Let Them Laugh: Using Humor in Math Class

Humor serves many functions in my life. Noticing the absurd. Playing with unexpected associations and enjoying the surprise. Sharing inside jokes with friends. Resisting, venting, and gaining temporary relief about abuses of power by belittling them with laughter.

I am obviously not unique in this. Humor is a crucial part of being human in a complex (and often ridiculous) world.

But humor –– especially in certain forms  –– is not always welcome in school. What does this mean for students’ expressions of their humanity? Think about the well known class clown archetype. Some educators use this label with derision, assuming students step into this role for negative reasons, like avoiding work or garnering attention that distracts from lessons, making the teacher’s job harder.

The essential role of humor for many people’s ways of being in the world is thus in tension with many ways of “doing school” that require deference to the teacher’s authority. This leads to dilemmas for those of us who want to build inclusive and humanizing classrooms. We get many messages from administrators, teacher educators, and other colleagues that a good class is an orderly class, one where the teacher leads and students follow, not one where spontaneous outbursts might be embraced and incorporated, where laughter might be happening in small groups, or where a class clown can have a legitimate place in learning –– even be an important member of the community.

Personally, I knew early on I was going to struggle to navigate my own predilections with these common images of “good teaching.”  During my student teaching, I accompanied a group of seventh graders on a field trip to the courthouse to observe a trial. Our guide welcomed us to the courthouse and looked at the docket to see what case we would be viewing.

“Oh, good. It’s a nice, clean trial,” she said.

“Dang!” said one student, leading me to snicker audibly.

Some nearby students turned to me in surprise: Wasn’t I supposed to scold him for his lack of decorum? Perhaps I was. But as a human, I loved his reaction for its honesty. In fact, much of my joy as an educator comes from engaging students’ clear eyed honesty, which extends to their disappointment about a “nice, clean trial.” I realized then that I would need to develop a way to enforce behavioral norms (many of which I myself find challenging to adhere to) and my genuine appreciation for his rascally reaction.

Let me be clear. I am not suggesting an “anything goes” approach to humor in schools. Educators resist humor for many reasons. Humor can be subversive. Humor can be exclusionary. And, obviously, humor can be mean. But shouldn’t we have a way of inviting some humor into our math classrooms? Can we pick and choose with a little more care?

Seeking a framework for humor in math classrooms

Recently, I have been working with my colleague, Dr. Nicole Joseph, and an undergraduate research assistant, Yasmin Aguillon, to look at humor in math classrooms.

Dr. J and I are working on this together for a few reasons. We share a commitment to welcoming students’ full humanity into the math classroom and seek to help teachers understand what that entails. In one of her recent articles, she and her colleagues heard from Black girls that they wanted math classes to be a place to be their full selves, including the silly and “goofy” (an adjective one of the participants used). We also share that our ways of being in the world do not require us to put our own silliness in a box when we are doing work. We can sing, tell jokes, code switch –– play –– while we are working on things we deeply care about. This, in our lived experience, is not at odds with deep and rigorous work. In fact, bringing our authentic selves to our work only enhances its depth and rigor.

Dr. Nicole Joseph and me, clowning with math at the Escher Museum

To build our framework, we draw on several sources. First, we have looked at research on humor in social life, especially classrooms. Second, we have looked at classroom level data from my recent research project where students are clowning around. Finally, we conducted a completely unscientific #MTBoS #iteachmath twitter poll on whether math teachers felt humor had a place in the classroom.

We are working on a fuller discussion of humor in math class, but for the purposes of this blog post, I want to suggest the following:

The playful –– and even the subversive –– aspects of humor belong in math class, not only because they allow students to more fully be themselves, but because they embody important mathematical habits of mind and allow entrée for students’ broader identities. At the same time, we recognize that inviting humor may require teachers to develop new forms of teaching and cultural competence. 

Playful and subversive humor belongs in math class

If you think about the pleasurable aha! of mathematics, you might notice its similarity to the pleasurable moment of getting a joke. The common experience is that of insight.

Mathematics has its share of jokers. Aside from famous people like Lewis Carroll who enjoyed playing with logic, we have mathematical entities that upend the order of things. Infinity and zero subversively violate our expectations of how operations and functions work: what do you mean we can’t divide by zero? why do functions change when we imagine taking them to infinity? The whole field of topology essentially invites us to imagine that objects as made of rubber, making a coffee mug “the same” as a donut.  These delights of insight and absurdity tickle any good rascal.

When we invite humor into math class, we also change the emotional tenor of what we are doing. Humor positively effects learning by releasing tension. When we laugh, we are at often more at ease. Humor has even been shown to improve students’ performance on tests. Maybe laughing sessions can improve study sessions. Humor can build rapport, either with individual students or with a classroom community. I know a teacher who strategically looks for something to laugh at with each of his classes, so that they can have a shared inside joke.

