see how they thought about 1 in a different way.
Change 2: Remove the expression and only give the context.
I find anytime there is a context it takes a bit longer because of the time to read and reread the problem so this change would take a bit more class time than the first. This change would give students more access to the problem and I could possibly learn how they make sense of a context, but I wonder what I would learn about their fractional thinking. Since the context pushes students to think about 1 as the whole pizza and also tells them that there are 6 equal pieces, the diagram, partitioning, and denominator are practically done for them.

Because of this, I may not learn if they know 1 is equivalent to ⁶⁄ 6 and may not find out how they represent fractions in a diagram because I imagine most would draw a circle. Since they could do the removal on the circle, I also wonder if I would learn much about how they saw this problem as an expression so I would add that as my follow-up question.
Change 3: Remove the expression and the numbers from the context.
This one ​would definitely take an entire class period as a numberless word problem and probably the longest to plan. Because it takes the longest to plan and implement I really have to think about what I learn above and beyond the two changes previously mentioned if I were to do it this way.

I imagine the scenario could sound like this:

"Sam ordered a pizza cut into equal pieces. He ate some of the pizza and put the rest away for later." or "Sam ordered a pizza. He ate some of the pizza and put the rest away for later."

When I do a numberless problem, my goals are to give students access to the problem and see how they make sense of a context without the numbers prompting them to feel like they have to do something. I have to plan for how I craftily find the appropriate time to let them notice and wonder and plan questions that elicit the subtraction from 1 that I hope to see. I also like to give students a chance to choose their own numbers for problems like these in order to see how they think about the reasonableness of numbers, which adds more time. The hardest part here is getting to the fraction work because I think students could stay in whole numbers as they talk about number of pieces. I can hear them wondering how many pieces it is cut into and how many he ate - neither of which guarantees fractions. So, while this has the potential to get at everything Change 1 and 2 do, a teacher must weigh how much the making sense of context portion meets the needs of his or her students.

All of this for one problem and I haven't even discussed the two most important questions we need to ask ourselves before even making these changes - what understandings are students building on? and what understandings are students building towards?

There is much to think about in planning that is often hard to think about all of the implications of one tiny change to a task, however, thinking about how each of these changes impacts teaching and learning is the fun and exciting part of the work!

After the post, Brian had another change that he posted on Twitter! I wanted to capture it here so it is not lost in the crazy Twitter feed:

I wonder if another change could be tweaking the problem to where you don't tell them it was cut into six slices: Sam ordered a small pizza. He ate 2/6 and put away the rest for later. How much of the pizza did he put away for later?

- Brian Bushart (@bstockus) February 1, 2018

Oh, and still get rid of the expression, in case that wasn't obvious. Perhaps after Ss share solutions, if no one uses 1 - 2/6, you could share it and ask how it also represents the situatuon, assuming that fits with your overarching goals.

- Brian Bushart (@bstockus) February 1, 2018

This entry was posted in 4th Grade, Fractions, Uncategorized on by .

I love when I read a blog post in which I can relate to how the teacher felt, learn from both the teacher and student thinking, want to hear what happens next, and leave with questions circling around in my head. This happened when I read Marilyn's recent post. I really appreciated how her recount of the lesson demonstrated the importance of number choice and the honest way we all have felt when we made a decision during a lesson that we wished we hadn't. It was really interesting to think about how changing the divisor from 4 to 5 changed what students experienced. I cannot wait to hear what they do when they try the original problem and see the remainder as 1 in the balloon context, 25¢ in the money context, 1/4 in the cookie context and .25 on the calculator. Awesome discussions could happen there!

I left this post still thinking about the math talk at the launch of the lesson. I was going to tweet about it, but because it seemed long and I have many questions of my own I want to play around with and revisit, I decided to put it here. I loved the connectedness of the number talk to the division task, and wondered how the recording of those strings could impact division patterns and structure students may see in future lessons. I started playing around with it in my journal in terms of how we think about recording a choral count.

I thought about:

How many problems in each row?

Does horizontal vs vertical recording impact what we see?

What might students notice about the remainders in each row? column?

What might students notice about the change in dividends in each row? column?

Not that I would launch the 4 Problems task with this following string, but I wondered what it would look like to change the divisor and what students might see here:

I think the number of problems with remainders at the bottom of the list versus the top is really interesting.

THEN, I started wondering about ways we could record the remainder and how that may impact how students interpret it? Not sure how this would work in terms of launch and facilitation, but I like thinking about the pink writing here.

Recording is one of those things I get so intrigued by and cannot wait to revisit this post, play around with patterns that could be elicited in different ways and think about tasks in which these talks could be connected.

Thank you, as always, Marilyn for sharing your work - you continue to be such an inspiration! My only hope is one day I can be in the room for one of your lessons!

This entry was posted in 4th Grade, Division, Uncategorized on by .

It is posts like Lisa's most recent one that make me long for more collaboration K-12. I have to admit, when I saw her Twitter post with the words pre-calculus and simplifying complex fractions, my inclination was to skim right by because I would not understand the post anyway. Literally, my only recollection of simplifying complex fractions like the one at the beginning of her post is through a set of procedures I was explicitly taught step by step. However, when I looked at the accompanying image that showed fraction division, I was curious how my understandings of fraction division connected to her pre-calculus work.

I loved reading Lisa's process of making the math accessible for her students because I am sure many would have felt like I did if shown the CPM opener from the very beginning. It is that same process of thinking about what students know and how we can build on it that made me get out my journal and start sketching out connections I was making as I read. In no time, my journal was full of problems, diagrams, concepts, questions and every tab on my computer referenced the progressions, standards, references linked in Lisa's post, and a blank email to Kate and Ashli to jot down my questions for them about the math. Talk about a wonderful rabbit hole to be going down.

The more I read and reread this post, the more I think it could lead to many more posts connecting how students are introduced to ideas in elementary school, the impact it has on later work, and the questions I have as I go. My questions revolve around not only the math, but also how these mathematical ideas build, how our representations impact student understandings, and how there are times when a problems lends itself to one way of thinking versus another.

During my first read, two things I wondered were:

1. How does the way the fractions are written impact the way I think about them?

2. What happens when I have two ways of thinking about fractions and two ways of thinking about division?

How does the recording of the fractions impact the way I think about them?
As the post progressed from an image of a complex fraction to one of fraction division, I felt like Lisa must have felt, wondering what students may know about the complex fraction and why they may struggle. My initial thought was they may not understand that a complex fraction is even division. This may not be the case for most, however based on what I remember from high school, I saw complex fractions as one thing I did operations on. As an elementary school teacher, it seems similar to the difference between seeing a fraction as a number (introduced in 3rd grade) versus seeing fraction as division (introduced in 5th grade). As I looked at CPM's complex fraction and how it was written, I only thought about it as multiplying the numerator by the reciprocal of the denominator because of how I was taught. However, when I looked at the fraction division problem written horizontally, I found myself attending more to each fraction as a number, using what I know about division to find the quotient. Less intimidating to me solely because of the way it was written on the paper. I wonder if this compares a bit to how we record computation problems horizontally versus stacked during number talks to encourage thinking about a problem versus always relying on the algorithm?

I know the fraction division problem means the same thing written either way, but how they are written impacts my thinking a lot. From an elementary perspective where we spend so much time attending to developing understanding of fraction as a number, I am not inclined to really think about what it means to divide the two terms when written as a complex fraction. To that end, I wonder if the opening problem written one way versus another evokes a different meaning for some students?

Knowing that there are things to be learned in between the problems listed below, but in terms of seeing the complex fraction as division where I think about the individual pieces as things in their own right, is one possibly a small transition to the other for me or students like me?

Lisa - I would love to hear more about the transition prompt between the fraction division problems the students were solving and the CPM problem. I think that is a really important piece of what you did so beautifully in this lesson.
What happens when we have two ways of thinking about fractions and two ways of thinking about division?
I think about fractions in the two ways I mentioned above: as a number and as division.

I think about division in two ways: how many groups? and how many in each group?

First, fractions: In 3rd grade, students learn a fraction is a number in which the numerator indicates the number of pieces and the denominator (as the denominator of a unit fraction) represents the size of the piece. For example, we say 3/ 4 is 3 pieces the size of 1/ 4 . This understanding and associated language are so beautiful when students use it to compare fractions and create equivalent fractions. In my 5th grade class, my students were comfortable using complex fractions such as 1/ 2 / 3 when talking about 1/ 6 because they were thinking ½ a piece the size of ⅓ is ⅙ . No division, just reasoning about the pieces and their size. When comparing 4/9 to 5/7, students would use the reasoning that four and a half ninths and three and a half sevenths are equivalent to a half so 5/7 is more than a half and 4/9 is less than a half. I saw a glimpse into how that thinking was not helpful when they asked what happened when there is a . This is where understanding fraction as division would have been more helpful. fraction in the denominator

In 5th grade students also learn about fractions as division . In terms of sharing situations, they learn that 5 things shared by 3 people results in each person getting 5/3 of the things or 5 divided by 3. In these situations, thinking about 5 pieces the size of 1/3 is not particularly helpful in solving, but division is. However, when it comes back to interpreting the solution, 5 pieces the size of 1/3 is needed.

Questions I am thinking about at this point:

How does the complex fraction in the post relate to either or both of these ways to think about fractions?

How does the way we represent fraction division relate to one or both of these ways to think about fractions?

Now, division: In 3rd grade, students learn division in two contexts: how many in groups and how many in each group. In 5th grade, students use those understandings to divide whole numbers by unit fractions and unit fractions by whole numbers. Those two meanings of division carry into middle school to divide fractions by fractions and conceptually understand the reason we multiply by the reciprocal.

After reading Kristin and Bill's series of posts on fraction division, I am now constantly thinking about how the context (interpretation) for division impacts the way students represent and solve a problem. I know changing the way I think about the division context changes how I represent the problem as well as how I operate with the reciprocal.

