A Love Letter to Great Math Teachers

My new book just came out. It’s called Motivated: Designing Math Classrooms Where Students Want to Join In. The book addresses the question: what can teachers do to design motivating environments for their students’ mathematical learning?

This is not the usual way that motivation is discussed. Typically, motivation is seen as characteristic of students themselves, with some kids being “more motivated” and others being less so. Drawing on social psychology, I turn this logic around by looking at motivation as a design issue. In other words, instead of seeing kids as more or less motivated, I offer a framework for teachers to design classrooms that are more motivating.

I knew I wanted to share the practice of math teachers I admire. I have had the great fortune to work with and learn from a number of amazing teachers over the years, usually in the context of research. Because of confidentiality agreements that are a part of those projects, I have had to mask teachers’ identities when I write about them, which sometimes makes me sad, because I want to give them due credit for their thoughtful work.

For this book, I crafted a process where I could ask amazing math teachers to share their practice as a way of illustrating the motivational framework –– and publicly give them credit. It involved lots of vetting, checking with them to make sure I told their stories accurately. This represented a tremendous commitment on their part, for which I am extremely grateful. The upside, for readers, is that you can find people whose thinking “clicks” with you, follow them on social media, find them at conferences, and continue learning.

The book is, in the end, a love letter to great math teachers. I have always been moved by great teaching, the way that some people might be moved by great art. When I am in a classroom watching an engaging lesson unfold, it is a profound experience for me. To me, lessons are engaging when students’ humanity is not put underground but is a part of instruction. Students can be a delight, and when learning is connecting with them, it is a joy to watch.

I wanted to challenge myself to illustrate a variety of ways talented teachers design motivating classrooms. In the book, I feature six teachers who work in very different secondary math settings. They are: Peg Cagle, Rafranz Davis, Sadie Estrella, Chris Luzniak, Fawn Nguyen, and Elizabeth Statmore. Additional examples of motivating design draw on the work of Anna Blinstein, Tina Cardone, Andrew Gael, Heather Kohn, Justin Lanier, Dan Meyer, Paul Salomon, Megan Schmidt, Anne Schwartz, Sara van der Werf, and Anna Weltman.

I encourage you to follow all of them, check out their blogs, and continue the conversation! Keep the love of good math teaching alive. Our hashtag is #MotivatedMath!

Update: Canadians, you can order it here.

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.

Choice Systems and False Agency

I have been reeling for the past few months at the startling erosion of some of the taken-as-shared narratives about public education as the backbone of democracy. I realize that these often came as platitudes and, worse, double-speak, but as long as there was some semblance of a shared commitment for high quality public education, I felt I had something to work with.

The current U.S.Secretary of Education has taken astonishingly hateful positions on protecting students’ civil rights in her valorization of “choice” and “states rights.” Our shameful history of Jim Crow has established latter as a well-known cover for government-sponsored racism. But I want to poke a few holes in “choice” as well.

The following text comes from a paper I published in 2004 in a study of a high school that allowed students “choice” about whether to take a traditional or reform math curriculum. I have edited it for this post.

The traditional US high school curriculum has famously been compared to the stores of a shopping mall, with a broad array of educational choices that provide something for everyone. In making choices, students are asserting a sense of themselves, the kind of socially-rooted self-understandings and social positions that constitute identities. Indeed, by the time students are in high school, because these understandings and positions may have been reinforced by an array of social and interpersonal forces, personal choice may be less of a choice than it seems: by labeling it as such, choice systems effectively erase the social categories that have been associated with different kinds of school curricula.

In the case of mathematics, different courses are often associated with different types of students. Teachers’ talk about courses reveals their notions of students and mathematical ability, and these ideas get built into the organization of the curriculum. For instance, when teachers are mandated to eliminate remedial courses but feel the need to accommodate “slow” or “lazy” students, they may effectively workaround the mandate and create separate tracks for these students. Students, likewise, may occupy the curricular space of a school in ways that are similar to the ways they occupy physical spaces: they gravitate to places of comfort, where their social identities find company among cohorts of similar peers.

It is not surprising then that one of the most robust findings in studies of relationships between curricular organization and student achievement is that a rigorous common curriculum ––– which minimizes such choices on the part of students –– distributes achievement more equitably (Lee, Bryk, & Smith, 1994). A narrow academic curriculum coupled with a strong organizational push for students to enroll in challenging courses leads to more equitable learning in mathematics (Lee, Smith & Croninger, 1997), with students from groups historically disenfranchised from schooling being especially advantaged by such structures (Lee, Bryk, & Smith, 1993). This research signals the kinds of curricular organization that correlates to higher achievement, giving us a broad look at what seems to matter.