Humor also invites students’ broader identities into their learning. Who we are is fundamental to how we make sense of the world; when we have to leave part of our selves at the door in order to be seen as “acceptable,” we abandon crucial sensemaking resources. Although laughing at my student’s response to the “nice, clean” court case may have not followed a proper teacher script, it appreciated him for the twelve year old boy that he was and his understandable wish for a meatier, dirtier case. In sanitizing the world for children, often in the name of protection, we omit important details. By the time they are twelve, they have often started to question who is actually protected through adult judgments of “appropriateness.” This kind of questioning signals curiosity and intelligence.

Why wouldn’t we welcome such attitudes in school?

Teachers may need new forms of teaching competence to productively support humor in class

Indeed, in our completely unscientific twitter poll of 741 people on the Internet, 94% of respondents agreed with our idea that humor belongs in math class. Among the 31 respondents who elaborated on their reasoning, we found that most people (64.5%) referred to math humor, like math puns and math jokes. The next most mentioned form of humor were intentional mistakes (12.9%), a pedagogical strategy where teachers present an incorrect solution and humorously play at not understanding to prompt student explanations. Most of the remaining responses referred to negative forms of humor such as sarcasm (9.7%), put downs (6.5%), or teasing (3.2%).

Breakdown of the 31 comments from our completely unscientific Twitter poll

What is interesting about these responses, as limited as they might be because of the nature of the poll, is that the positive examples of humor are teacher-centered; that is, they are controlled by the teachers’ choices about curriculum and pedagogy. The negative examples signal adverse relationships, whether from students to the teacher, teacher to students, or among students themselves. But, in the examples offered, we do not see examples of how students themselves can be a positive source of classroom humor.

We suspect this is because student-derived humor is trickier teaching terrain. (Or it may be due to the limitations of our completely unscientific Twitter poll.)

Building off of the first interpretation, we acknowledge that there are many shades of gray when we laugh with students. How do we navigate the ideas of “appropriate” that can vary so much within a classroom, let alone a broader school community?

Once we open the door to students’ humor, how do we ensure that positive humor does not cross over into the exclusionary humor that can lead to hurt feelings and negative social dynamics?

How do we feel if students are laughing at us? Or even if we are not the target, how do we feel if they are sharing a joke that, for reasons of generational, linguistic, or cultural differences, we don’t get? What would it mean for us, as teachers, if we are on the outside of the humor?

We suspect that productively managing humor requires a unique form of teaching competence. As the outsider example suggests, this also includes forms of cultural competence.

We are just starting to figure out how to name and describe what these teaching competencies might look like, using the classroom data examples. From our initial look, we think humor competence for teaching involves a form of self-knowledge –– knowing yourself, being comfortable with your own identities, having the humility to laugh at yourself. But it also involves relational skills of reading students’ reactions, developing rapport to invite open communication, and having strategies for repairing relationships when lines get crossed or feelings hurt.

We would love to hear more about how you use humor in math class to make it a more humanizing space. Math class could use a few more laughs.

This post was written as a part of The Virtual Conference on Humanizing Mathematics (#VConHM on Twitter)

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Modeling Mathematical Aesthetics

Fractal-Geometry-HQ-Desktop-Wallpaper-24806Note: This post was written by my two doctoral students, Lara Jasien and Nadav Ehrenfeld, as part of the Virtual Conference on Mathematical Flavors. This essay responds to the prompt  “How do teachers move the needle on what their kids think about the doing of math?” It is also part of a strand inside that conference, inspired by an essay by Tim Gowers,Two Cultures in Mathematics.

Before we begin responding to Gowers’ essay, we’d like to share a little bit about what draws us to this conversation. As budding researchers of mathematics teaching and learning, we spend our days watching teachers go about their daily routines with their students. We look for the ways teachers support their students to engage in meaningful learning and position them as capable, curious thinkers. Our work is fundamentally concerned with the ways classroom culture shapes what it means to teach, learn, and do mathematics. Gowers’ essay provides us with an interesting new lens on the role of culture in mathematics. We want to share (what we think is) a problem-of-practice worth considering and then point to an often overlooked teaching move that we recently saw a teacher use in ways that counteracted this problem-of-practice.

As educators and learners of mathematics, our experiences usually involve engaging students in correctly solving mathematical problems that are predetermined, handed down to us over generations through textbooks and pacing guides (with slight variations). This means that students have few opportunities to engage in a core element of mathematics — finding and articulating problems that are interesting to solve. We think this is intimately connected to a missing aspect of mathematics culture in typical math education: the mathematical aesthetic. Mathematicians’ aesthetic tastes and values lead them to pursue some problems, solution strategies, and forms of proof write-ups over others. When mathematicians’ inquiry is driven by their aesthetics, they engage in exploration, noticing, wondering, and problem-posing.

The mathematical aesthetic is the mechanism by which mathematicians distinguish between what they experience as meaningful, interesting mathematics and trivial, boring mathematics. In his essay, The Two Cultures of Mathematics, W.T. Gowers identified two groups of mathematicians who find each other’s work equally distasteful (with a little dramatic flare): problem solving mathematicians and theoretical mathematicians. Typically, school instruction exposes students to problems that fit both cultures of mathematics: Some school mathematics is done for mathematics sake, some is done for the purpose of real life or pseudo real life (word problems) problem solving. Yet, when do students have the opportunity to develop aesthetic preferences for different ways of engaging with and thinking about mathematics?

