Rochelle Gutiérrez:
I have an abuela rule in my classes and in my work, and the abuela rule is this. We need to be able to communicate with each other in ways that are understandable to our grandmothers. I am not saying that we need to water down or distill or somehow dumb down the ideas that we have, but our abuelas carry such wisdom in this world. We do not want to waste their time with jargon when we can just say things simply.

Alec Patton:
This is High Tech High Unboxed. I’m Alec Patton, and that was the voice of Dr. Rochelle Gutierrez giving the closing keynote speech at the 2023 Deeper Learning Conference. Rochelle is a professor of curriculum and instruction at the University of Illinois, and her research focuses on mathematics. I had a chance to talk to her at the Deeper Learning Conference and it was totally fascinating. Here’s our conversation.
On the most one sentence level, what should folks know about you?

Rochelle Gutiérrez:
Folks should know that I come from a long line of activist women who have political clarity and know how to fight for our people and for a better world.

Alec Patton:
When you say our people, who’s …

Rochelle Gutiérrez:
Our people meaning people who’ve had lands stolen from them, people who have been misunderstood, people who have been touched by this grief of diaspora. Really most of us have been touched by this grief of diaspora where we’ve been pulled from lands and where we either don’t know our own lands for people who came as slaves or people who’ve been forcefully removed from their lands, and also people who deeply know their connections to lands. But it’s this idea that society has not been built in a way that honors all people, and so when I say our people, I’m talking about my people in particular, people who are Chicana and my ancestral roots are [inaudible 00:01:55], which is from the Copper Canyon region of Mexico.

Alec Patton:
Thank you. All right, let’s go back now. When you think about your childhood, what memories surface when you think about your experience of math as a student?

Rochelle Gutiérrez:
It’s so complex. On the one hand, I have incredibly joyful memories of mathematics growing up, learning sewing through my mother and my grandmother. I think about all of the complex ways that women who sew know how to think about different permutations of colors and bringing things together and thinking about orders of operation and angles. There’s so much that’s kind of embedded in this way of making something and also being in relation to our plant relatives, knowing that yarn and fiber and thread comes from plants. But as a child, when I was in the gifted program, we did amazing mathematics. I mean, we were working in other bases. We were often adding and subtracting and multiplying in base two, base three, base eight, something else. This was so playful for me, this felt like it was so much fun. I remember learning about base two and thinking, “This is great.”
I would write down on a piece of paper, I’d tell my mom, “I want this many cookies”, and I’d write one one and she would look at me like, “I’m not going to give you 11 cookies,” and I would say, “In base two-”

Alec Patton:
What is one one in base two?

Rochelle Gutiérrez:
The way that bases work is that base two would be the first digit one would represent one, and then once you get to two, it’s … one zero would basically be two cookies, and then one one would be three cookies. I was basically asking for three cookies, but when I wrote it down, I would write one one and she’d be thinking base 10, and I’d be thinking base two and I just thought that this was so much power as a child to be able to play with the rules. I had all of these amazing experiences like that where we were doing logic problems and we were working on other bases.
We were doing point perspective drawings as part of math class. So thinking about how can you look at a street and then take a point that’s either a diminishing point or thinking about what would need to be the angles for the windows to look like they were actually perpendicular, but they’re not going to look like rectangles anymore. Now they’re going to have to have this angled line. Just all of that was so much fun, and that was math class. That was what we did in math class. But then I also had points in my life where in seventh grade I had this math teacher who was really strict on rules, and so you were not allowed to say something like 3.6. You had to say three and six tenths. If you said 3.6, he took out a permanent marker and he put a dot on your forehead and that was to remind you that you don’t say point.
Imagine as a young girl, this was in middle school, having somebody put a permanent marker on your forehead. This was such a horrific and traumatic experience for a person, and how is that supposed to be humane at all for helping people think about what really is just accustom in mathematics? That’s an aesthetic choice that we took to say three and six tenths so that we can help people understand place value, so we can say we’re in a base 10 system, and so when we have something that’s after the decimal point, that’s tens and something that’s two after that’s hundreds and something that’s thousands. But that was a choice that was made by somebody in history in mathematics that we followed, and then that was a choice of that one teacher.
My parents, I mean, my mother was furious and pulled me out of that math class and put me in a different teacher and tried to get that teacher sanctioned because there were going to be other parents who didn’t know this was happening, or other children who didn’t feel that they had any power to stand up to that teacher. There were things like that. There were things later in my life … I don’t have a degree in mathematics, I have a degree in biology and later in my life I remembered thinking, “Maybe I need to get a degree in mathematics. Maybe this will help me better be able to explain to mathematicians when I’m talking about when I’m trying to say rehumanizing or I’m trying to think about these other worlds that we can have in mathematics.”
There was a point when I thought, “Well, maybe I should go get a master’s degree in mathematics. My university will allow me to do it. It’ll be free. I can go take courses.” But in thinking about that decision, I had to ask myself … when I was looking at the courses I would take, I thought, “Is this really going to be the mathematics that’s going to help me envision another world? Or is this really going to be the mathematics that’s going to help me get credentialed by other people and have this mark of approval that I’ve also been able to do this thing that we call Western mathematics in math departments that focus on research?” So really then I decided … I guess my math story goes into the fact that I had a lot of friends who were mathematicians, and I could learn from them. I could learn from books.
I was most interested in the history of mathematics because I thought there’s no way that we can reimagine another kind of mathematics if we don’t know where we’ve come from. Currently, the mathematics we’re teaching in schools is this western modern mathematics. I mean, even the mathematics we teach, when we say geometry, we’re telling students this is geometry. This is Western Euclidean geometry. This is not all the geometries that actually are practiced in the world or that have been practiced over time. For me, my story has been … it’s kind of meandered from these joyful experiences to these awful experiences to these questioning myself experiences, and I think that’s all part of what many of us go through in mathematics.