When students make “choices” about courses, this narrative represent a false sense of agency and autonomy. More accurately, choices reflect their assertions (or assignments by parents, counselors, or others who advise in these decisions) of identity: an “honors” student, a “pre-algebra” student. Choice, as a force that propels students through the curriculum,  is problematic: the supposed self-determination of course-selection is actually a mechanism for perpetuation of the status quo. By narrowing the array of courses, the social meaning of courses must correspondingly broaden. Rigorous academic courses are no longer the province of some students to the exclusion of others; they must expand to become habitable places for all students.

The same can be said for taking the logic of choice to a district level, only worse. Not only will “choice” provide a false narrative of agency and autonomy, these systems will thrive on families with insider information. I see it in my own children’s education, where parents wonder at each transition from elementary to middle school to high school, how to navigate the system. Choice systems privilege parents with resources, such as access to  insider knowledge and flexibility to drive children around town.

Good public schools matter. Providing all children access to a high quality education matters. When I hear choice, I hear opportunity hoarding and re-segregation.

I cannot stand by silently while we gut our best tools for democracy.

 

Public Education Matters for Democracy

I have struggled to find my place in my online communities as the political ground has so dramatically shifted in the past few months. The US Presidential election fostered a climate that counters so much of what I stand for as an educator and a citizen. My twitter feed has been taken over by politics as I watch so many institutions struggle to uphold our democracy, institutions designed to safeguard cherished ideals like free speech, the right to assemble, and the pursuit of happiness.

I am grateful to have spent the week between Martin Luther King, Jr. Day and the Inauguration with a group of like-minded scholars at a Spencer-sponsored conference at UCLA. We share a commitment to preparing asset- and equity-oriented educators, so it was a great moment to figure out what that might mean in the years ahead.

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Left to right: Manka Varghese, Matt Diemer, me, Lauren Anderson, Mariana Souto-Manning, Dorinda Carter Andrews, Thomas Philip, Jamy Stillman

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Clockwise from left: Gloria Ladson-Billings, Mariana Souto-Manning, Lauren Anderson, Elizabeth Self, Thomas Philip, Matt Diemer, me, Alfredo Artiles, Marilyn Cochran-Smith, Sharon Feiman-Nemser, Jamy Stillman, Josephine Pham, Dorinda Carter Andrews

It was a productive week for clarifying my values and commitments. We even wrote an editorial together arguing against Betsy DeVos’s appointment as Secretary of Education.

I think I will need to return to these commitments a lot over the coming years. For this reason, I am going to go back to basics and sharing those commitments with you.

To be sure, I have no illusions that the prior administration upheld my educational values. Market-based reforms have been a centerpiece of educational policy for the past several administrations. President Bush’s landmark legislation No Child Left Behind certainly advanced this agenda, but President Obama’s Race to the Top put it on steroids. By tying teaching and learning to narrow metrics, discourses of desirable educational outcomes became less about children’s growth, their humanity, and their potential as future citizens. On the whole, national goals for children’s learning slid to the bottom of Bloom’s Taxonomy.

I believe in education as a public good. We live in a time and place where the ethos of individualism prevails. In this logic, if my kids are okay, everything is fine. In contrast, if we see education as a public good, our concerns must extend beyond our own children into our communities, states, and country. Whether we realize it or not, we have a vested interest in the solid education of all our citizens. To take an example that begins with  individual needs, I want the nurse administering my chemo to know the difference between .5 L and .05 L when pulling the dose. Beyond that basic skill, I also want him to be an empathetic person who can talk my family and I through our fear. There are countless situations where our personal interests depend on others’ competence and humanity.

Market-based reforms emphasize competition between institutions. This corrodes the ideal of schools as places that should be serving children and communities, contributing to their development and well-being. In a market-based framework, there are winners and losers, successes and failures. Despite meritocratic ideas, these winners and losers are not determined by raw talent but rather the status and resources of children and their families. I recognize that education has always been an unevenly distributed resource, especially in the U.S. I felt I could do my work as an advocate, because there were enough shared commitments to democratic ideals of opportunity. I am not so sure at this moment.

I believe that meaningful learning engages the whole person. It is not just knowing but also becoming. In my work, I study what it means to teach in ways that allow children become mathematicians. I also study what it means for teachers to become humanistic educators who engage with children’s experiences, build learning communities, and respond thoughtfully to children’s ideas.