Questions I am thinking about at this point:

Does one context of division connect more closely with the CPM complex fraction problem?

Does the visual fraction model of the the division problem impact the way students approach the complex fraction problem?

Is an array representing both fractions being divided helpful in this complex fraction?

Is one bar model representing both fractions on one helpful in this complex fraction?

Is one way of representing it more helpful than the other?

Obviously, I have a lot to read about how a problem such as the one Lisa posed progresses after middle school but after seeing the division of fraction problem, I am even more intrigued to see how these ideas progress from the time they are introduced. I am so curious when certain ways of thinking are more helpful than others and how we can construct learning experiences that help all students have access to the mathematics in a lesson in the way Lisa did.

This entry was posted in 5th Grade, Division, Fraction Division, Fractions, Uncategorized on by .

I never tire of conversations about the 3rd - 5th grade fraction progression because after each one, I leave with the desire to reread the Standards and Progressions with a new lens.

A few weeks ago, a conversation about 3rd grade fractions sent me back to the Standards with a #pairedtexts type of lens. Unlike the hashtag's typical MO of pairing contrasting texts, I was looking for standards that connected in a meaningful, but maybe unexpected way. By unexpected, I don't mean unintentional, I mean the two standards are not necessarily near-grade or in the same strand, so the connection (to me) is not as obvious as one standard building directly toward another.

The conversation focused on this standard:

With that standard in mind, imagine a 3rd grade student is asked to locate 3/4 on a number line on which only 0 is marked.

I expect a student would mark off the 1/4's starting at 0 and write 3/4 above the point after the third 1/4 segment. What exactly is the student doing in that process?

Is the student adding?

Is the student counting?

Is the student doing both?

How does adding and counting look or sound the same in this scenario? different?

This is where I find pairing two standards fun and interesting to think about because it demonstrates how important seemingly unrelated ideas work together to build mathematical understandings. It is also really fun to think about how a standard in Kindergarten is so important for work in grades 3-5 and beyond.

In this scenario, I think we instinctively believe students are adding unit fractions when asked to place 3/4 on the number line because the standard is in the fraction strand and therefore we consider all of the work to be solely about fractions. We also sometimes impose our thinking on what students are actually doing in this task. For example, you could imagine the student marking off the fourths, stopping after the third one, writing 3/4 and say the student was adding 1/4+1/4+1/4 to get to the 3/4 because they moved along the number line. If this is the case, then the standard would pair with this 4th grade standard:

Don't get me wrong, those standards definitely pair as students move from 3rd to 4th grade, however, since the scenario is about a 3rd grade student, pairing it with a higher grade level standard doesn't seem to make sense in terms of what students are building on. Right here, it is really interesting to pause and think about how building fractions from unit fractions, locating a fraction on a number line, and adding unit fractions are slightly different things a progression.

When I think about the student locating 3/4 in 3rd grade, I hear counting (with a change in units) and would pair that 3rd grade standard with this Kindergarten counting and cardinality standard:

However, because the 3rd grade work is on a number line and the arrangement and order does matter, I would have to add this 2nd grade measurement standard into the mix, but take off the sum and differences part:

So, instead of a #pairedtext, I now think of it more as a #CCSSMashup to create this standard:

With that mashup in mind, I went back to the progressions documents to look for evidence and examples of this.

In the 3rd Grade NF Progression these parts jumped out at me as being representative of this standard mashup:
The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions. The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1 /3 .
There is also a great "Meaning of Fractions" video on the Illustrative Mathematics site that explains this idea with visuals.

There are so many of these great mashups in the standards, especially in the fraction strand, that I find incredibly helpful in thinking about how students coherently learn mathematics.

I look forward to hearing your favorite #CCSSMashup!

This entry was posted in 3rd Grade, Fractions, Standards & CCSSMashup, Uncategorized on by .

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2023 Math Collaborative Summer Book Club Resource Deck STREET DATA

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2023 Math Collaborative Summer Book Club Resource Deck STREET DATA

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Blog | The Educator Collaborative Community

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by The Educator Collaborative Fellow Shawna Coppola

What are the professional texts that seem to "live" at your desk-that rarely make it back to their home on your shelves or tucked away in your digital folders? So often, these texts-if they are of the analog variety-are our most loved and appreciated ones, made soft from so many hands handling their weight, their edges worn and curled from incessant travel from desk to bag to desk again.

For me, Street Data: A Next-Generation Model for Equity, Pedagogy, and School Transformation by Shane Safir and Jamila Dugan is one such text. While it is focused, overall, on initiating transformation at the school and/or district level, I have found it to be just as useful when working toward transformation at the classroom level.

In Chapter 1, Safir, a self-described "white woman of Irish and Jewish descent," writes how complicit she was as a young person in both "buying into the white supremacist structure of schooling" and "subscribing to the myth of meritocracy." She wonders (much like author Alfie Kohn wonders in his popular 2003 essay "What Does It Mean to Be Well-Educated? ") what a "radical reimagining" of an educational system would look like that would, essentially, upend how we perceive of both "data" and "success." The "street data" paradigm that she and Dugan conceptualize centers around a framework that incorporates the use of three distinct "levels" of data that Safir first created in her 2017 book The Listening Leader: Creating the Conditions for Equitable School Transformation . These three levels of data are:

SATELLITE DATA (conceptually furthest from the learner), such as that we gain from high-stakes standardized assessments, graduation rates, or school health records. Satellite data "illuminate [large-grain] patterns" around achievement, efficacy, or equity and can point educators and other stakeholders (e.g., families) "in a general direction." However, satellite data is most likely to privilege dominant ways of knowing and/or of "doing school" and thus is highly likely to perpetuate White supremacy, linguistic discrimination, and anglocentrism.

MAP DATA (closer, but still somewhat removed from the learner), which can "point us in a slightly more focused direction" and which we often gather from sources like running records or curriculum-based assessments (e.g., locally-designed performance assessments or end-of-unit tests). Considering that the majority of teachers working in the U.S. identify as White , map data is also highly vulnerable to the perpetuation of dominant ideas around what kinds of knowledges and skills (and how we demonstrate those knowledges and skills) "count."

STREET DATA (closest to the learner), which offers the most potential for contextualizing student learning and which, in theory,** is most likely to be both asset- or strength-based and culturally sustaining. We can most effectively gather street data from close, sustained observation; from interviewing or conferencing with students; and/or from inviting students to share their own learning experiences, habits, and processes. "Street data," Safir and Dugan write, "disrupt [the] dynamics" of both white supremacy and white saviorism by "bringing student voice to the forefront" of educational data collection.

In my work as a consultant, I have found this conceptualization of "data" to be transformative for the educators with whom I am privileged to work-particularly for those who entered the profession long after Gretchen Owocki and Yetta Goodman brought the concept of "kidwatching" to the educational masses (or who never had the opportunity to learn about the Reggio-Emilia approach to education). Rather than lean too heavily on the kinds of data that are less contextualized and less reflective of students as whole humans , we can use street data collection as part of a qualitative research approach that helps us tell a more humanizing, more holistic "story" of our learners -that, as Safir and Dugan write, centers students as "human beings whose experiences are worthy of careful study and deep listening."

However, as with any educational paradigm, framework, or approach that pushes back against dominant practices that work to maintain the status quo, many of the educators with whom I work have envisioned the possibilities that some stakeholders ( e.g., colleagues, administration and/or families) may perceive the information gleaned from street data to be less valuable than that gleaned from the other, often more highly-regarded, levels of data. They have requested some language that they might use to help them illuminate for others why such data is so important. I share this set of slides here in the hope that the language offered within may be useful to others committed to incorporating street data collection into their educational practice.

This being a dynamic document, though-and me being a fallible human with a lifetime of learning to do-I am well aware that revisions may need to be made to some of this language in order to provide the best argument for systematically collecting street data as part of a more equitable practice of assessment. To that end, I invite anyone reading this to offer your own suggestions in the comments below. Our advocacy around a more holistic and humanizing approach to data collection that centers our students, and all of their complexity, will be infinitely stronger the more collective our efforts are.

**NOTE: I say "in theory" because the work of ensuring that our data collection efforts are more culturally sustaining is dependent on our commitment to interrogating our own biases and assumptions when it comes to teaching and learning.

If you enjoyed this post, be sure to check out from our Spring 2023 Gathering as well as her upcoming book on anti-oppressive literacy education from Routledge, due out this fall.

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Advocating for a More Humanizing Approach to Data Collection - Blog | The Educator Collaborative Community

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by The Educator Collaborative Fellow Shawna Coppola What are the professional texts that seem to "live" at your desk-that rarely make it back to their home on your shelves or tucked away in your digital folders? So often, these texts-if they are of the analog variety-are our most loved and appreciated ones, made soft from [...]

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2023 Math Collaborative Summer Book Club Text

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2023 Math Collaborative Summer Book Club Text

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Street Data: A Next-Generation Model for Equity, Pedagogy, and School Transformation

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Street Data: A Next-Generation Model for Equity, Pedagogy, and School Transformation [Safir, Shane, Dugan, Jamila] on Amazon.com. *FREE* shipping on qualifying offers. Street Data: A Next-Generation Model for Equity, Pedagogy, and School Transformation

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Math Collaborative 2023 Summer Book Club Zoom Info

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Math Collaborative 2023 Summer Book Club Zoom Info

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https://us02web.zoom.us/j/4219107593

Meeting ID: 421 910 7593

We will meet on ZOOM!

Monday, July 3, 2023
Chapters 1 & 2

Monday, July 24, 2023
Chapter 3 & 4

Monday, August 7, 2023
Chapter 5, 6 & 7

Monday, August 14, 2023
Chapters 8 & 9

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A free way to practice multiplication! You can set up a class account and keep track of practice & fluency.