Alec Patton:
The first thing you mentioned was sewing. When did you recognize that as mathematical training and mathematical experience?

Rochelle Gutiérrez:
Oh my goodness. I would say maybe graduate school. I never really thought of that as mathematics. I mean, I thought that this was really cool. My mother encouraged me to make a lot of small objects for my dolls. Trying to think about proportions and reasoning and all that would be involved in having a pattern of something much larger and then trying to reduce it so that it still had the form and the shape and the proportions. I don’t think I thought about that because there weren’t people around me who … I mean, I think also from an indigenous perspective, math isn’t this extracted thing. We are in relation to our worlds, and we think about pattern and logic and just our spacial reasonings structure. Those are all embedded in kind of everyday ways of how we move through the world. It’s not like this thing I’m doing is mathematics now. It’s kind of like it’s math and it’s storytelling and math is storytelling and it’s bonding, and it’s all these other kinds of these rituals.
I feel like … I don’t know. I think maybe when I started to question who gets to decide what counts as mathematics when I was in graduate school, maybe that started making me think about there’s all these ways people already do mathematics that we actually just don’t sanction.

Alec Patton:
I mean, it sounds like you were, mixed experience, but pretty stoked on math. I mean, wasn’t asking for cookies in base two, certainly when I was growing up. I’m curious, why didn’t you study math in college?

Rochelle Gutiérrez:
There’s four children in our family and I was a third child, and when I got into Stanford University, I did well in math. I took AB and BC calculus and at that time, there weren’t a lot of schools that offered BC calculus. I was on the math team. I did all the kind of geeky things that you would do. I tutored people a lot of times in math in my free time, but at that time, I felt like the narrative was if you are going to use science or math to do something, you go to med school. I didn’t know anybody who was a mathematician. I didn’t know anybody who did that kind of work, but a doctor, that was an identifiable thing. So I thought, “this is what I should be doing.”
But it really wasn’t until I was in my junior year and I was applying to med schools. I had taken MCATs, was in secondary applications, was all in, and I had to take a class to fulfill one of these other requirements that was not supposed to be for my biology major because Stanford tries to make you a well-rounded person, and I resisted as much as I could. But there was this one class I had to take that was on art or art expression or something like that. I took a class from Elliot Eisner on the artistic development of the child and that whole class we fought because I didn’t agree with some of the things in the class, and I thought that it assumed certain things about the people. After the class was ending, he said to me, he said, “Rochelle”, he said, “You’re really smart.” He said, “You’re really analytical,” and he said, “Why not try something really hard like education instead of med school or whatever?”
It really hit me that, “Yeah, why don’t I think education is the harder thing to do? Why have I somehow been convinced by people that if you’re smart, you go become a doctor?” Then I couldn’t apply right away to school. It was too late to apply to get a master’s degree in education or something like that, so I started teaching in the migrant education program at Stanford, and it was like my light turned on. It was like, “This is what I’m supposed to be doing.”