In our work, we have seen classrooms with cultures that support students in posing questions to their peers — questions like, I wonder if there is a reason for that? or What’s your hunch?. In these classrooms, we see students begin to be interested in and passionate about mathematics. In our minds, when students develop such passionate tastes about meaningful mathematics, we are on a good track for empowering our students for success.

The questions we just mentioned are actually questions we recently overheard a teacher asking her students as her last statement before exiting small group conversations. We consider her enthusiastic questions to be a form of modelling mathematical aesthetics, prompting students to be curious, explore, wonder, and use their intuition. While ideally the classroom culture would eventually lead to students asking themselves and each other these kinds of aesthetic questions, we know that our own authentic intellectual curiosity as educators does not go unnoticed by our students. Importantly, this teacher did not ask these questions and then hang around and wait for student answers. She left the students with juicy questions that they could investigate together.

As teachers, we rarely get feedback on how our exit moves from small group conversations affect their conversation or the classroom culture. Of course, some exit strategies ­–– such as telling students the answer or funneling them towards it –– will clearly lead to cultures where students see mathematics as a discipline of quick-and-correct answer finding. This view of mathematics can preclude opportunities for students to develop as autonomous doers and thinkers of mathematics. Fortunately, options for productive exit strategies and modelling of intellectual interests are many. These options also present new decision-making challenges to teachers as what happens when we exit the conversation becomes far less predictable. Our students do not need to have the same mathematical tastes as we do, but we do want them to feel empowered and intellectually curious in our classrooms. By foregrounding noticing, wondering, and problem-posing as authentic mathematical practices, we can support students in developing their own mathematical aesthetic. Of course, doing so requires us to model genuine intellectual curiosity, make room for uncertainty and ambiguity in our tasks (groupworthy!), create access to multiple resources for pursuing mathematical questions (Google is acceptable!), and scaffold for conversation rather than bottom-lines (exit moves!). Leaving students with a juicy, natural question is a start.

You Are More than Your Achievements

Last night, I was invited to address the new members of the National Honors Society at my daughter Naomi’s high school. I am sharing the text of my speech here.

Good evening. First, let me congratulate you on all the hard work and talent that is reflected by your being here tonight. Your presence in this auditorium means that have worked hard. You have earned good grades. You have demonstrated a commitment to your community.

No doubt, you deserve to be congratulated –– to be honored ––  for all of these things.

I too am honored to be here this evening, to share in your accomplishment, to beam alongside your parents.

As Naomi mentioned, in addition to being a parent here, I am a professor. My research is in the field of math education — secondary math education, in fact. So I have spent a lot of the last 25 years in public schools, especially urban high schools. In that capacity,  get to work with and talk to kids and teachers about their experiences.

I do my research anthropologically: that is, I study schools as cultural systems and try to understand what makes them work, what values they convey through their customs and rituals, with an eye toward helping them become places that serve our country’s democratic ideals. When U.S. public education was founded in the 1800s, one of its central purposes was to  build citizens who can make good lives for themselves and contribute to our society.

After 25 years of working and living in schools all across the U.S. (and even a few of them abroad), I can say: schools are wonderful and terrible.

Sometimes even one school is both at the same time.

At their best, schools serve the democratic ideals I just described. The open up opportunities, promote social mobility, and develop informed and thoughtful citizens.

When schools operate in this way, they fill me with hope and optimism. I see young people discovering what is possible, preparing themselves as the next generation of leaders.

At their worst, schools work against democratic ideals. They operate in ways that preserve opportunities for some while shutting others out. They contribute to social reproduction. They fall short of their mission to develop informed and thoughtful citizens.

When schools operate in that way, I imagine them as heartless machines, pumping kids through, sorting them on a conveyor belt toward paths of success or failure.

My observations about the dual possibilities of schooling are not meant to denigrate your important accomplishment tonight. Your induction into NHS means that you have been  sorted into the academic successes.

To the contrary, my observation only increases my accolades and admiration, because it means that, in addition to having talent and working hard, many of you have very likely managed to advocate for yourselves, size up a system that is not always kind, and maintain a strong academic record in spite of various blows to your humanity along the way.

Every one of you has a story. Every one of you can tell me about how you got here tonight. And if your stories are like the stories of other high achieving students I have met over the years, along with your talent and hard work, your presence here tonight very likely also shows resilience and savvy. This deserves its own congratulations and it will also no doubt serve you well in your future.

Becoming a member of NHS marks you as an achiever. The culture of achievement to which you are being formally inducted here is a wonderful thing. It gives your families cause for pride and celebration. It will open doors to jobs and colleges for you.

Spending time in schools and studying adolescents in education, I know that (perhaps for some of you more than others), there are also personal costs for joining the culture of achievement. I want to take a moment to consider those, out of a sense of care and concern for you and your long term wellbeing.