Some children, however, are given more opportunities to become themselves than others. This starts with issues of language and culture, with some children’s home language and culture fitting into the social patterns of school, providing an important resource for their success there. Aside from such cultural capital, parents actual capital allows them to navigate the system in radically different ways. I have noticed a pattern in middle class parents’ rationale for sending their children to private schools. Most of the time, they are working to preserve their child’s competence. A child’s anxiety increases untenably in a test-prep focused school. A child’s difficulties with memorization lead to failing grades in a narrow curriculum. A child’s artistic strengths are not given adequate play in the school day. A child is inadequately challenged by a constantly changing cast of temp-work teachers.

I believe that strong community schools can anchor families and bring neighborhoods together around common concerns. Community schools, at their best, provide gathering places. They allow neighborhoods to feel like neighborhoods, with children getting to know the people around them. When there is a shared concern, community schools provide a space for people to come together around that concern. Dissolving community schools feels like another part of the effort to weaken the collective impact of people with shared interests.

In short, I believe that public education is central to meeting the ideals of our democracy. I know that a lot of work needs to be done to have it meet these ideals, but if we gut it completely, we will only be further behind on meeting the potential and promise of America.

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.

Supporting Instructional Growth in Mathematics (Project SIGMa)

Good news to share: another research grant has been funded by the National Science Foundation. Yay!

For this project, my research team and I will be working with Math for America in Los Angeles to design a video-based coaching method for their Master Teacher Fellow program.

sigma logo

This is what we pitched to the NSF:

This study addresses the need to develop processes for adequate and timely feedback to inform mathematics teachers’ instructional improvement goals. In this study, we propose using design-based implementation research to develop and investigate a process for documenting mathematics teachers’ instruction in a way that is close to classroom practice and contributes to teachers’ ongoing pedagogical sense making. The practical contribution will be a framework for formative feedback for mathematics teachers’ learning in and from practice. The intellectual contribution will be a theory of mathematics teachers’ learning, as they move from typical to more ambitious forms of teaching in the context of urban secondary schools. Both the practical and theoretical products can inform the design of professional development and boost other instructional improvement efforts.

In a recent Spencer study, my team and I investigated how teachers used standardized test data to inform their instruction. (That team was Mollie Appelgate, Jason Brasel, Brette Garner, Britnie Kane, and Jonee Wilson.)

Part of the theory of accountability policies like No Child Left Behind is that students fail to learn because teachers do not always know what they know. By providing teachers with better information, teachers can adjust instruction and reach more students. There are a few ways we saw that theory break down. First, the standardized test data did not always come back to teachers in a timely fashion. It doesn’t really help teachers adjust  instruction when the information arrives in September about students they taught last May. Second, the standardized test data took a lot of translation to apply to what teachers did in their classroom. Most of the time, teachers used data to identify frequently challenging topics and simply re-taught them. So students got basically the same instruction again, instead of instruction that had been modified to address central misunderstandings. We called this “more of the same,” which is not synonymous with better instruction. Finally, there were a lot of issues of alignment. Part of how schools and districts addressed the first problem on this list was by giving interim assessments –– basically mini versions of year end tests. Often, the instruments were designed in-house and thus not psychometrically validated, so they may have not always measured what they purported to measure. Other times, districts bought off-the-shelf interim assessments whose items had been developed in the traditional (and more expensive) manner. However, these tests seldom aligned to the curriculum. You can read the synopsis here.

Accountability theory’s central idea  ––  giving teachers feedback –– seemed important. We saw where that version broke down, so we wanted to figure out a way to give feedback that was closer to what happens in the classroom and doesn’t require so much translation to improve instruction. Data-informed action is a good idea, we just wanted to think about better kinds of data. We plan to use a dual video coaching system — yet to be developed — to help teachers make sharper interpretations of what is happening in their classrooms.

Why did we partner MfA LA? When I reviewed the literature on teachers’ professional learning, they seemed to be hitting all the marks of what we know to be effective professional development. They focus on content knowledge; organize their work around materials that can be used in the classroom; focus on specific instructional practices; they have a coherent and multifaceted professional development program; and they garner the support of teacher communities. Despite hitting all of these marks, the program knows it can do more to support teachers.

This is where I, as a researcher, get to make conjectures. I looked at the professional development literature and compared it to what we know about teacher learning. MfA may hit all the marks in the PD literature, but when we look at what we know about learning, we can start to see some gaps.

*Conjecture 1 Professional learning activities need to address teachers’ existing concepts about and practices for teaching.

 

Conjecture 2 Professional learning activities need to align with teachers’ personal goals for their learning.

 

Conjecture 3 Professional learning activities need to draw on knowledge of accomplished teaching.