Alec Patton:
What’s that? What’s the migrant-

Rochelle Gutiérrez:
The migrant education program at Stanford was? They would bring in students of migrant families for the summer, and we would teach them math and computer science at the time. That was really where I think for the first time, I thought, “God, this is what I do in my free time and somebody would pay me to do … this is what I love. This is what I …” It was really that beautiful experience of being able to be in my community, being able to be with people who looked like me. Both of my parents had grown up picking in fields and so this just felt like I could pull all my selves together. I could be one person and I could be doing this work.

Alec Patton:
All right. I got a couple follow up questions here. What were you and Elliot Eisner arguing about?

Rochelle Gutiérrez:
We argued about the readings, sometimes the assumptions, that were made about how people experience school. I felt like there was this kind of universalistic experience that was presented. Like students learn in these ways or students do these kinds of things, and I just thought, “God, there’s all kinds of ways that people learn. People learn by observing people learn by dreaming, people learn by singing, by making, by …” I was kind of like, “This feels like we’re just kind of stuck in this one person’s view of what learning counts as.” He was amazing in terms of expanding my views about what does it mean to be somebody who, when you are an expert in a field, how you take in different things, you pay attention and you notice different things. At the time, I had done 16 years of ballet and he was helping me realize that when you go to a ballet performance, what I’m paying attention to might be something different than what somebody next to me is paying attention to.
It’s really that level of expertise that makes you focus in on particular things and notice and value and think about what counts as beauty. So when he was talking about art, he was saying that there’s all these different things that come into play about what we find aesthetically pleasing and also what we value in art. I felt like I took that idea from him and said, “Well, aren’t we kind of doing that in education? We’re also kind of saying there’s these aesthetic choices we make in education and we value certain things, and we look for those things in learners, but maybe there’s other things that learners are doing that we’re missing.”

Alec Patton:
All right. There’s one more piece here because the narrative that I have is you’re going to be a doctor, and you said, “And then what took me into studying how math is taught was taking a class on art education”, which one might imagine would lead you naturally to studying how art is taught.

Rochelle Gutiérrez:
People often ask me, “Well, you have a degree in biology. Why didn’t you go to science education? That seems like it makes a more natural connection.” I think at the time, what I understood though was the power of mathematics as it formats our lives. I think I understood that mathematics served as a gatekeeper so that people who I wanted to be able to go into sciences wouldn’t be able to do that if they couldn’t show that they learned their math in the ways that school sanctions it. I felt like if I’m going to make a big difference in the lives of other people, I’m going to have to address this thing that’s mathematic, and it was the thing that I loved doing kind of in my free time.

Alec Patton:
Thank you for that. I want to take now the abuela rule and tweak it slightly to the teacher rule. If a teacher’s listening to this and they’re thinking like, “Wow, that makes a lot of sense. I can see that math is like this gatekeeper. I can see that kids have different approaches and that doesn’t get honored,” what should they be thinking about?