May is Mental Health Awareness month, and we know right now that young people are facing greater rates of anxiety and depression than they have previously. The last statistic I saw reported about 1 out of 5 college students experiencing these things. That’s about a 33% increase over previous rates.

People my age like to think this is, in part, because of social media — you put your social lives online. On Snapchat and Instagram, you are constantly comparing yourselves to others. We even have a word –– FoMO –– to describe the special angst of missing out on things you see others doing that, in my generation, we most likely would have never known were happening.

I think there is more going on than that. I think there are vulnerabilities here in this room that are not often enough addressed.

I am bringing up mental health because it matters for your long term happiness. I teach in a university. I see students, just like you, who have successfully navigated the perils of schooling, who got sorted in the right ways. They, like you, are creative, inquisitive, driven. Sometimes, they end up in my office, and I offer tissues as they share their concerns and their worries. I see that too often they are led to believe that their value as a person is directly tied to their achievement.

So much emphasis in high school is on getting into the best college. And what happens if you manage to get there? We all hear of the benefits: Great education, valuable social network, important internship opportunities, a “brand name” university on your resume.

What we hear about a lot less are the risks to you, as a person and a human being.

Jamie O’Keeffe, an educational researcher at Stanford University, studied high achieving undergraduates at a top-tier university and how they contend with the pressures to achieve. She describes that when students “make it” to a selective university, they feel pressure to maintain the image of a “total winner”: the ideal self-made individual who excels without effort, confidently demonstrating genuine intellectual passion and desire to make a difference in the world, while effectively realizing increasingly greater achievements.

Paradoxically, this pressure makes it harder to maintain a sense of wellbeing. They experience stress, insecurity, and self-criticism, as any sense of struggle or failure feels like it is evidence against their legitimacy there. We have a name for that feeling. It’s called impostor syndrome. These feelings are amplified for first generation college students or students from historically marginalized groups.

So what is my message? Do I recommend slacking off? Bailing on NHS and aiming for the middle?

No. Absolutely not. As I started out saying, what you have done is commendable, maybe more so than many even realize. It is worthy of the honor you are receiving.

All of you sitting here belong here.

I suggest you follow Albert Einstein’s advice: “Try not to become a person of success, but rather try to become a person of value.”

I just want you to remember that you are more than your achievements. You are more than what you can list on the Common App next Fall. In fact, much of what is so precious about you won’t ever make it into that form.

I also am asking you to remember this. Aim to be honorable, not only in the ways that got you here tonight, but honorable to your friend who needs you to listen. Be honorable as children to your parents who have cared for you all these years. Be honorable as members of your communities –– your churches, synagogues, mosques, or whatever you consider your community. Be honorable to strangers in need. Be honorable to the students who struggle more than you do, the ones who are not here today.

Being honorable means doing what is right. And doing what is right is not always easy. It is not always obvious. Part of being honorable involves really knowing what you stand for and what you care about. It means knowing who you are. That should, ideally, be a part of your education too.

What does this have to do with mental health? In my experience, people who know who they are tend to hit those challenges, those setbacks, those inevitable and often unexpected bumps in the road with, if not grace,  with inner strength. They get knocked down, but they see a bigger picture, a larger purpose. They have people they can lean on, a sense of their own value beyond whatever went wrong. In this way, they maintain their perspective, even when they are shown to be less than a “total winner.”

From honor –– true honor –– comes strength in the face of adversity.

So congratulations on this honor. Be sure to cultivate it, and I wish you all a good life.

 

Teacher Education in the New Economy

This week, I am spending some time with my colleagues who comprise the Teacher Education Collective. This past year, we have been working together to address critical issues in teacher education. We met previously in January, which I wrote about here, and we are meeting up again.

Teacher Ed Collective 2 (1)

Clockwise, from top left: Jamy Stillman, Mariana Souto-Manning, Lauren Anderson, Matt Diemer, Dorinda Carter Andrews, Thomas Philip, Manka Varghese, and me

As a part of our agenda, I asked if we could read and discuss Tressie McMillan Cottom‘s book, Lower Ed. Her book examines the expansion of for-profit higher education in the 21st century. (You can read an excerpt here.)

Among other things that make this book a valuable read for anybody who cares about education and inequality, Tressie does a phenomenal job of characterizing the social and economic forces that have served to accelerate the growth of for-profit colleges.

As I read the book, I was struck by how these same forces have operated in slightly different ways on teacher education to facilitate the proliferation of alternative licensure programs –– and even the complete abandonment of teacher credentialing in states like Utah –– and the concomitant devaluation of university-based teacher education.

Here are some ways I see Tressie’s analysis explaining what we see in teacher education.

The Education Gospel

Tressie draws on Grubb and Lazerson’s idea of the Education Gospel, the collective faith that education is “moral, personally edifying, collectively beneficial, and a worthwhile investment no matter the cost, either individual societal.” Teacher education has always had a funny relationship to that Gospel. On the one hand, teachers are clergypeople in the church of education. On the other hand, teacher education has always had a shadowed status in relation to universities, in part, because of their vocational mission.