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A free way to practice multiplication! You can set up a class account and keep track of practice & fluency.

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These beautiful flash cards use spaced repetition to teach multiplication facts. Achieve fluency with just five minutes of practice per day!

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Unleash your creativity with the world's best virtual manipulatives!

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The problem with timed multiplication fact tests

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The problem with timed multiplication fact tests

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Download PDF Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts By Jo Boaler, Professor of Mathematics Education, co-founder youcubed With the help of Cathy Williams, […]

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Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning (Corwin Mathematics Series)

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Amazon.com: Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning (Corwin Mathematics Series): 9781544374833: Liljedahl, Peter: Books

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High-quality K–12 Curriculum and Assessments

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Math Collaborative 2022 Summer Book Club Text

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Math Collaborative 2022 Summer Book Club Text

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Sparking Connections, Joy, and Wonder Through Read-Alouds and Discussion

Available Now!

Exploring the intersection of math and literacy, Mathematizing Children’s Literature asks the question, “What if we approached any story with a math lens?” Throughout the book, we learn how to select picture books to meet our math and literacy goals, and how to plan and facilitate three different kinds of read-aloud discussions. Mathematizing Children’s Literature includes dozens of suggestions for mathy read-alouds as well as guidance on choosing books in your own collection to read again with a math lens.

You can create a quote by adding products to your cart and selecting Quote in checkout.

This title is also available through major book retailers, including Amazon.

Shipping Details

In stock. Free ground shipping. No minimum.

“Allison and Tony have crafted such a joyful resource for educators to leverage together a love for literature, a love for math, and an intense desire to better listen to and honor the voices and lived experiences of the children who grace our classroom spaces.”

Angela Chan Turrou
co-editor of Choral Counting & Counting Collections

Many teachers use traditional counting and shape books in math class. But what would happen if we approached any story with a math lens? How might mathematizing children’s literature give learners space to ask their own questions and make connections between stories, their lives, and the world around them? These are the questions Allison Hintz and Antony T. Smith set out to explore in this book as they invite us to consider fresh ways of using interactive read-alouds to nurture students as both readers and mathematicians.

Inside Mathematizing Children’s Literature, you’ll learn how to do the following:

Select picture books according to the goals of the read aloud experience.

Plan and facilitate three kinds of read aloud discussions—Open Notice and Wonder, Math Lens, and Story Explore.

Utilize Idea Investigations—experiences that invite students to pursue literacy and math-focused ideas beyond the pages of the read aloud.

Connect with students’ families and communities through stories.

Along the way, Allison and Antony offer a wide range of picture book suggestions and appendices that include ready-to-use planning templates, a note-taking form, and a bookmark of guiding questions. Mathematizing Children’s Literature is a practical resource you’ll find yourself referring to frequently.

About the Author(s)

Table of Contents

Foreword by Christopher Danielson

Acknowledgments

Note to Readers

Chapter 1: Celebrating the Joy and Wonder of Children’s Thinking

Chapter 2: Mathematizing Interactive Read-Alouds

Chapter 3: Book Types and Selection

Chapter 4: Open Notice and Wonder Reads

Chapter 5: Focused Reads: Story Explore and Math Lens

Chapter 6: Idea Investigations: Extending the Read-Aloud Experience

Chapter 7: Learning Together as Educators

Chapter 8: Family and Community Connections

Appendices

Appendix A: Open Notice and Wonder Planning Template

Appendix B: Questions as Refrain Bookmark

Appendix C: Focused Read Planning Template

Appendix D: Idea Investigation Planning Template

Appendix E: Listening to Students at Work Note-Taking Form

Professional Resources and References

Children’s Literature Bibliography

Index

Credits

Reviews

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Mathematizing Children’s Literature | Stenhouse Publishers

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Available Now! Exploring the intersection of math and literacy, Mathematizing Children’s Literature asks the question, “What if we approached any story with a math lens?” Throughout the book, we learn how to select picture books to meet our math and literacy goals, and how to plan and facilitate thr…

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Math Games for Tutoring

If you are a teacher or tutor with a student who needs more practice with a particular skill, this is the place for you! Rather than using tedious flashcards or worksheets, use these games to provide that same practice in a more engaging and enjoyable way. This is a curated collection of EFM games and puzzles that are quick to set up and explain, and they provide lots of practice in the indicated skill. Have fun!

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Math Games for Tutoring - Early Family Math

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If you are a teacher or tutor with a student who needs more practice with a particular skill, this is the place for you! Rather than using tedious flashcards or worksheets, use these games to provide that same practice in a more engaging and enjoyable way.

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Eyes that Speak to the Stars (Cotsen Inquiry)

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Eyes that Speak to the Stars (Cotsen Inquiry)

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A Padlet dedicated to own voice authors and books in the Asian community.  Theme of intergenerational relationships also surfaces. Final columns mathematize culturally relevant read alouds, using them as authentic contexts and shared schema for problems and number sense and reasoning routines.
https://padlet.com/smathewson2/b9hqyq65x9lrsh87

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Problem Sets with Context Derived from Culturally Relevant Read Alouds Set 3

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Problem Sets with Context Derived from Culturally Relevant Read Alouds Set 3

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Problem Sets with Context Derived from Culturally Relevant Read Alouds Set 1

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Problem Sets with Context Derived from Culturally Relevant Read Alouds Set 1

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Gr 1  Separating Probs Dec 21-mid Jan 22

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Problem Sets with Context Derived from Culturally Relevant Read Alouds Set 2

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Problem Sets with Context Derived from Culturally Relevant Read Alouds Set 2

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Gr. 1 Problems MidJan 2022-MidMarch 2022 CRRA (varied unknowns)

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Gloria Ladson-Billings - Critical Race Theory

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Using Fingers to Count in Math Class Is Not ‘Babyish’

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Evidence from brain science suggests that using the stigmatized technique is essential for developing mathematical thinking.

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Math Collaborative 2021 Summer Book Club Text

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Math Collaborative 2021 Summer Book Club Text

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Rachel Lambert's collection of math + disability resources

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Insπre: Rethinking the Purpose of Math Education | Tisha Jones | TEDxGeorgiaStateU

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It Starts With... ME! CMC — South 2019 Ignite feat. Dr. Kristopher J. Childs

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Subitizing! This ability to instantly "see" how many is a critical component in young students developing number sense. There are two types:

1. Perceptual subitizing -the ability to see a small number of objects and know how many there are without counting. For example, this skill allows us to realize we "rolled six" on a die without counting. It is usually an instantaneous realization, and generally applies to small amounts of objects.

2. Conceptual subitizing -the ability to see more than one group and add them together to find the total. This one takes a little more effort as it requires recognizing smaller groups and adding them up.

Studies cite the use of subitizing in primary grades as the key to students being able to pass algebra. It builds the skills needed to move toward proficiency in a sense of equality and balanced equations, spatial sense, and number sense. We know subitizing is an important part of building procedural fluency through conceptual understanding, but how do we prepare students for this?

Students need tangible experience working with number sets-and if we can create a tangible experience that is also FUN, success will come much more quickly. A common strategy is to work with dot arrangements, but any simple symbol can be used, such as stars or squares. To add the fun factor, I expanded on an idea from my colleague, Mary Gong, who used spaceships in her primary classroom. Her "spaceships" were nothing more than paper plates with large dots made with stickers or Bingo dauber, but were effective nonetheless. You can get as creative as you like here but be sure to stick with simple shapes and vary the patterns on each. First, I randomly provided students with paper plates with small sets of 3-5 stars. I asked them to silently maneuver their "spaceships" to another student with a matching number of dots. I then led a discussion, using the prompt, "How did you know the set has the same number when it looks different?"

After this activity, I placed students in two groups to explore more concepts. I asked Group 1 to "fly" to a student in Group 2 with a spaceship that had fewer dots than their own. Then, I asked Group 2 to fly to a student in Group 1 with a spaceship that had more dots. I also had each student hunt for a copilot, whose spaceship combined with his/her own, totaled a certain number. You can easily do this in a whole group setting and mix up the instructions however you like.

Let's get started! Here are a few examples of how you can help young learners begin to access subitizing.

Step 1 Matching equivalent quantities-build the idea of conservation of numbers, same number different representation

Step 2 Comparing number and combining multiple sets to create sums

Step 3 Building relationships by linking Real World, Linguistic, Representational, and Symbolic using the above activities

Once students become familiar with subitizing, you can initiate lively discussions to use as springboards for other subitizing challenges and whole number sense routines. For example, display a dot arrangement, making sure numbers are appropriate, for 3-5 seconds then ask, "What did you see and how did you see it?" I used arrangements of eight.

Many of my students responded with eight, so It was very important for me to accurately represent what they saw to inform our discussions. Here are some examples:

These kinds of activities are precursors to helping students have access to algebraic concepts. They help create a sense of equality, conservation of numbers, decomposing, and more. Being able to recognize equations as being balanced on both sides instead of just where the one correct answer goes is a huge development and deepens number relationships for students. After students have had some time to work with dot arrangements like this, let them have opportunities to explore true or false statements such as 4 + 2 + 2 = 6 + 1 + 1.

As students gain confidence in subitizing and their number sense, it is important to consider ways to differentiate for students based on their points of understanding. These could easily be created during centers or learning stations. Having Bingo daubers or pre-cut, one-inch squares will allow students to quickly create their own arrangements based on the numbers they need to practice. Some students can be at five while others are at nine, etc. Slide these into a sheet protector so students can easily exchange their arrangements to find the different ways they see to make the total number represented. If ready, their equations should match their drawn models. One caution, make sure students are actually subitizing. Circling ten individual dots to say 1+1+1+1+1+1+1+1+1+1=10 is not actually subitizing! Here is a sample launch: Use eight color tiles to make an arrangement. What numbers match your arrangement?