Rochelle Gutiérrez:
I think that we need to be more explicit with students about how we got to this form of mathematics that we’re doing. I feel like we need to be more consistent in offering students opportunities to say, “Well, in the 17th century when we were thinking we were only in two dimensions, we got these Cartesian coordinates, and that helped us think about geometry in these ways. But then once we realized we were on a curved surface and we had different assumptions about the world, then we got to non-Euclidean geometries.” But how do we help students see that all along over time, we’ve made decisions and those decisions have led to the kinds of things that we value in terms of structure and reasoning and patterns and logics that we call mathematics, but that we could have other things?
When do we give students opportunities, both to see that the mathematics that’s been created is a reflection of our human practice, it’s a reflection of our relations in the world and seeing that, “Oh, in fact, we’ve learned many of these patterns from our more than human relatives.” So pine cones and Nautilus shells are doing are performing these versions of mathematics that we’ve learned from, and then how do we offer opportunities for students to say, “What if you had different assumptions? What are some different assumptions you would come up with?” For example, if we have younger children and we say … and young kids are great because they haven’t been locked into this idea that there is one mathematics. They’re still in this kind of playful stage of you can invent anything that you want. If you said to a kid, “We have four operations right now that we use in math class. We have addition, we have subtraction, we have multiplication, and we have division. Why don’t you come up with a fifth operation for us? What would that operation be? Why would it be useful for you as a learner? What would be its constraints? Would it work with only integers?”
You can get kids into a space where they could be playful and they can say, “Well, if I’m in this other world where my world …” So my other world was other bases and so I can imagine if somebody had given me a prompt to say, “We’re doing this thing, what if you wanted to do it differently?” and I thought, “Okay, if I’m in base two, I’m going to do it in this way.” Or “If I imagine a world where maybe we’re not on a sphere or a globe, maybe we’re in some kind of a donut shape”, it’s giving students that opportunity to not only understand the rules and follow the rules, but to break the rules.
It’s like in music. In music, when you are taught … or I’ll use dance, classical ballet. You are taught classical ballet, and it’s really you have to ask whose classics, but classical ballet teaches you a very kind of everything up high, everything is in your chest and your head and everything is lifted. The lines of your body and everything are lifted up. The angles of everything that you’re doing, they’re always this very kind of upward space. But in other cultures … so my husband’s Indian, and when I think about dancing with Indian dance, everything is low and it’s centered. Or even in salsa or [inaudible 00:17:02] or something else, how are your values of stomping the ground? Again, thinking about what is our relation to lands that when we are stomping the grounds, we are literally keeping the momentum going in the cosmos. This is part of our relation to lands.
This is why we need to be here and loud and stomping and lower, closer to the ground, not this high upper lifting, whatever. But if that’s my way of think of thinking about the world, and then you tell me to come up with a dance move, I’m not going to be locked in this everything has to be airy and uplifted and light and whatever. If we think about that mathematically, what could be the ways we could be inviting students to riff off of worlds they can imagine or things that they currently experience? Pointing out for them that makes so much sense that from your world or from where you’re coming from that you would decide that that would be a useful thing to do in mathematics. It’s kind of like when we think about community knowledge and we’ve come up with mean, median and mode as forms of representation in statistics.
So why those? Might there be some other form of representation that we would want that wouldn’t be one of those? What would it be? Why would that form be useful for us mathematically? What would that communicate in a visual or in a name? When people say like, “Oh yeah, the mode that’s most often people understand, oh, that’s useful because we might want to know something about why that was a popular thing for people to choose as opposed to the average thing for people to choose, which might not make as much sense, but what would be other things that we might want in mathematics?” It’s this idea of … I don’t want to say it’s just playful because when I talk about rehumanizing mathematics, I’m talking about centering the very people who have been most harmed by the dehumanizing forms of mathematics in schools.
That’s things like the binary logics that make it seem like a number is either even or it’s odd, which goes against two-spirit experience or people who are who are queer recognize that genders actually very fluid. Our more than human relatives teach us that, black bears and Clownfish and all these other relatives who can transform into other genders or multiple genders. I think it’s giving students those opportunities to say, “Here’s a rule or here’s this thing you’re going to learn and now we want to try to expand it. How would you expand it?” Because then we’re saying how can students be authors on mathematics rather than just consumers of mathematics? I think that’s a big piece of how we think about changing math itself, not just who gets to play this game of mathematics.

Alec Patton:
Got it. I think you’ve kind of touched on this already, but I read that you use the phrase a spiritual turn in mathematics education. What does that mean? Unpack that.

Rochelle Gutiérrez:
In 2010, I wrote this piece. It didn’t get published till 2013 because of the way that journals work, but I published this piece called The Sociopolitical Turn, and I was saying that we needed to be more conscious of issues of identity and power in mathematics. I wasn’t the only one talking about this, so I was really just pointing it out as this is a turn our field is making. But I feel like now we’re in a position … and in 2002, actually, I wrote a piece that was talking about how we need to have other relationships between humans, mathematics, and the planet.
But at the time, I didn’t have an actual thing to point to and so I feel like we’re in a spiritual turn now because we’re in a moment where people are recognizing that their relations to the world, I would say, our more than human relatives, but other people might say nature or they might say our living world, or they might use whatever terms kind of make sense for them, that there’s a space for us to recognize that we need to be in better relation with each other as humans in terms of what’s going on politically in the world and we need to be in better relations with lands and waters and skies, because of what’s going on with climate change and there’s kind of an awakening.
But I also think that this is a time when we’re recognizing globally that we’ve all been affected by this idea, what I was saying, this kind of grief of diaspora. We’ve all been taken from places, and so we’ve been scattered and in all different traditions, we have ways of talking about finding the unity in the world and bringing things back together. I feel like that’s the spiritual turn. The spiritual turn, it’s not about religion. It’s about how do I come to see a version of you and me and a version of me and you? I feel like the spiritual part is saying, how do we think about seeing each other as kin, bringing each other together?
The role of mathematics in that is if mathematics is really just about pattern and structure and logic, how are we pattern? How are we pattern in the world? How are you, again, like me, but not me? How are we structured together? How do we have opportunities for students to see each other as, “Oh, the way that you’re approaching that problem is like the way I was approaching this problem, but then it’s not the way I was approaching this problem, and so actually, that’s really interesting the way you are doing it just like me, but not like me, and somebody in another part of the world is also doing something like that.” So spiritually, how do we come to see this work as about attachment to each other and not just about problem solving in either school systems or even problem solving in technology spaces?

