Invasions and Invitations

Lately I have had several conversations with colleagues and friends about implicit bias and racism. As a scholar of race and schools, and specifically urban education, I am drawn to studying ways that beliefs impact practice. The topic of microaggressions comes up often in my conversations.

Some people are well-versed in what microaggressions are, and can share anecdotes either from their own lived experiences or experiences shared by peers. Others are in a beginning place about the concept and might be skeptical — they can recognize overt forms of racism (hate speech, violent acts) but cannot yet see how a seemingly innocuous compliment is actually a microagression. For instance, I have received compliments about my hair that are veiled inquisitions unto my body: “your hair so so different and pretty! Is it like that naturally, or to you have to do something to it to make it so curly? Can I touch it?” Sometimes the question is left off, and I have had others touch my hair without asking permission. In another instance, I was asked if I “really see myself as a woman of color.” This translates to the classic “you’re so articulate for a person of color” transgression, the sting was much worse because it came from a close and trusted colleague.

An invasion is when someone does not take social cues and assumes the right to ask about another person’s cultural heritage and background or (as in the case of my hair, will go so far as to touch me without permission). An invitation, on the other hand, is the act of inviting a conversation about one’s identity and lived experience. An invasion is like pushing through a door to walk through it, whereas an invitation involves a person coming to the door and opening it for another to enter. As a scholar of urban schools, I’ve been thinking about the types of microaggressions students deal with on a daily basis. I worry about these microaggressions because children do not always have the conversational tools or vocabulary to challenge an adult in a way that sheds light on the microaggression. Children are experts in understanding fairness, and can sense when questions and comments do not feel fair or out of bounds, but often don’t know exactly how to speak up about how something makes them feel.

Below are two examples of exchanges in which a caring adult interacts with a student on the basis of getting to know one another a little bit better. The first is a a conversation I recently heard between a well-intentioned white adult and a ninth grader who is a person of color.

Adult: What’s something you like about being in this school?
Student: We don’t have homework at this school! It’s great, I had way too much homework at my old school, I just moved here.
Adult: So, where are you from? And where’s your family from?
Student: Harlem. Now we’re in Brooklyn.
Adult: Oh, that’s quite a distance to travel to get to school in Manhattan. But where is your family from?
Student: Uhh, yea it’s pretty far I guess. My family is from Mexico.
Adult: Okay! That’s so great!

(conversation ended here)

Compare the above with the invitation below. In the conversation I heard, a well intentioned white adult had a conversation with a student of color at the same school as above, and with the same goal of getting to know one another a little better.

Adult: So what’s your favorite food?
Student: I really love Caribbean food, it’s delicious.
Adult: Oh, I really enjoy Caribbean food. I really like eating the seafood curries, I wish I knew more about Caribbean cooking. What’s your favorite Caribbean dish?
Student: I love eating everything. I especially love my mom’s jerk chicken and rice. She is from Jamaica.
Adult: Oh, okay! Your mom is from Jamaica, do you have family there?

(conversation continues until student passes to lunch in the cafeteria)

An invitation feels natural and builds a bridge between people. When an invitation goes well, sometimes the question about one’s cultural or ethnic heritage might not be needed because the information is raised naturally in dialogue (“She is from Jamaica.”). An invasion is usually met with discomfort or confusion on the part of the student, who is likely to find themselves* in a precarious situation. Power and status hierarchy are at play. A student has to consider the risk imposed if they are seen “talking back” to the teacher/adult in the situation, so the least harmful course of action is to remain neutral and safely exit the conversation with a slightly awkward moment. The most harmful course of action could result in punishment if the student’s response is unacceptable to the adult.

When a student offers information about their identity, slow yourself down and check in with your preconceived notions about that student — we all have them, it is what you do about your preconceptions that will make the difference. Then, pick up on the invitation. Stay on the subject, don’t overgeneralize, and share enthusiasm if you genuinely feel it. Avoid empty praise and gratitude (“That’s so great!”) and do not assume to know more than what is offered. In the second exchange, it would have been a mistake to say, “Where is your dad from?” because that isn’t information the student has offered for the discussion. If as educators, you are able to actively engage and practice authentic conversation with young people, you may find that you experience a shift in your preconceptions about people, and your actions may begin to more wholly support your beliefs about equity and equality.

In the math classroom, invitations and invasions look similar. In one instance, an eighth grade student of color responded to a question posed to the whole group about what they noticed in a table of values. The table contained information about incomes for men and women in the United States. When the student shared that incomes for men are “much more greater by almost double,” the teacher interrupted the student prematurely and then only recorded the word “greater” on the board as his noticing. The student is of West African decent (I know this because I have had conversations with this student) and speaks with an accent. The teacher did not slow down to hear the most important mathematical contribution the student made, he observed one value is nearly double of another. The teacher committed an invasion by interrupting the student, and as I learned later, this invasion was intentional because the teacher felt as though they were not going to understand what was said next. In that moment, I raised my hand to get some clarification on what I heard, and the teacher updated the student noticing to state, “greater, almost double.” This moment was troubling to me for two reasons. First, how will the teacher learn that what they did was silencing to the student, and did my actions do anything to help the student speak for themselves? I used my power and privilege in the classroom as a math coach and I knew I would be called on immediately and that my contribution would be heard and recorded. In this case, I followed up with student and teacher about the moment afterwards, but how many of these interactions go unchallenged?