Student samples:

Note, the first student was able to accurately subitize and create equations based on what he/she saw. The second student shows great knowledge of working with ten and flexibility of thinking, but is no longer actually subitizing. It looks like this student is ready for a higher number!

Communication and productive discourse are essential to help students clarify information, deepen their reasoning, and move learning to long-term memory. Consider helping them learn with some sentence frames such as these:

I agree with _____ because _______.

I do not understand _________. Can you explain this again?

I disagree with _______ because _______.

How did you decide _______?

I am thinking about this problem differently. Here is the way I solved it: _____________.

My way is different from_____. Instead of____, I_____________.

Can you explain the difference between ________and ________?

Interested in learning more about counting, subitizing, and other critical important foundations for math learning? You can explore Lisa's presentation from the recent NCTM 2019 conference, Tool Time: It's All About the Connections!, by visiting https://mathsolutions.com/contact-us/speaker-presentations/

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Subitizing! This ability to instantly "see" how many is a critical component in young students developing number sense. There are two types: 1. Perceptual subitizing -the ability to see a small number of objects and know how many there are without counting. For example, this skill allows us to realize we "rolled six" on a die without counting.

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Learning from One Another: Lessons in (Educational) Excellence | Dr. Betina Hsieh | TEDxCSULB

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The CGI Math Teacher Learning Center provides math teacher professional development seminars nationwide. CGI, or Cognitively Guided Instruction, is the preeminent method for teaching math for understanding.

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Teaching, Learning, and Loving Mathematics

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"Let's talk about the common misconceptions kids have with fractions."
I cringe.

I, like many others, have been thinking a lot about the language we use to describe children in the mathematics classroom. Words matter. Words convey and confirm how we treat each other. When it comes to mathematics, seemingly innocent words have a lot of power.

The word that made me cringe was misconceptions. Why this word? When you say misconception, you are suggesting that someone is thinking about something wrong. If you are trained as a teacher to notice students' misconceptions, you will find yourself listening for what they don't understand.

Knowing what children do not yet understand is useful to a point, because it can give you ideas of what to teach. However, the balance of research in learning says that children are not blank slates. That means that they already know something, and odds are even if you have not yet taught a particular concept to your kids, like say subtraction above 10 with your first graders, that they know something about the operation of subtracting, situations when you separate and compare things, and something about counting and addition that will help them to make sense of what you want to teach. In short, as the authors of the Cognitively Guided Instruction series have said for years now, "Children are sense-makers." They are always making sense of new information in relation to what they know.

So rather than looking for the holes in kids knowledge, teachers should be listening and looking for what sense-making kids are engaged in, and what makes sense to the student.

Let's consider Julieta - a 9 year old emergent bilingual student from Ecuador who has been in the United States for a couple of years (not a real person, an amalgam of kids I've met over the years). Her teacher approaches her during a lesson on division. Julieta is to show how to "share 12 cookies equally between 4 people." She counts 12 counters, then proceeds to push them around on the desk until she has 4 piles, of 3, 3, 2, and 4.

Uh oh, her teacher is thinking, Julieta doesn't understand that she has to make each pile have the same amount of cookies. Julieta has a misconception about how division works. The teacher then asks Julieta if she's solved the problem. Julieta nods. The teacher now asks, " Is each pile the same?" And Julieta says " no". The teacher, says "make each pile the same", satisfied she addressed the misconception. Julieta takes a moment, then reshuffles her counters, dealing two out to four piles, and puts the rest off to the side.

Glancing at her work, the teacher says, "we'll look at how other people solved it later and it might give you ideas" and moves on, making a mental note that Julieta doesn't understand division.

What is the consequence of the teacher's thoughts over time? Julieta's teacher is going to start to form ideas of the kinds of mathematical people her students are, using whatever patterns of thinking she has been accustomed to. If her major sorting of students is who "gets it," and who has misconceptions, it's not out of reach to start calling kids "good" or "bad" at math. It's a slippery slope towards unwittingly condemning students as deficient, or lacking skills.

Julieta's teacher could approach her in another way. If she pays attention first to what kids are doing, and the sense they are making, Julieta's teacher might first notice that:

1. Julieta understands that this situation calls for counting out 12 cookies, which she does, next

2. Julieta understands that 4 people are going to all get some cookies, and she does that well too, as everyone of four piles got some cookies, and finally

3. Initially, Julieta gave out all the cookies, and only when the teacher asked her a question that she may or may not have understood did she put some cookies aside.

So what can we conclude? Julieta understands a lot about partitive division: start with a total, give out the total to the number of sharers, use all the total. This is evidence of her current conception of division. Now, we can be curious about her thinking. Imagine if the teacher had approached her and said, " Tell me more about your 12 cookies?" Julieta might have explained that she gave them all out. Continuing to presume that Julieta is making some sense out of the situation, the teacher could have continued, "Tell me more about your piles." Julieta might then have said, "I get three, and my sister gets 3, but my brother is small so he gets 2, and so my mom gets 4." Ah-ha, the teacher might now say to herself, I see how she's making sense of this situation. Now the teacher can make a choice on how to use Julieta's excellent sense making to help her further her understanding of division, and also to talk with the class about how dividing up in mathematics isn't always the same as how we divide up at home. And the mathematics goes on.....

By training ourselves to notice what students do understand, and what actions they take based on their understanding, we can challenge deficit notions of students. By making it our job to elicit and understand student thinking, we can help ourselves move away from the damaging language that labels some kids as "good at math" and others as "bad at math."

Here's my first challenge for us all as we prepare to get back into school mode: Let's take a stand. Let's all agree that there are no misconceptions in mathematics. There are current conceptions. And we can all revise our thinking with new evidence, leading to new conceptional understanding.

Here are some ways you can help yourself move from deficit to asset-based language in the mathematics classroom:

When planning your math lessons, Instead of anticipating "popular misconceptions," prepare to focus on what kids actually do, and how they describe why they did what they did. If you need to talk about what students understand about a concept, you can use the sentence starter The current conception seems to be...

When circulating while students work, ask yourself how is the way this student is thinking useful in another setting? For example, Julieta was thinking reasonably about sharing cookies based on her experience that her little brother shouldn't get as much as she, her sister, or her mom. This is perfectly logical outside of the arbitrary rigidity of school mathematics. This provides insight into current conceptions.

Put the emphasis on difference, not deficit. Help students make sense of how different situations call for different kinds of reasoning. This will help us stop sorting kids into piles of getting it or not getting it, and help us think about what kids actually understand and how to support them to make new connections or go deeper into their own thinking.

I'll also plug the book Schooltalk: Rethinking What we Say About -and to- Students Everyday, by Mica Pollock. I'm reading it right now with some people and so far it's great. This quote from the introduction has a lot to do with what I'm saying about doing away with deficit talk such as "misconceptions", and Mica Pollock ties attending to how we talk as a key part of equity, in particular in how we understand who our students really are:

"In our schooltalk, all of us sometimes repeat common, habitual comments about young people or their communities that are fundamentally inaccurate and under-informed. Such 'scripts' are the simplifying, familiar claim we reach for when we talk about young people and about education. Much of the time, these 'scripts' have a shallow grasp on facts about young people's lives and instead adhere to premade ideas about young people. Dangerously, scripts misrepresenting young people in general can keep adults from fully supporting actual young people - and keep young people believing falsehoods about themselves and their peers."

This school year, let it be the school year that we end deficit thinking and talk in all its forms, and that we all grow more aware of the impact of what we say and don't say about and to students.
Sources:
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B.(2012). Children's mathematics: Cognitively Guided Instruction, 2nd Ed. Ports-mouth, NH: Heinemann

Pollock, M. (2017). Schooltalk: Rethinking What we Say About -and to- Students Everyday. New York, NY: The New Press.

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Back to School Part 1: Challenging Deficit Language in the Math Classroom

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"Let's talk about the common misconceptions kids have with fractions." I cringe. I, like many others, have been thinking a lot about the language we use to describe children in the mathematics classroom. Words matter. Words convey and confirm how we treat each other. When it comes to mathematics, seemingly innocent words have a lot...

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Ed-Talk: Learning with an Emotional Brain - Mary Helen Immordino-Yang

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Play with Deck of Cards, source available! Pure vanilla JS - no dependencies.

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Are you ready to count beyond one hundred? This online 120 chart makes counting to one hundred and beyond fun!

How does it work? You can color different squares of the 120 chart. Just click on a color and then tap the squares. Show a pattern or count by multiples of 5s or 10s. Or, just practice counting by 1s and keep track by coloring the squares as you go. You can also black out some or all of the numbers with the color black. Then, you can ask students which numbers are missing. Show the numbers again with the color white or another color of your choice. Clear the board by pressing the "clear" button. Make the 120 chart blank by pressing the "hide" button. Show the numbers from bottom to top by pressing "Up".

Once students are comfortable counting to one hundred, it's time to show them how the pattern continues. A 120 chart shows that perfectly! With a 120 chart, students can also practice addition, subtraction, and multiplication operations. There are many creative ways to use the online 120 chart on its own or alongside a printable 120 chart. Here are a few ideas:

Ask children to fill in missing numbers that you've blacked out.

Ask children to mark a given number on the 120 chart. Then, have them add 10 or subtract 10. What patterns can they notice?

Ask students to skip count by 3s, 5s, or 10s from 1 to 120.

The 120 chart is great for whole-class lessons, group work, centers, or individual learning. You can even ask students to use it for homework assignments!

Visit classplayground.com for more hundred chart activities and printables.

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120 Chart | Free Virtual Manipulatives | Toy Theater

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Use this interactive 120 chart to practice counting to one hundred and beyond. It's also great for modeling math problems, skip counting, and more!