Alec Patton:
It strikes me that it feels like a lot of this aligns with your arguments with Elliot Eisner, that … it kind of feels like rather than thinking about there’s a universal that things align to or not, you’re kind of saying there’s a multitude of different ways, but they’re all connected.

Rochelle Gutiérrez:
Yes, yes, yes. I would say that’s [foreign language 00:23:21] , it’s an Aztec word that basically means kind of neither and both at the same time. It’s embracing that contradiction. It’s saying that mathematics is both incredibly universal and not universal at the same time. I think we have universal shared things that we experience in the world in terms of pattern and structure and logic, and so that unites us. But also we know that that mathematics has arisen through place-based ways of being in the world. Different people have developed attending to certain kinds of patterns based on whether you’re on the planes or whether you’re in a Saharan space or whether you’re in mountainous regions, or whether you’ve got lots of water around you or whatever. That’s going to call our attention to different kinds of things and for different purposes.
That doesn’t mean that somebody’s mathematics is better than somebody else’s mathematics, but it’s recognizing that tension that we actually all do the same mathematics and then we don’t do the same mathematics. I just want to make a plug for my friend Linda Faruto, who runs the Ethno Mathematics Program at the University of Hawaii. Only a few years ago did we get a full-fledged program where you can get a master’s degree in ethno mathematics. I think there’s people who have never even heard of that word, ethno mathematics, but it’s basically just pointing out that there are many mathematics that are practiced throughout the world. Ethnomathematics is a tradition that goes back to 1985 with Ubi D’Ambrosio. I don’t want to make it sound like any of these ideas bringing up are brand new, but I think that we are in a point right now where I think we’re just in a different moment where people are willing to understand and take up these ideas in ways that maybe they weren’t 10 years ago.

Alec Patton:
We are at 30 minutes. I have a couple other things I’d love to ask, but we can also stop it here if you got to get going.

Rochelle Gutiérrez:
Let’s just maybe take one more.

Alec Patton:
No problem. This is sort of a double question, but recommended reading.

Rochelle Gutiérrez:
I would say this is your creative insubordination. I said one more, and you’re like, “Okay, I have one question with two parts.”

Alec Patton:
You can choose either one of these. You can choose either one of these, is recommended reading and/or recommended board game.

Rochelle Gutiérrez:
Oh, wow. Oh my gosh. Just one board game? Recommended reading. A recommended reading for whom? For teachers, for people just entering this?

Alec Patton:
For teachers and just like when you’re going like, “Hey, you know what? More people should read this book.”

Rochelle Gutiérrez:
I feel like I don’t know that educators have as much time to read books as they have time to read shorter things.

Alec Patton:
This is true.

Rochelle Gutiérrez:
I would say maybe read the … I have a piece that came out in June, the Journal of Urban Mathematics Education in 2013. This was a long time ago, but I’m kind of pointing out what’s political about everything we do as math teachers. It’s a seven-page article. It’s really short, and somebody could just pick that up. Or there’s the introduction to the rehumanizing Mathematics for Black, Indigenous and Latinx students. That’s a five-page introduction that gets people to understand a little bit about what is this rehumanizing mathematics. I feel like those are kind of short, easy ways into something.

Alec Patton:
Great. Well, thank you so much

Rochelle Gutiérrez:
Thank you for having me. This has been fun. Take care.

Alec Patton:
Super fun. High Tech High Unboxed is hosted and edited by me, Alec Patton. Our theme music is by Brother Herschel. Huge thanks to Rochelle Gutierrez for taking the time to talk about her work. You can find links to her recommended reading in the show notes. Thanks for listening.