In my separate debriefs with the teacher and student we talked about strategies that could be used to slow down in the teacher’s case and speak up in the student’s case. The teacher admitted to being a little confused by the student’s accent, and we talked about how to slow down to help avoid an invasion, and ask for students to repeat themselves loudly so that they are heard. It is okay to say, “I didn’t hear you, can you please repeat that?” Research shows, unsurprisingly, that by listening to an accent that is different from yours for longer periods of time, your brain maps the words you hear onto the words you already know and comprehension improves. And just think about the types of invitations that may occur as you hear your students speak freely with you! The lesson is, listen to your students more. For my debrief with the student, I asked them how they felt when they were misunderstood, and the student shared feelings of confusion and sadness. We talked about how it is okay to say, “that’s not exactly what I said.” It is not a perfect strategy because it does not level the power structure between teacher and student, but it does at the very least give more voice to the student for a brief moment.

We cannot go through with our daily work feeling proud of our equity-based stance on mathematics education, it is not enough and we must actively work on changing our preconceived ideas about groups of people who are different than us. Well-intentioned educators of various racial, ethnic, gender, and other social identities commit microaggressions and invasions all the time. In my experience, and largely in the literature on microaggessions, invasions are more frequently committed by people from the dominant culture – straight, cisgender, financially stable, white people, however, nobody is off the hook. What I propose in this blog is that there is a great need for math educators to actively practice engaging in and picking up on invitations into dialogue with students, because students in most instances do not have the power or privilege to challenge invasions levied upon them. To avoid committing invasions my advice is simple — slow down, check in with preconceptions, and listen with care.

*Non-binary pronouns are intentional.

They Can Fit

I was working in a middle school classroom in Harlem on a problem called “chairs and tables.” It is a classic problem:

The Italian Villa Restaurant has square tables that the servers can push together to create longer tables and seat larger groups of customers. One one chair fits along a side of the square table.

Students were invited to model the problem using flat tiles and dried black beans. Each bean represented a person in a chair and each tile represented a table. Students played with the model for a little while and shared how they would lengthen tables and add more chairs. Here is an example of how one child thought about the problem:

student work tables and chairs.png

As seen in the image, the student envisions moving the chair on the right around to the side of the second table, adding two chairs to the top and bottom of the second table, and then the tables get “put together.”

Students worked on solving and creating an algebraic rule for the problem, based on building the model and then keeping track of the data on growing patterns in a t-chart. Most students found the rule 2t + 2, where t=tables. The purpose of the task was to model a pattern and make the leap to algebraic representations for tables of any length.

I could go on and on about marvelous and brilliant student strategies for solving this problem. The most important moment in working on this lesson for me as an educator took place during a conversation I had with one student named Damon. Damon was perplexed by this problem and struggled to model the addition of a square table to the existing table. To step back a moment, I started the problem by having students share about a time that they went out to a restaurant to eat with their families, and we talked about how sometimes we might see the servers moving tables together so that families could be together. To Damon, this problem presented a challenge that did not match his lived experience and caused an obstacle to seeing the problem as described.

Damon worked and worked, and kept adding four chairs every time he added a table. I asked him to restate the rule governing the problem, that there could only be one chair on one length of the square. But Damon insisted, “just put the extra person between the tables, they can fit.” To Damon, it was unthinkable that the servers would leave two people out every time they moved tables together. When we described the problem and asked students to envision the act of putting tables together, Damon envisioned his family and he was not thinking in chairs and tables, he was thinking about people. If there are already people sitting around a table, and then we want to slide tables together, to him it would be impossible to leave out the people on the ends and the servers should squeeze everyone in. For two tables, Damon drew this:

What I learned in that moment is that context and clarity really matter, especially if we want students to connect to the mathematics and grow their critical thinking skills. This problem in the most academic form is a standard linear growth problem — for each table add two chairs. Damon’s interpretation of the problem based on my explanation of families going to dinner together led him to see the problem in a different and beautiful way. Damon needed to include everyone, because that’s what happens in his family. I am indebted to Damon and the light he shined for me that day. I learned that how I talk about a problem needs to be clear — if I clearly talked about adding chairs before the family sat down together to eat, I wonder if Damon would have made sense of the context that I planned for. Second, I reflected and thought about how for some students who come from families and cultures that focus on inclusion and community, these types of problems that rely on shared cultural norms may not be interpreted in the same way when our cultural norms and beliefs vary. Can I go so far as to say this is evidence of a colonized curriculum in mathematics? Probably not. But I am pausing to think about how I can blur the lines between “school math” and reflecting students’ lives purposefully in the mathematics curriculum.

As a woman of Indian cultural heritage, I too can relate to how Damon understood this problem. For some communities and cultures, the norms around eating and communal gathering dictate that everyone should be included; and we can always say, “they can fit.” Growing up in an Indian family, our outings and social gatherings always included everyone, the elderly, the children, and the parents. We did not leave anyone out, and if we needed to move a little this way or that way to ensure everyone had space to sit and eat at a table, we did.