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Mental Math - visualize adding and subtracting 10s

Common Core Connections - Grades K to 2

Add and subtract with 100.
Count to 120, starting at any number less than 120.
Given a two-digit number, mentally find 10 more or 10 less than the number.
Subtract multiples of 10.

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Interactive Hundreds Chart | Math Playground

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Interactive Hundreds Chart at Math Playground.com! Explore addition, subtraction, patterns, and more.

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Launch is how we use the first 5-15 minutes of our math block for purposeful, discussion-rich number sense routines in our classrooms. Learn more!

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Marvelous Measurement Div! Feb. 23, 2021 - Wakelet

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We built Wakelet using the latest technologies to make it faster and easier to use. Unfortunately, your browser doesn't support those technologies.

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January 26, 2021 Celebrating Student Thinking - Wakelet

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We collaborated during our Math Collaborative Zoom to design a problem &amp; develop differentiated sets for grades TK-5. During this Twitter chat, we celebrated student thinking by noticing the brilliant details of their thinking.

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Steve Wyborney\\\'s Blog: I\\\'m on a Learning Mission.

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For my educator friends who have access to Google Slides, but do not have access to PowerPoint, this post is for you. Since Google Slides does not have the animation that makes the Splats become transparent, the Splat! reveal does not work in Slides. Now - with these materials - it will. I've rewritten the first 20 Splat! lessons and the first 20 Fraction Splat! lessons with a different method so they will now include the reveal when using Google Slides.

The content of the lessons is the same as the original lessons.

Visit these links to see the original videos about how to use the and the Fraction Splat! slides.

Quick Note: The file sharing power of Google Drive is impressive and highly useful. However, for the purposes of sharing these resources, I need to use a method that I can personally maintain, update and support - and I've learned that the best way to provide resources from my blog is through the download links and the sign-up forms.

I hope you enjoy these Splats! formatted to work in Google Slides!

Splat Through 10
How many dots are under the Splat?
SPLAT 1.1 - numbers through 10 (Google Slides compatible version)
SPLAT 2.1 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.2 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.3 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.4 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.5 - with numbers 12-20 (Google Slides compatible version) SPLAT 1.2 - numbers through 10 (Google Slides compatible version)SPLAT 3.1 - multiple splats (Google Slides compatible version) SPLAT 3.2 - multiple splats (Google Slides compatible version) SPLAT 3.3 - multiple splats (Google Slides compatible version) SPLAT 3.4 - multiple splats (Google Slides compatible version) SPLAT 3.5 - multiple splats (Google Slides compatible version)
SPLAT 4.1 - instant multiple splats (Google Slides compatible version) SPLAT 4.2 - instant multiple splats (Google Slides compatible version) SPLAT 4.3 - instant multiple splats (Google Slides compatible version) SPLAT 4.4 - instant multiple splats (Google Slides compatible version) SPLAT 4.5 - instant multiple splats (Google Slides compatible version) Note: These lessons use a single splat along with a mixture of unit fractions and whole numbers. In Level 11, you control the Splat! SPLAT 11.1 - fraction splats (Google Slides compatible version) SPLAT 11.2 - fraction splats (Google Slides compatible version) SPLAT 11.3 - fraction splats (Google Slides compatible version) SPLAT 11.4 - fraction splats (Google Slides compatible version) SPLAT 11.5 - fraction splats (Google Slides compatible version) SPLAT 1.3 - numbers through 10 (Google Slides compatible version)Note: These lessons use a single Splat! which appears at the same moment as the whole numbers and fractions. Each question instantly presents missing information. SPLAT 12.1 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.2 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.3 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.4 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.5 - Instant Fraction Splats (Google Slides compatible version) Multiple Splats! with Fractions Note: These lessons include multiple Splats! and you control the timing of the Splats! The entire screen can be seen, the total can be determined, and then Splat! Remember: Each Splat! that is the same color must be covering the same number. SPLAT 13.1 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.2 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.3 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.4 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.5 - Multiple Fraction Splats (Google Slides compatible version)
Note: These lessons use multiple Splats! which appear at the same moment as the whole numbers and fractions. Each question instantly presents missing information. Remember: Each Splat! that is the same color must be covering the same number. SPLAT 14.1 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.2 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.3 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.4 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.5 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 1.4 - numbers through 10 (Google Slides compatible version)
SPLAT 1.5 - numbers through 10 (Google Slides compatible version)

Splat Through 20
How many dots are under the Splat?

Multiple Splats
Note: Splats that are the same color must cover the same number.

Instant Multiple Splat
Note: An additional feature of this level is the opportunity to ask, "What could the total be?"

Instant Fraction Splat!

Multiple Instant Splats! with Fractions

After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.

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Splat! for Google Slides - 40 Lessons - Steve Wyborney's Blog: I'm on a Learning Mission.

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For my educator friends who have access to Google Slides, but do not have access to PowerPoint, this post is for you. Since Google Slides does not have the

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Next Twitter Chat will be TBD @missmath8 #MathCollabChat #MathCollaborative 

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Next Twitter Chat will be TBD @missmath8 #MathCollabChat #MathCollaborative 

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We will be admiring the work of students in our shared problem.  Please join us!

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"If I love you, I have to make you conscious of the things you do not see." -James Baldwin

Anti-racist work in all schools is essential. It is the exercise of hope, the practice of undoing and dismantling systems of oppression, the practice of freedom and of truth-telling. Anti-racist work is the practice of healing and of restoring; it is a practice of love.

We live in a time where it is increasingly clear that Black lives are deeply undervalued and violently treated. And as the words anti-racist and equity become dangerously trendy, educators should pause and grapple with what they truly mean.

It is important that we have a shared understanding of what anti-racism in schools looks like, that we have an understanding of what it is not, and that we embrace the understanding that anti-racist work is never completely finished, nor does it always look the same. I hope, however, we may begin to think about anti-racist work in schools in ways that are holistic and practical.

Anti-racist educators believe in love.

Many educators operate from the premise that they love their students. In fact, many harmful and violent teachers operate from a place of "love." But anti-racist educators understand that love and silence are deeply contradictory. The love that underpins our practice is not the form of love often associated with passivity and inaction. For anti-racist educators, love is action. Love is sharp. Love is truth-telling. Love is fighting for what it is right. Love is doing what is right.

Anti-racist educators, particularly those who teach Black students, understand that part of the act of love is understanding what this country has intentionally done-and continues to do-to Black bodies. It is love that compels our practices.

Anti-racist educators understand that, in love, they must never be silent. Ever. Anti-racist educators understand that their positions as teachers, leaders, policymakers and social workers are positions of great privilege and power, and that they have the ability to leave this world better than they found it.

Anti-racist educators study and are committed to deepening their critical consciousness.

Anti-racist educators understand that they must be lifelong learners. They believe in the power of developing a critical consciousness, which comes from reading, from studying, from deep scholarship, from humility, from listening to and engaging with others, and from constantly examining and re-examining their own ideas, beliefs and truths.

Educators who are seeking to be anti-racist must understand the nuanced, complex, deliberate design of racism in the United States and across the diaspora. Educators in the United States who are seeking to be anti-racist have to acknowledge and recognize that this country was built upon racist principles. They must be willing to commit to a lifestyle of studying what racism looks like in all its forms.

This means that we have to know how to identify racism; how racism is carried out and built in systems and structures; how people hold, sustain, maintain and perpetuate racism; and how people of color can internalize racism.

Anti-racist educators know and continue to seek out the names of those who are never called upon and the work that is hardly ever revered. They recognize that we should study more deeply the tactics of the Black Panther Party and the Freedom Schools. Dr. Gholdy Muhammad is a true leader and scholar-warrior in this regard, as her work focuses on the ways the Black Literary Societies of the early 1800s operated and creates a framework for how we should be using the power of literacy to teach and empower Black students.

Anti-racist educators believe in the intellectual power of teaching. They believe they have to honor the practice and take teaching very, very seriously because they know and understand how schools have been used either to oppress or to liberate.

Anti-racist educators move away from checklists and embrace a holistic approach.

Anti-racist education and anti-racist schooling cannot be packaged or prescribed, arranged into a checklist, rubric or formula. Anti-racist educators understand that anti-racist work begins with the self. They begin by grappling with their beliefs, mindsets, philosophies and biases about the world, education and their students. They work to become conscious of the intentional, multiple ways schools mirror society and how all aspects of school systems are designed to uphold the oppressive aims of the society in which they operate.

Anti-racist educators recognize that schools are doing exactly what they were built to do in this country: Exclude. Silence. Erase. Promote white supremacy. They recognize, therefore, that upending racism in schools will end schooling practices as we have come to know them.

To think about all of the parts of schooling, not just the classroom, anti-racist educators take a holistic approach and consider many dimensions of schooling through an anti-racist, anti-oppressive lens. These dimensions include:

The demographics of staff, particularly in schools with predominantly BIPOC students

School leadership and paths to school leadership

School governance (e.g., network and district leadership)

School curriculum

Special education

Teaching and learning practices

Definitions and measurements of academic success

Definitions and measurements of teacher success

Professional development

New-teacher training and support

The wellness of teachers, staff and students

School mission and vision statements

School, network and district policies

School culture and approaches to discipline

School infrastructure

Allocation of resources and budget

It is important we recognize this list is not exhaustive. In order to be anti- racist educators, we must be steadfast and consistent and consider all aspects of schooling.

We have to commit to continuously evolving, pushing back, studying, engaging in critical discourse with others, examining what is happening in our minds and bodies, and listening to students. Anti-racist educators understand that an anti-racist approach to schooling could very well mean an ending to schools as we know them. They understand this is how we begin to get free.

Anti-racism is healing and love in action.

The work of anti-racism in schools is incomplete without an understanding of the integral role that healing must play in the practices of freedom. Healers are people who remove harm and reduce the impact of violence, people who restore and repair.