So teachers kind of benefit from our faith in education because they serve it (thus the halo around being a teacher and the tendency for teacher narratives to smack of saviorism). At the same time, their own education is put in doubt for being of lesser value than higher status professions like law or medicine.

In other words, the Education Gospel, because it disavows the relationship between education and jobs, leaves room for tremendous skepticism about the value of university-based teacher education. That skepticism has been exploited and amplified in the last few decades, making way for the market imperative.

The Market Imperative

Tressie spends time addressing how the idea of higher education as a “marketplace” has become naturalized in discussions about traditional non-profit and for-profit institutions. Under this rubric, for-profit institutions have been praised for being “nimble” and “responsive” to consumer demands.

This is also the case with K-12 schools. The market imperative has become a go-to rationale for many “disruptive” innovations in public education, like vouchers and school choice.

The market imperative assumes that the best forms of practice –– whether they are charter schools or teacher preparation programs –– will rise to the top in response to market pressures.

The common sense appeal of the market imperative erases other possible solutions to endemic educational problems. For instance, the market imperative underlies credential expansion (“alternative certification,” or “alt cert”) by leveraging the skepticism about university teacher education without addressing the status quo inequalities in schools. The market imperative can be seen in policy press toward easier teaching credentials (or no credentials) in lieu of better working conditions for teachers. Likewise, when calls for professionalization are met with training for short term teacher labor in charter networks, that is the market imperative at work. Finally, we see the market imperative when we ask for educational justice and, instead, we get ‘missionary tourism for elites’ before they go onto their real careers.

Dependence on Inequality to their Survival

One of the most devastating conclusions Tressie draws in her analysis is for-profit higher education’s deep reliance on social and economic inequality for its own survival.

I suspect many of the alternative credentialing pathways in teacher education likewise depend on continuing inequality to justify their place in the marketplace. What if, tomorrow, the economic deprivation of minoritized communities was redressed? What if students and educators were given adequate resources within schools –– no more GoFundMe’s for classroom supplies? What if teachers working in urban settings earned legitimate middle class salaries? What if they were provided with adequate buildings and instructional materials? What if the communities they served had a full range of social services and high quality health care? What would it do to the narrative on which organizations like Teach for America or RELAY are based? If these organizations’ narratives depend on the economic and social deprivation of the communities they serve, that signals that they rely on social inequality to justify their existence.

Labor Conditions in the New Economy

Tressie lays out the labor conditions in the New Economy. These include:

 

  • Job mobility: Unlike earlier generations, workers can expect to change jobs many times over the course of a career.
  • Labor flexibility: The economy is increasing its reliance on temporary labor.
  • Declining internal labor markets: Employers no longer see it as their role to re-train workers, leaving workers themselves with the costs of both time and money to maintain their skills.
  • Risk shift: Workers are shouldering more responsibility for their job training, healthcare and retirement.

 

She also describes the astronomical increases in debt-burden of college graduates since the 2000s.

These same conditions have impacted teacher education. By assuming more debt for undergraduate education, continuing on in higher education for a teaching credential becomes an irrational choice. Why would a college graduate assume even more debt for a credential that can be obtained for little to no financial burden, especially if it requires staying out of the labor market even longer, when fast track options allow them to work quickly for the same wages?

Teaching used to be a means for social mobility as a secure middle class profession, since it was less subject to fluctuations in the economy and came with strong benefits, especially in unionized states. Stagnation in teacher wages has no longer made that the case. It has become, especially in high priced urban centers, economically tenable primarily for single childless people.


The proliferation of alt cert pathways has been explained, using market logic, as a failure of university-based teacher education. Within the context of the New Economy, it becomes clearer that it is equally reasonable to view it as a response to shifting social and labor conditions.

 

 

 

 

Don’t ask if it’s “good” without adding “for whom?”

Once upon a time, a friend and I were talking about a math teacher our daughters shared. She said, “You know, his lessons weren’t so exciting, but at least he’s got the content knowledge, so the kids could work it out.”

I grew quiet, and an awkward pause ensued.
This had not been my daughter’s experience at all.

My friend’s daughter is the quintessential math kid, the kind you can provide with some math content, and she will eagerly work through it. She loves patterns and puzzles, sticks with new things until she gleans key insights, with or without a teacher. She is a bright and thoughtful kid, and any teacher would be glad to have her in class.

My kid? The social world is her main interest in school. Honestly, she is a fabulous little social scientist, drawing me sociograms of the lunch tables to explain the different friend groups, designing “crush scales” to help her friends quantify the extent of their infatuations, providing me with details of the social markers of popularity at her school. She finds school fascinating, but the school curriculum does not always engage her. Her academic interests depend largely on her relationship with a given teacher.