The conversation with Damon was so powerful for me because I could draw a distinction between an academic interpretation of the chairs and tables problem and the reality of children and families from diverse cultural backgrounds going out to eat. I continue to wonder about how to bring in more context from students’ lived experiences to add richness and love to the math classroom. When children feel that their lives are reflected in the content in math, they are more likely to engage and persist in problem-solving. I will be more careful in the future, as I think about the tables and chairs problem and other problems like it.

Socio-Politics of Numeracy

The Socio-Politics of Numeracy: What 6th Graders in the Bronx Wonder About

By Anjali Deshpande and McKenna Shaw

In February, 2018, McKenna’s school participated in New York’s Black Lives Matter Week of Action In Schools. This middle school is located in the north Bronx and serves a diverse student population, with the large majority of students qualifying for free and reduced price lunch. Many students’ families are living in poverty. The school’s administrative team ascribes to progressive ideals about community, service, and citizenship. The teachers work very long hours, often working weekends and nights. Similarly to other Bronx middle schools I’ve worked in, the students are almost all Black and Brown, and the school is subject to a revolving door of reforms that are implemented and sometimes abandoned in place of new policies. While it is a safe and welcoming environment, the school continues to look for ways to support students with diverse learning needs.

To start to the week of action, McKenna and I revisited the Black Lives Matter Guiding Principles and reflected on how the principles spoke to the reasons we became educators. I helped McKenna find a way to open the unit of study by asking her 6th graders: “What connections do you see between Black Lives Matter and our work in Math and Science?” This question was met with silence, blank stares, and one or two cocked heads and raised eyebrows. Then we asked students to write. We received numerous reflection papers where the sentence starter, “A connection between math and science and Black Lives Matter I see is . . . “ went unfinished. Several question marks, a few doodles. We received a limited number of substantial responses, one student wrote, “All of us in the classroom, we are a family,” and another wrote, “ one of the principles was called a ‘village’ meaning people coming together to make a family larger than your own.” Some students shared that they felt safe to make mistakes in math and science, and have their questions answered. Students saw McKenna as a teacher who “had their back” and would advocate for them socially and academically.

We learned two things. On the one hand, McKenna had created a safe classroom environment where students could engage respectfully and appreciatively of the diverse identities in the room. Students recognized and named the enactment of many of the Black Lives Matter principles in the classroom. It was rewarding to hear from sixth graders that they feel safe to learn especially when some students are coming to school to escape potentially difficult conditions at home. But it was evident that the students did not yet see math and science as real tools that can be used to measure, discuss, and affect social and political change. “Math politics? Is that a thing?” one student asked.

Mathematical literacy, or numeracy, must be primary goal for our students, it is a requirement for justice and citizenship. PISA (2013) defined the mathematically literate person as “one who understands the role mathematics plays in the world, makes well-founded judgements and uses mathematics to meet their needs as a constructive, concerned, and reflective member of society.” In Horace’s Compromise, Ted Sizer (1984) claims “Numeracy means the ability both to use numbers, arithmetically and algebraically, and to understand the concepts, relationships, and logic embedded in mathematical thought. A modern citizen cannot make critical judgments without these skills.” The math curriculum at the school asks students to reason mathematically with real-life contexts, and while using unit rates at the grocery store to identify a better buy is important and necessary, the curriculum is not designed to push students to become mathematically literate and empowered to make critical judgments about their worlds. Calculation is not enough.

In retrospect, McKenna and I are thinking of ways to shape next year’s statistics unit around current events. What better place to help students see the real and powerful applications that math has in the socio-political world? McKenna decided to pilot a new routine this year. She introduced the graph “Gun Deaths in Florida” and created a space for students to ask questions, think critically, and make connections. Take a moment toanalyze this graph; notice the y-axis (vertical axis) is inverted and the values decrease from bottom to top. Did you think gun deaths went down prior to “Stand Your Ground” law? At first, so did we. After having a chance to unpack features of the graph, one student asked, “Wait – wasn’t Trayvon Martin killed in Florida? What year was he murdered?” Trayvon Martin was killed in 2012, and it was powerful to hear a sixth grader ask this question because this student was making an important connection. A great math lesson about interpreting graphs will help students make sense of data, but an empowering lesson about graphs will motivate a student to research, question, analyze, and critique the way things work in the world.

McKenna is now reflecting on the power to build numeracy into her daily teaching and learning practice. Rather than saving rich content for the very last unit of the school year, where statistics usually lives in the sixth grade curriculum, she plans to make “Math in the News” discussions happen once a week. Even if these discussions last for 5-10 minutes, the hope is to establish the norm of discussing politics, social justice and awareness, and news in math class. The hope is to help students ask the most important questions about their worlds. Students today find themselves in politically charged spaces, take the #Enough movement and gun control walkouts for example, and it is our job to give them the tools, critical and analytical skills, and the language they need to affect change. Instead of teaching math as a “school” subject and hoping students will make connections outside of the classroom, we must make space for those connections to happen during the school day.