The greatest of healers consider the whole person and seek to get to the heart of the matter. Anti-racist educators understand that all forms of oppression are connected, that it is not possible to care about racism and disregard the violence in sexism, classism, homophobia and more. The hearts of anti-racist advocates push us to be concerned for all violations of human rights.

The ways that we as educators must care for ourselves and create educational spaces that center healing our students' bodies is crucial. We must create educational spaces that tend to the harm and violence that has been enacted against BIPOC bodies and minds-specifically those of our children. Anti-racist educators recognize that this is the work of undoing, of dismantling, of liberating, of healing and of truth-telling. This is our collective work.

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What Anti-racism Really Means for Educators

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"If I love you, I have to make you conscious of the things you do not see." -James Baldwin Anti-racist work in all schools is essential. It is the exercise of hope, the practice of undoing and dismantling systems of oppression, the practice of freedom and of truth-telling.

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"The primary purpose of assessment is to improve student learning" -Anne Davies, PHD.

The above quote sums up why I challenged my assessment practices. In 2013, as a High School Math teacher, here is the model I used:

Teach concept

Various quizzes during the learning journey.

Summative unit exam.

Repeat

After 8 years of teaching in this model, I realized there was an issue...students who entered my class with passion towards mathematics, were leaving the class beaten down, sometimes dropping out, and ultimately not having the desire to learn more math.
Originally, I thought this was normal!! When I went to high school, math classes always ended with less students than it started with. As a student, I remember daily expectations of having to do the odd (or even) numbered questions on page X, multiple worksheets, and having to prepare for weekly quizzes or tests. This was my normality. This was the machine I wanted to perpetuate when I entered teaching.

Why?

Because this worked for me. I am intrinsically motivated by mathematics, and I find prime, fibonacci, and complex numbers inherently interesting...because they are!! However, too many people have not had the chance to struggle, discover, and play with these awesome, and other, mathematical ideas.

In 2013, as an educator, I saw the true problem...my assessment style was more about ranking, sorting and grading, not at all about learning. Even furthermore, I was more focused on preparing students for AP, or diploma exams, instead of creating an environment which allowed students to bring their passions and interests in, next to their pencils and paper.

Similar to the comic, my grades were focused on what was easy to test, grade, and report on, instead of what was important.

This had to change. If I was differentiating my instruction, why was I focusing on standardizing my assessment?

In 2013, I made a stand:

I will only assess in a way that increases learning; if my assessment isn't increasing learning then the assessment needs to change.

In this year, my late friend Joe Bower reminded me "the word 'assessment' comes from the latin word 'Assidere': to sit beside"; an action that was rarely taken when I was assessing my students.

Here is my journey, and the steps I took, to explore what it mean to provide differentiated assessment.
1. Manageable outcomes:

In consultation with University professors, colleagues, and teachers across the province, I looked at every course outcome through the "Rock, Sand, Water" analogy: If you plan for the rocks first, then sand and then water, it will all fit, however if you simply plan a course to cover all outcomes equally, all the outcomes will rarely fit.

During this process I combined parts of one outcome with another, broke up some outcomes into smaller chunks, and then I created a list of "Rock, Sand and Water" objectives:

Rock outcomes (outcomes that pass the endurance, leverage and readiness test) - Expect ALL students to master.

Sand outcomes - Expect MOST students to master.

Water outcomes - Expect SOME of my students to master.

I then ensured that these decisions reflected in my long range outcomes, course outlines, and daily plans. I planned my courses in a way to ensure that the essential learning outcomes were weaved throughout the entire year, while less essential outcomes were covered through the lens of a higher leverage outcome. Of course, I still taught all the outcomes, but I decided to only report on the essential ones, regardless of how difficult it might be to do so.
2. Change the tests
After having a smaller list of outcomes to report on, I decided to ensure my summative assessments matched this philosophy. Instead of giving tests grouped by question type, I grouped questions based on outcome. Any assessment that covered more than 1 outcome would be given back to students with more than 1 grade. Each grade represented the learning of the student on a specific outcome; no longer did I average 2 or 3 outcomes into one mark and call it "Unit X Test". I then changed the categories on the online reporting program to "outcomes" instead of "Quizzes, tests, homework, etc". Every mark, on a specific outcome, was reported in the corresponding "outcome" category.

3. Ensure learning is the focus on every assessment
During this time, my summative assessments where 1 part multiple choice, 1 part numerical response and 1 part written. Simply sorted by outcomes not by question type.

I quickly realized that when my students answered a multiple choice question wrong (or even when they guessed right) I was clueless as to how to support them from their current understanding to mastery. If I wanted learning to be the primary focus, I could not administer multiple choice exams.

In 2014-2015 I moved to an entire written response assessment strategy grouped by outcomes. Instead of 1 part multiple choice, 1 part numerical response, and 1 part written, I only assessed with questions that forced students to make their understanding visible. It was in this year, I truly started to sit next to my students and provide them written and verbal feedback that pushed their learning forward instead of simply saying "here are the X questions you answered incorrectly and here are the correct answers". My feedback was focused on learning, not on which questions did they answer wrong.

I was writing grades and comments on everything my students handed in. This was the inherent problem; I was giving both grades and comments.

Every time I handed back an assessment with a mark, I quickly noticed that students focused on their individual grade, their friend's grade; and how they ranked within their peer group; and most completely ignored the comments. Students were not asking "How do I understand this better?" but instead "How do I get an A (or 90% or excellent)?"

I had invested a lot of my time into giving useful and effective feedback however these comments were being overshadowed by mark. Grades were the commodity of my classroom not learning. This had to change.
4. Challenge the grading system
THIS IS THE GAME CHANGER!!

Simply put, I stopped writing grades, learning levels, or any other ranking system on student work. Instead I only provided feedback and asked questions that pushed learning forward. Even if a student demonstrated "mastery" of an outcome, I would still provide feedback or leave them with a question that pushed them beyond the scope of the outcome.

This was the most profound transformation I have ever experienced in my entire career.

Students truly became engaged in their learning, not their grade (or ranking). As well, I was able to truly push my students forward when they made mistakes. When I looked at the work of my students, I simply focused on 3 essential questions:

What does success on the essential learning outcome (rock outcome) look like?

Where is the student now?

How do we close gap?

This ensured that the feedback I was providing to students was truly "learning focused". Every written comment was also input into our online reporting system.

This meant that when a parent, or a student, logged into our online reporting tool, they didn't see grades, but instead comments for any outcome. Instead of seeing "80%" on an outcome, parents (and students) would see what they need to work on to close the gap...without the support of a grade.

Even the conversations I was having with parents were learning focused and not grade focused. Incredible shift! At the beginning of the year, parents were apprehensive of not receiving any marks as feedback, however when parents saw the products their children were bringing home from my courses, parents were quick to become allies of this new model of assessment.
5. Differentiate the assessments
I finally started a "Differentiated Assessment" model. After teaching outcome X, I would have an assessment on the outcome, however, I started to use this time to also assess each students' understanding on a previous outcome. For example, students who were also being retaught ideas on outcome 2 would have questions around outcome 2 on their sheet, the ones who were working on outcome 3 would see questions around this outcome...truly every single assessment was tiered to the individual student and what he/she has been working on in the previous weeks.

The assessment Jimmy and Jane received on this day, would only match if they were working on identical material with identical errors and misconceptions; which was rarely the case. Even when designing the questions from previous outcomes, the questions were focused around the feedback the learner had received on their last assessment. For example:

Jane might be tasked to demonstrate understanding of a specific part of a certain essential outcome, because I saw only a minor learning gap when I previously assessed her on this outcome.

Jimmy, however, might have more questions around the same essential outcome, because when I assessed him previously I saw major learning gaps.

This is when learning became the focus of every single assessment I gave. I can honestly say that every assessment had learning as the only priority!
Looking back, I have always believed that every child can learn math to the highest levels, but only in the past 3 years did I take a differentiated approach to what happens when they don't.

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"The primary purpose of assessment is to improve student learning" -Anne Davies, PHD. The above quote sums up why I challenged my assessment practices. In 2013, as a High School Math teacher, here is the model I used: Teach concept Various quizzes during the learning journey. Summative unit exam.

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Browse through over 60 complete K-12 math units of study chalked full of problem based investigations, 3 act math tasks, visual number talks, consolidation prompts and purposeful practice worksheets to Spark Curiosity & Fuel Sensemaking in every student.

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Real World Math Lessons | 3 Act Math Tasks | Math Worksheets & Practice

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Contextual problem based real world math lessons with rich math tasks, multi-day 3 act math tasks, math practice materials and math worksheets

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Home | Spark Curiosity to Fuel Sense Making

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Home. Spark Curiosity With Tasks That Are Contextual, Visual and Concrete

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Math Is Visual: Let's Teach It That Way | Free Visual Math Talk Prompts

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Access free Visual Math Talk Prompts to build fluency and flexibility with number sense and operations for your K-12 Math Class or at Home.

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Subitizing

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Subitizing! This ability to instantly "see" how many is a critical component in young students developing number sense. There are two types:

1. Perceptual subitizing -the ability to see a small number of objects and know how many there are without counting. For example, this skill allows us to realize we "rolled six" on a die without counting. It is usually an instantaneous realization, and generally applies to small amounts of objects.

2. Conceptual subitizing -the ability to see more than one group and add them together to find the total. This one takes a little more effort as it requires recognizing smaller groups and adding them up.

Studies cite the use of subitizing in primary grades as the key to students being able to pass algebra. It builds the skills needed to move toward proficiency in a sense of equality and balanced equations, spatial sense, and number sense. We know subitizing is an important part of building procedural fluency through conceptual understanding, but how do we prepare students for this?