With the math teacher in question, she had a strong strike against her. She is a doodler, and this same math teacher was squarely anti-doodle.

mr math man

In a parent conference, he told me that he viewed her doodling as a lack of seriousness. I tried to push back and help him see her more clearly, to little avail. In the end, the same perceptiveness that led my kid to understand the subtle workings of her school’s social dynamics led her to the conclusion that this teacher did not think very highly of her as a potential learner in his class. Not surprisingly, her engagement reflected that.

The differences between my friend’s and my experiences with the same teacher has me thinking about how limited we are when we talk about the quality of education. Often our discourse focuses on whether things are “good” ––  whether it is teachers, classes, or schools –– as if it is an essential property of the thing.  In this case, my friend would probably say that this teacher was “good enough,” because that was true –– for her daughter. As you may surmise, I would say otherwise.

Our impoverished way of sorting the educational world into good/not good has consequences for the systems we have created. I would go so far as to say that it contributes to inequality. Many middle class parents are probably familiar with the conversations about “good schools.” Parents seldom actually go and visit the schools, but there are signifiers that stand in for goodness, many of which are problematic: test scores (which correlate to parent income and education), student populations that skew White/Asian/affluent. As the consensus grows about where “goodness” lies, in our crazy U.S. system, property values respond accordingly, reinforcing the demographic factors that underlie those assessments.

The thing is, I go to a lot of schools. I have been to schools that are deemed “good” by powerful parents and have found myself distressed at the socio-emotional climate or underwhelmed by the substance of the academics. I have been to schools that are deemed “bad” and have conversely been bowled over by the widespread care and wowed by thoughtful teaching and learning.

Our discourse around “goodness” in education cuts off the essential qualifier –– for whom? In doing so, it reinforces goodness as inextricable and erases important questions about whose learning is being supported.

Auditing Your Classrooms for Competence and Status

This past weekend, I had the great pleasure of giving a keynote address at the Mathematics Council of Alberta Teachers (MCATA) Conference.

First things first: @minaclark did sketch notes of my talk!  I am delighted because I have always wanted somebody to do that. She did a fantastic job too.

During the breakout session afterwards, I talked about how we can audit our classrooms to support better interactions. In particular, we need to pay attention to issues of mathematical competence and student status. (I have written a lot on these topics since they are critical to fostering positive relationships between students and the subject. You can read earlier posts here, here, and here.)

Here are my audit questions.

Competence audit:

  • What kinds of competencies are valued in your classroom? Where do students have a chance to show them?
  • Consider the last few activities you have done in your class. Did they provide multiple entry points toward a rich mathematical idea? If not, can you use the table below to adapt them to become a low ceiling/high floor question?
  • When you look at your class roster, can you identify at least one way that every student is mathematically smart?
  • When you think of students who struggle, do they have competencies that you might better support by redesigning some of your class activities?
  • When you think of students who have a history of high achievement, do they value other ways to be smart aside from quick and accurate calculation? Do they value other competencies in themselves? In others?

table

Some low floor-high ceiling question types. (Adapted from Will Stafford’s “Create Debate” Handout)

Status audit:

  • When you think of the students you worry about, how much of their challenge stems from lack of confidence?
  • How much do students recognize the value and contributions of their peers?
  • What small changes could you make to address status problems and support more students in experiencing a sense of competence?

Please feel free to add others or offer your thoughts in the comment section.

Renegotiating Classroom Treaties

Many classrooms are governed by tacitly negotiated treaties. That is, students trade in their compliance and cooperation –– student behaviors that alleviate the challenges of crowded classrooms ––  for minimal demands for engagement by the teacher. When I have worked with teachers trying out open-ended tasks for the first time, I will often hear about “pushback” or “resistance” from the students: “I tried using this activity but the kids balked. They complained the whole time and refused to engage.”

These student responses indicate that teachers are violating their part of the treaty by going beyond minimal demands for engagement and increasing intellectual press. Put differently, by using an open-ended task, teachers raise the social risk, leaving students open to judgment since they can not rely on the usual rituals of math class to hide their uncertainty. Treaties may, as their name suggests, keep the peace, but they reflect norms of minimal engagement that interfere with deeper learning.