Students need tangible experience working with number sets-and if we can create a tangible experience that is also FUN, success will come much more quickly. A common strategy is to work with dot arrangements, but any simple symbol can be used, such as stars or squares. To add the fun factor, I expanded on an idea from my colleague, Mary Gong, who used spaceships in her primary classroom. Her "spaceships" were nothing more than paper plates with large dots made with stickers or Bingo dauber, but were effective nonetheless. You can get as creative as you like here but be sure to stick with simple shapes and vary the patterns on each. First, I randomly provided students with paper plates with small sets of 3-5 stars. I asked them to silently maneuver their "spaceships" to another student with a matching number of dots. I then led a discussion, using the prompt, "How did you know the set has the same number when it looks different?"

After this activity, I placed students in two groups to explore more concepts. I asked Group 1 to "fly" to a student in Group 2 with a spaceship that had fewer dots than their own. Then, I asked Group 2 to fly to a student in Group 1 with a spaceship that had more dots. I also had each student hunt for a copilot, whose spaceship combined with his/her own, totaled a certain number. You can easily do this in a whole group setting and mix up the instructions however you like.

Let's get started! Here are a few examples of how you can help young learners begin to access subitizing.

Step 1 Matching equivalent quantities-build the idea of conservation of numbers, same number different representation

Step 2 Comparing number and combining multiple sets to create sums

Step 3 Building relationships by linking Real World, Linguistic, Representational, and Symbolic using the above activities

Once students become familiar with subitizing, you can initiate lively discussions to use as springboards for other subitizing challenges and whole number sense routines. For example, display a dot arrangement, making sure numbers are appropriate, for 3-5 seconds then ask, "What did you see and how did you see it?" I used arrangements of eight.

Many of my students responded with eight, so It was very important for me to accurately represent what they saw to inform our discussions. Here are some examples:

These kinds of activities are precursors to helping students have access to algebraic concepts. They help create a sense of equality, conservation of numbers, decomposing, and more. Being able to recognize equations as being balanced on both sides instead of just where the one correct answer goes is a huge development and deepens number relationships for students. After students have had some time to work with dot arrangements like this, let them have opportunities to explore true or false statements such as 4 + 2 + 2 = 6 + 1 + 1.

As students gain confidence in subitizing and their number sense, it is important to consider ways to differentiate for students based on their points of understanding. These could easily be created during centers or learning stations. Having Bingo daubers or pre-cut, one-inch squares will allow students to quickly create their own arrangements based on the numbers they need to practice. Some students can be at five while others are at nine, etc. Slide these into a sheet protector so students can easily exchange their arrangements to find the different ways they see to make the total number represented. If ready, their equations should match their drawn models. One caution, make sure students are actually subitizing. Circling ten individual dots to say 1+1+1+1+1+1+1+1+1+1=10 is not actually subitizing! Here is a sample launch: Use eight color tiles to make an arrangement. What numbers match your arrangement?

Student samples:

Note, the first student was able to accurately subitize and create equations based on what he/she saw. The second student shows great knowledge of working with ten and flexibility of thinking, but is no longer actually subitizing. It looks like this student is ready for a higher number!

Communication and productive discourse are essential to help students clarify information, deepen their reasoning, and move learning to long-term memory. Consider helping them learn with some sentence frames such as these:

I agree with _____ because _______.

I do not understand _________. Can you explain this again?

I disagree with _______ because _______.

How did you decide _______?

I am thinking about this problem differently. Here is the way I solved it: _____________.

My way is different from_____. Instead of____, I_____________.

Can you explain the difference between ________and ________?

Interested in learning more about counting, subitizing, and other critical important foundations for math learning? You can explore Lisa's presentation from the recent NCTM 2019 conference, Tool Time: It's All About the Connections!, by visiting https://mathsolutions.com/contact-us/speaker-presentations/

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Subitizing! This ability to instantly "see" how many is a critical component in young students developing number sense. There are two types: 1. Perceptual subitizing -the ability to see a small number of objects and know how many there are without counting. For example, this skill allows us to realize we "rolled six" on a die without counting.

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Math Collaborative 2022 Summer Book Club Zoom Info 

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Math Collaborative 2022 Summer Book Club Zoom Info 

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https://us02web.zoom.us/j/4219107593

Meeting ID: 421 910 7593

We will meet weekly on ZOOM!
Wed., July 6, 2022 through Wed., August 10, 2022 at 6:00 PM Pacific.
Draft Reading Plan:
Wed., July 6 Reconnect and get organized; make sure to order your book!

Wed., July 13 Chapters 1, 2 & 3

Wed., July 20 Chapter 4

Wed., July 27 Chapter 5

Wed., August 3 Chapter 6

Wed., August 10 Chapters 7 & 8

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This is a simple activity that teachers can use to promote children's spatial language and ideas. Children are asked to view a simple drawing, a picture (sometimes taken from a storybook), or a complex geometrical shape, and think carefully about what they notice in the picture.

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Steve Wyborney\\\'s Blog: I\\\'m on a Learning Mission.

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Introducing Esti-Mysteries! Each image invites students to wonder what number is represented by the image. As you click through the each Esti-Mystery, clues will appear that will allow the students to use math concepts to narrow the set of possibilities to a small set of numbers. In the end, the students will need to call upon their estimation skills to solve the mystery and find the missing number.

NOTE: In addition, to these original Esti-Mysteries, you may also want to visit the brand new post 51 Esti-Mysteries.

Here is an example:

Download the Esti-Mysteries, try them in your class, and let me know about your experience.

Update: Here are 4 Printable Esti-Mystery Charts you may enjoy using with Esti-Mysteries.

Update: Here is the Full Set of Esti-Mystery Clues (without images) to allow you to browse through the Esti-Mystery Concepts.

As always, you can find me on Twitter @stevewyborney

After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.
Esti-Mystery #1 The First Mystery Esti-Mystery #2

Esti-Mystery #3

Esti-Mystery #4 ***This is the one where the numbers on the dice are first used as a clue.
Esti-Mystery #5

Esti-Mystery #6
***The most challenging one yet!

Esti-Mystery #7 "The Dice Jam"

Esti-Mystery #8 "Bright Beads"Esti-Mystery #9 "Cupside Down"Esti-Mysteries #10 "Yellow Yarn"
Esti-Mystery: The Smaller Numbers Request
Esti-Mystery: "Sinker?" (from The Smaller Numbers Series)Other Posts and Resources Which May Be of Interest
***This one that may lead to the greatest debate and surprise at the end.

Tiled Area Questions
The Smaller Number Series

This next Esti-Mysteries was requested for a kindergarten class. Enjoy!

***This is a great one to use with kindergarten students.

51 Esti-Mysteries - 51 (fifty-one!) brand new Esti-Mysteries which I've already started posting .

Splat! (50 downloadable, animated lessons)

The Fraction Splat! Series (with 20 more downloadable lessons)

Splat! for Google Slides (This is the one you want if you don't have PowerPoint but do have Google Slides.)

Cube Conversations (80 downloadable, animated lessons)

Thank you for visiting my blog!

All my best,

Steve

P.S. Please feel free to post your reflections in the comments below.

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Esti-Mysteries: Estimation Meets Math Mysteries - Steve Wyborney's Blog: I'm on a Learning Mission.

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Introducing Esti-Mysteries! Each image invites students to wonder what number is represented by the image. As you click through the each Esti-Mystery, clues

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Same But Different is a powerful routine for use in math classrooms. The images can be used for teaching a way of thinking: grayscale thinking, categorical thinking; helping students to build a network of ideas and an approach for learning and thinking about mathematics.

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Math flips, tips and more!

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"I love the low-floor high-ceiling aspect of number talk images, particularly for parents with multiple children learning at home," says 2nd grade teacher Stefanie Mathewson. "Making space to see the nuances in the pictures cracks certain images wide open as fertile soil for early counting and sophisticated groupings at the same time."

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About

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Whiteboard.fi is created by Sebastian Laxell and Jaakko Virtanen, and is run by Digital Teaching Tools Finland Ltd - a Kahoot! company. We have experience in teaching and IT-management - so we know the struggle in finding good digital tools.

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Whiteboard.fi - Free online whiteboard for teachers and classrooms

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Whiteboard.fi is an instant formative assessment tool for your classroom, providing you with live feedback and immediate overview over your students.

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Math Collaborative Twitter Chat, November 9, 2020 - Wakelet

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We built Wakelet using the latest technologies to make it faster and easier to use. Unfortunately, your browser doesn't support those technologies.

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Problem Strings

The number one instructional routine for building strategy in a community of learners. Below you will find amazing example Problem Strings.

Each Problem String comes with

Video of the String in a classroom

A page dedicated to understanding the mathematics and structure of the string

Teacher moves to make the most out of the Problem String

Transcript annotated by the expert teacher giving their insights

What are Problem Strings? Click HERE to learn more.

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The number one instructional routine for building strategy in a community of learners. Below you will find amazing example Problem Strings. Each Problem String comes with Video of the String in a classroom A page dedicated to understanding the mathematics and structure of the string Teacher moves to make the most out of the Problem String Transcript annotated by the expert teacher giving their insights What are Problem Strings?

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The Make Math Moments Academy

YOUR ROADMAP TO AN INSPIRING MATH CLASS
Create memorable math moments daily with access to curiosity rich tasks, in-depth PD video trainings, and a system for making these moments a reality. Join the Making Math Moments Academy now. Your first 30 days are for free.

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Make Math Moments | Empowering Teachers with Math Activities

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The Make Math Moments Academy YOUR ROADMAP TO AN INSPIRING MATH CLASS Create memorable math moments daily with access to curiosity rich tasks, in-depth PD video trainings, and a system for making these moments a reality. Join the Making Math Moments Academy now. Your first 30 days are for free.