In my own observations, I see teachers struggle to move students past their initial reluctance to participate and make it clear that active involvement is required in their classrooms. Renegotiating classroom treaties requires a clear vision for what student participation can look like, structures to support that vision, along with the determination to see it through. The teachers I interviewed for my forthcoming book all emphasize how critical the first days are for setting these expectations for their students, particularly since their expectations may differ from what students are used to in math class. “It’s entirely intentional that I begin setting norms and structures on the first day of school,” Fawn explains. By launching the new school year by showing students what it means to do math in her class, Fawn renegotiates the classroom treaty through norms and structures, introducing the Visual Pattern and other discourse routines from the start. She says, “I need to provide students with ample opportunities to experience the culture that we have set up. We need to establish and maintain a culture that’s safe for sharing and discussing mathematics, safe for making mistakes, and a culture that honors each person’s right to contribute. There needs to be a firm belief among everyone that mathematics is a vital social endeavor. Building this culture takes time.”
Starting the school year with clear expectations is important, but guiding individual students’ participation is an ongoing project. The teachers I interviewed have numerous strategies for monitoring and building positive participation throughout the year. Students students who hide or students who dominate make for uneven participation. The teachers describe how they contend with these inevitable situations.
When figuring out how to respond to quiet students, the teachers try to understand the nature of students’ limited participation. Not all quiet students are quiet for the same reasons. At times, quietness is rooted in temperament: some students inclined to hang back until they feel confident about what is going on, but they are tracking everything in class. These students do not contribute frequently, but, when they do, their contributions add a lot to conversations. This kind of quiet is less of a concern and can even be acknowledged: “Raymond, you don’t talk a lot, but when you do, I always love hearing what you have to say.”
Other times, quietness signals students’ lack confidence. That is, students indicate some understanding in their work or small group conversations, but they do not have the confidence to participate in public conversations. With these students, the teachers seek out individual conversations. Chris calls these doorway talks, while Peg calls them sidebars. (“Trying to deal with calculators and rulers at the end of class, I couldn’t make it to the doorway!” Peg tells me when I note the different names.) “I might say to a kid, ‘You know, you had really good ideas today, and I would have loved to have heard more of them in the conversation we had a the end. I think you have a lot more to contribute than you give yourself credit for.’” Sometimes, there are ways of encouraging good ideas to become public that do not directly address the student. Chris explains that he might say something like, “I haven’t heard from this corner of the room.” He then asks other students to hold their ideas while waiting for a contribution from the quiet group.

Of course, some students are quiet because they really do not know what is going on. This could be due to a language issue, in which case, the teacher needs to modify instruction to give them more access to the ideas. If there are other learning issues going on, this might suggest the need to check in with colleagues about the students performance in previous years or in other subjects.

eager-students
Talkative students pose another kind of challenge to the expectation that everyone participates.  On the one hand, they can provide wonderful models of sharing their thinking. They can be the “brave volunteers” who explore their thinking publicly, and teachers can lean on them to get conversations started. On the other hand, they can be domineering, making it difficult for other students to get a word in. The quiet students who temperamentally need to think before they speak have their counterparts in some talkative students: these are the students who think by talking. Asking for their silence sometimes gets heard as asking them not to think. When I have had students like that in my own classes, I make sure to assure them that I value their engagement but that I need them to find other strategies for processing so that other students can be heard. Sometimes, students with impaired executive functioning, like those with ADD, have a hard time with the turn-taking aspect of classroom dialogue, so not only do they talk a lot sometimes, they struggle to take turns. Again, teachers can respond by valuing students’ ideas while helping them participate more effectively: “I know you get excited, but we need to take turns so that we can hear each other.” Finally, domineering behavior can get expressed through a lot of talking: students who are highly confident in their understanding and want to explain to others. Teachers need to judge the extent to which this is altruistic, a sense of trying to share knowledge, and the extent to which it shuts conversations down. In the first case, students can be coached towards asking questions of their classmates, channeling their impulse to talk into something constructive. In the second case, the dominance can be corrosive to the classroom culture and the students might need stronger redirection. For all of this feedback, similar strategies of direct address (via sidebars or doorway talks) and indirect address (“Let’s hear from somebody else”) can help teachers manage participation.

Structure Can Change Agency

One great privilege of the work I do are the many opportunities I get to share the things I care about with different groups of people. If you do it enough, you get a chance to clarify your own ideas, learn from others, and notice connections.

This past weekend, I had the honor to give a keynote talk at the Carnegie Math Pathways Forum. If you don’t know about their work, it is worth checking out. Briefly, their work addresses the enormous blockage in the math pipeline as students transition from secondary to post-secondary. A staggering number of students get placed in developmental math classes, and often, these courses become a holding bin students cannot get out of. The Carnegie folks have worked primarily with community college instructors to re-think developmental math curricularly and pedagogically. It’s fascinating and important work.

My talk was about the relationship between structure and agency, how both contribute to inequalities in mathematics education. When we are teaching in a classroom, it is easy to see problems of inequality as they look locally: high enrollments in developmental math, over-representation of students coming from poverty and students of color, a sense of student apathy. To make progress, however, instructors can learn by linking the local to broader social processes: the maldistribution of qualified math teachers, STEM classrooms that are hostile environments to minoritized students, a K-12 curriculum that often reflects the institution of schooling more than what it means to do meaningful mathematics. I argued that if we frame these problems through what we see locally, we give ourselves, as teachers, less leverage to make progress on them. I shared two key concepts for linking these social processes to what we see in our classrooms: social risk and status. I have written about both of these (click the links if you are curious), but briefly, social risk refers to the threats people feel are posed to their status in a community while status describes the perception of students’ academic capability and social desirability. Both of these ideas link the social process explanations for inequality to what teachers see in their classrooms locally.