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Sara VanDerWerf - Inspiration and resources for mathematics teachers

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UPDATE - SPRING/SUMMER 2020 - During the start of the pandemic, I recorded a video for families titled "What is Math?". Too many adults and students I meet say "I can't do math" or "I am not good at math" - When they say this, I think, 'Are they basing this belief on the real definition of math - or on a misconception based on memorization and speed." If you prefer to get the idea of this post from a video instead of reading, click below. Otherwise - keep reading for more details for the classroom.

I started teaching in 1991 when NSF funded reform curricula entered many schools around the nation. My district started using IMP (someday I will write a blog post, "A love letter to IMP") and for 13 years I taught math out of these texts. Many times during my first years of teaching I would think "Oh, this is why math works". Here I was a math teacher and in some ways I was discovering math and what it means to be a mathematician for the first time. I honestly was a bit pissed off about how I had been taught math throughout my entire education. I received A's in every math class I ever took, but I knew in my heart this was just because I was good at the system of school. I was OK with not knowing why and just doing what they told me to do. When my friend became frustrated when her questions of "Why does this work" were unsatisfactorily answered I would quietly think "she needs to let that go and just do what they say". My poor friend. Math classes were not built for her then. The sad thing is 30+ years later many still do not work for those like my friend. I was lucky, my early years of teaching were full of best practices PD that I lovingly say brainwashed me into the way I would think about mathematics and what mathematicians do for the rest of my career.

For the last 20 years of my teaching career I've given the same homework assignment sometime during the first week of school. One of my goals week one of school is to convince students that all of them, yes ALL, are mathematicians. Many students enter my room not believing this. In recent years there has been lots of people talking about this and many, many resources created to help students believe that they can get smarter in math. Jo Boaler, Stanford and the YouCubed team have created some of the best resources with their week of Inspirational Mathematics. We are lucky to live in the era of easy access to newly created resources we can immediately bring into our classroom.

Despite all the resources created recently, I think there is something that most math teachers miss in their early messages to students about math. In my experience many students - even those in HS - have a misconception of what mathematics is and what mathematicians do. Once I clear up this misconception and give consistent messaging around our work in mathematics class, I've found that more students engage in the work of becoming mathematicians. This messaging to students has ultimately changed me as a teacher and what I choose to put in front of them as they learn mathematics.

What is Math? You need to do this week 1!

I have a couple of week 1 posts with a ton of views in the last year. Tons of you like my group work 100# task for week 1 and have used my name tents to build relationships with students. What follows though is maybe more important to me week 1. This messaging follows through all school year. It is how I set the tone for what the work of y classroom will look like all school year. Here is how it plays out during the first week of school. I'll give you a script of sorts.

For the next 10 days of school I put something in front of students to notice. Every day I repeat the definition of math and the 3 things mathematicians do. Every day I ask, "What do you notice?" and give students space to notice. I walk around the room and catch students noticing and I say "you are a mathematician! You described what you noticed...". My goal is to catch every student acting as a mathematician and name it when I see it. I want to change how they see themselves as mathematicians.

This messaging is also pretty much the sum total of my behavior management plan for my classroom. Anything that gets in the way of all of us acting as mathematicians for the full 50 minutes is annoying to me. Tardies therefore are annoying. Being on social media is annoying. Sleeping is annoying. Any disengagement is annoying. I tell students anything outside of being a mathematician for 50 minutes is not OK with me.

My messaging about the definition of math also transforms my planning as a teacher. I must have multiple opportunities every class period for students to notice and wonder. I often work with principals and others who observe teachers and I tell them - any time you observe a math lesson, if the teacher does not ask some form of "What do you notice?" of students, then the math lesson is not as good as it could be. They are not empowering students to be mathematicians and own their own learning of math. When you plan a lesson this school year, I encourage you to start with this question, 'What mathematics will I ask students to notice today?' and/or 'How will I set up this math concept so my students have something to notice?'.

A word of caution. It is easy to ask students to always practice what I am talking about using number sequences. Be careful of this. If you do this than student's powerful ability to notice patterns will tell them that math is only about number sequences. For this reason, I mix things up the first few weeks. I use all kinds of things. Some are related to the content we are studying and some are silly things I find on twitter or Facebook. Here are just a few examples. Most use up 5-10 minutes of class maximum. My goal is to convince students that they are mathematicians and they do have the ability to notice and wonder.

1. First a visual I got from Christopher Danielson. What do you notice? How many avocados? How did you see that number of avocados?

2. I found this sign in some tweet. I ask, "what do you notice? " I then doctor the photo and ask " How much would 'Hello, one small coffee' cost?" "Justify this cost".

3. I put up the first picture of the green shirt Kristen Fouss tweeted out and ask, "What do you notice?" I follow up with Jim Pardun's question, "If the pattern continues, how many tickets could you get for $40?" and have partners work on justifying their amount on white boards.

4. I give student partners the following 3 graphs & equations in factored form and ask, "What do you notice?" I ask students, "How could we find the coordinates of the vertex if all we have is an equation in factored form?" It does not take long for students to notice that the vertex is in the middle of the 2 x-intercepts/roots. In a 10-minute discussion students discover a visual method for finding the vertex without having to convert the form of the equation.

5. This visual has I lot of patterns. I made it from screen shots from this cool animated factorization site. Can you see the patterns in the rows and columns? I use this at the start of my unit on exponential and power functions. Lots to notice and describe....and generalize. Lots of chances to practice being a mathematician.

6. If you are not following the hashtags #cthenc (contemplate then calculate) and #connecingreps (connecting representations) - 2 instructional routines from the free online New Visions math curriculum - on twitter, then you are missing out on tons of things to have students look at to notice, wonder and then calculate.

7. Any 'Which one doesn't belong?' is great to use to practice being mathematicians.

8. I found this picture posted by a pet-loving relative on facebook. I covered up some of the information. Ask students to notice & wonder. You can't help but try and make some meaning out of this. I then follow up with more information using one of theses photos at a time.
When I reveal the 2nd photo (2 pieces of data), students want to make this linear and find a rate of change - even without me telling them too. The third photo is the reveal later and students are annoyed (which I love)....but they are acting as mathematicians (and I name it for them) and thinking critically about making sense out of data.

9. One huge thing I've learned from Annie Fetter, Max Ray and all the other amazing people at The Math Forum is to turn tasks into scenarios (remove all question and some details) and ask what do you notice & what do you wonder. Here is the original task. In my experience with 5 minutes of noticing and wondering, Students solve all these questions without me giving them first. They also see more things than this. They ask/say, "are these right triangles?", "I wonder if we can use the Pythagorean theorem." I give information like, 'these triangles are similar' as students notice the details. I give the question to solve after they've said it first. Try it for yourself. Create 'scenarios' instead of questions. See how that changes your students ability to see themselves as mathematicians.

10. Any pattern from Visual Patterns would be amazing.

Noticing Patterns Goes Wrong

One of my main messaging day one, week one, month one is that 'We are all mathematicians. We all have the power to notice, describe and generalize patterns. You have all had this ability since birth.' One way to be convinced of this is based on how powerful students misconceptions in math are. I would argue most if not all misconceptions come from acting as a mathematician and noticing patterns given to them by their teachers. Here is an example of what I mean.

Twenty years ago my friend, Terry Wyberg ( U of MN prof), told me that 30% of my HS students would get this problem wrong. I did not believe him and tried it out. He was wrong (only about the % - not the underlying problem), only about 20% of my HS students got this wrong, but OMG, 20% got it wrong. 20%. 20%. No wonder my students struggled to solve math equations. The most common answer was 19. Students saw the equal sign as meaning 'do whatever is on the left side of the equal sign'.

In the last 20 years there has been a ton written about improving students understanding of the concept of equality, but I find many teachers don't know our part in creating this misconception. Students often created their misconception of the equal sign from noticing the patterns in the work we gave them. If most of our work for students look like these...

..than no wonder students created a misconception about the equal sign. They simply had noticed that to solve math problems I just do what is on the left.

The good news about our students errors is they know how to notice patterns. We need to see ours students assets, not their deficits. The challenge then for us math teachers is to put things in front of our students to notice that don't create misconceptions when our students notice patterns. Since realizing this many years ago I am always writing equations in a variety of ways. For example y=2x + 3 vs. 3 + 2x = y so that they don't see linear equations as having to start with y=.

Here are a few other classic ways we - the teachers - create math misconceptions.

Google 'area of triangle worksheet' and take a look at the vast majority of problems given students. No wonder students think 'base' means 'bottom' of the triangle.

Be careful not to give examples that can give correct answers by doing the problem incorrectly. Here are 2 classics....

A Challenge for the new school year.

This school year I challenge you to start the school year by defining math and the role of of a mathematician. Give your students the big picture and don't start with the details. Us math teachers tend to treat the definition of math as a set of details and don't honor our students who need to see the big picture of what mathematics is to be successful.

I've been reading Tracy Johnston Zager's "Becoming the math teacher you wish you'd Had" book this summer and it is amazing. You need to order it and read it today. The first chapter of the book touches on the definition of math starting with her own mother's definition. There is more great stuff throughout this book to support what I've talked about above in much further detail.

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Math as the science of patterns from the Mathematical Association of America (MAA)

Keith Devlin's Mathematics as the Science of Patterns book. "Mathematicians now see their work as the study of patterns-real or imagined, visual or mental, arising from the natural world or from within the human mind." Using this basic definition as his central theme, Devlin explores the patterns of counting, measuring, reasoning, motion, shape, position, and prediction, revealing the powerful influence mathematics has over our perception of reality. Interweaving historical highlights and current developments, and using a minimum of formulas, Devlin celebrates the precision, purity, and elegance of mathematics.

Follow-up: My friend Laura Wagenman posted this great 'Math is' video on Facebook today. Love when others find resources connected to my passions. Give it a view.

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What is Math? What do Mathematicians do? - Sara VanDerWerf

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UPDATE - SPRING/SUMMER 2020 - During the start of the pandemic, I recorded a video for families titled "What is Math?".