Teachers can then work to design classrooms that reduce social risk by, in part, attending to status dynamics. In other words, to connect structure and agency, we need ways to think across scale and look at the social origins of problems too often narrated as individual issues. Instead of, for example, blaming students for being apathetic about mathematics learning, we need to recognize what their history has likely been in our current system and accept their apparent apathy as a reasonable response. Our task shifts from finger pointing (“My students just aren’t motivated!“) to having the productive challenge of honoring their experience while trying to change their ideas about math and learning.

In the end, then, structure can help us change agency in two ways. First, by recognizing that it is there, along with the social processes it holds in place, we can arrive at more productive framings of the problems we face locally. Second, we can leverage the structural designs in our classroom to invite students’ agency.

I have written about designing structures to promote agency before. If you don’t feel like reading that (I realize it’s summer!), maybe watch this video instead. It is quite a joy.

And don’t we all need more of that right now?

 

Why Meaningful Math Learning Matters

What Meaningfulness Means

Learning and schooling are not the same thing. There are children who are great learners but terrible students. These young people are full of ideas and questions, but they have not managed to connect their innate curiosity with their experiences in school. There are many possible reasons for this. Children may find school to be a hard place to inhabit, due to invisible expectations that leave them feeling alienated. Sometimes, school curriculum just seems irrelevant: their personal questions about the world do not find inroads in the work they are asked to do.
Although many parenting books extol children’s natural curiosity and emphasize its importance in their learning and development, schooling too often emphasizes compliance over curiosity. Thus, it is not surprising that children who are great learners and weak students have their antithesis: children who are great students but who are less invested in learning and sense making. Make no mistake: these students hit every mark of good organization, compliance, diligence, and timely work production, but they do not seek deep engagement with ideas. Given the freedom to develop a question or explore an idea, they balk and ask for more explicit directions. I have heard teachers refer to these children as “teacher-dependent.”
Too often, meaningfulness falls through this gap between learning and schooling. There is a fundamental contradiction at play: meaningfulness arises from and connects to children’s curiosity, yet “curious children” is not entirely synonymous with “successful students.” Meaningfulness comes about when students develop an appreciation for mathematical ideas. Rich and meaningful learning happens when students draw on prior knowledge and experiences to make sense of ideas and explore problems, invoke their own strategies, get to ask “what if…?”  In short, meaningful learning happens when students’ activity connects to their own curiosity. To make meaningfulness central to math teaching, then, teachers need to narrow the gap between being curious and being a good student.

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Meaningfulness: When students connect their own curiosity and experience to ideas, thereby developing an interest in and appreciation for mathematical content.
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Why Meaningfulness Matters

Every math teacher, at one one time or another, has been asked the question, “When are we going to use this?” While this question often gets cast as students’ resistance to learning, it can be productively reinterpreted as a plea for meaningfulness. When the hidden curriculum of math class –– the messages that are inadvertently relayed through classroom organization and activity –– consistently communicates that meaning does not matter, we end up with hordes of students who no longer reason when they are doing math. They instead focus on rituals, such as following the worked example, and cues, such as applying the last learned procedure to the current problem.
As researcher Sheila Tobias explained in her classic exploration of math anxiety, a lack of meaning exacerbates many students’ negative experiences learning mathematics. When math class emphasizes rituals and cues that rely on memorization over sense making, students’ own interpretations become worthless.

For instance, they memorize multiplication facts, and, in a search for meaning, they decide that multiplication makes things bigger. Then, they learn how to multiply numbers between 0 and 1. Their prior understanding of multiplication no longer works, so they might settle on the idea that mulitiplication intensifies numbers since it makes these fractional quantities even smaller. Finally, when they learn how to multiply negative numbers, all their ideas about multiplication become meaningless, leaving them completely at sea in their sense making. The inability to make meaning out of procedures leaves students grasping and anxious, as the procedures seem ever more arbitrary.
In contrast, when classrooms are geared toward supporting mathematical sense making, they reap multiple motivational benefits. First, students’ sense of ownership over their learning increases. Students see that multiplication can be thought of as repeated addition, the dimensions of a rectangle as related to its area, or the inverse of division. When they learn new types of multiplication, the procedures have a conceptual basis to expand on. Relatedly, their learning is more durable. Because they understand the meaning behind the mathematics they are learning, they are more likely to connect it to their own experiences. This, in turn, provides openings for their curiosity and questions. Beyond giving students opportunities for sense making, meaningful mathematics classrooms provide students chances to identify and explore their own problems. Indeed, in a systematic comparison of teacher-guided and student-driven problem solving, educational researchers Tesha Sengupta-Irving and Noel Enyedy found that the ownership, relevance, and opportunities to engage curiosity in student-driven problem solving supported stronger outcomes in student affect and engagement.[1]

The challenge, then, for teachers is how to help students engage in meaningful mathematical learning within the structures of schooling. I would love to hear your ideas about how to achieve this.


[1] Tesha Sengupta-Irving & Noel Enyedy (2014): Why Engaging in Mathematical Practices May Explain Stronger Outcomes in Affect and Engagement: Comparing Student-Driven With Highly Guided Inquiry, Journal of the Learning Sciences, DOI: 10.1080/10508406.2014.928214