 

*Conjecture 4 Professional learning activities need to respond to issues that come up in teachers’ ongoing instruction

 

*Conjecture 5 Professional learning activities need to provide adequate and timely feedback on teachers’ attempts to improve their instructional practice to support their ongoing efforts.

 

Conjecture 6 Professional learning activities should provide teachers with a community of like-minded colleagues to learn with and garner support from as they work through the challenges inevitable in transformative learning.

 

*Conjecture 7 Professional learning activities should provide teachers with rich images of their own classroom teaching.

 

The conjectures with * are the ones we will use to design our two camera coaching method.

We need to work out the details (that’s the research!) but  teacher’s instruction will be recorded with two cameras, one to capture their perspective on significant teaching moments and a second to capture an entire class session. The first self-archiving, point-of-view camera will be mounted on the teacher’s head. When the teacher decides that a moment of classroom discourse illustrates work toward her learning goal, she will press a button on a remote worn around her wrist that will archive video of that interaction, starting 30 seconds prior to her noticing the event. (As weird as it sounds, it has been used successfully by Elizabeth Dyer and Miriam Sherin!)  The act of archiving encodes the moment as significant and worthy of reflection. For example, if a teacher’s learning goal is to incorporate the CCSSM practice of justification into her classroom discourse, she will archive moments that she thinks illustrate her efforts to get students to justify their reasoning. Simultaneously, a second tablet-based camera would record the entire class session using Swivl®. Swivl® is a capture app installed in the tablet. It works with a robot tripod and tracks the teacher as she moves around the room, allowing for a teacher-centered recording of the whole class session. Extending the prior example, the tablet-based recording will allow project team members to review the class session to identify moments where the teacher might support students’ justifying their reasoning but did not do so. The second recording also captures the overall lesson, capturing some of the lesson tone and classroom dynamics that are a critical context for the archived interactions. Through a discussion and comparison of what the teachers capture and what the project team notices, teachers will receive feedback on their work toward their learning goals. We will design this coaching system to address the starred conjectures in the table

Anyway, I am super excited about this project. I am working with amazing graduate students: Grace Chen, Brette Garner, and Samantha Marshall. Plus, my partners at MfA LA: Darryl Yong and Pam Mason.

I will keep you posted!

 

 

 

Playful Mathematics Learning

I have had the great pleasure of spending the last several days at the Minnesota State Fair.

math on a stick

My colleague Melissa Gresalfi and I got a National Science Foundation grant to study a very special exhibit there called Math On-A-Stick. We have an awesome team of graduate students helping us with the research. They are Lara Heiberger, Panchompoo Fai Wisittanawat, Kate Chapman, and Amanda Bell.

PML Logo

Math On-A-Stick is the brainchild of Christopher Danielson,  educator, promoter of talking math with your kids, and mathematical toy maker.

christopher danielson

That is Christopher on the right. The woman in the pink jacket is a former math teacher. She made the beautiful quilt for Math On-A-Stick.

The exhibit is just a delight. Not only is it a lovely respite in a shady, relatively quiet corner of the fairgrounds, it is filled with math play. Here are a few of the stations in the exhibit.

On the left is a tile station. The tiles are half black, half colored, and children can make all kinds of patterns with them. The center image comes from the pentagon station. I could spend all day there myself. I made that creation. On the right are tessalating lizards and turtles.

Everyday there are visiting mathematicians and mathematical artists. The first day I was there, Megan Schmidt brought some of her spiral magic. Yesterday were hexaflexagons.

Today, Christopher was the Visiting Mathematician. He built a giant pattern machine that children could play with. It is made up of little “pattern machines,” and the buttons pop up and down, making a satisfying clicking noise.

Melissa and I are interested in studying two things about children’s encounters with the exhibit. First, we are interested in the design, investigating how the various activities support mathematical interactions between children, the exhibit, the mathematics educators, parents –– and each other. Second, we are interested in children’s engagement. We want to examine how the children engage with different parts of the exhibit, looking for relationships among children’s ideas about mathematics, reported experiences in math class, and the exhibit design.

Our primary data come from recordings the kids make while they are playing. We outfit them with GoPro cameras so we can see how they interact with the exhibit, recording their interactions, their general gaze, and the time they spend at the various stations.

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Melissa and Fai set up a stationary camera, while Lara pretends to be a kid at play.


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A couple of kids getting outfitted with GoPros

This is supplemented by entry and exit surveys, brief interviews, and stationary recordings of the stations (e.g. a camera positioned at the Pentagons so we can see how a cross section of children play with that station and compare that to the activity of our focal children).

IMG_9925

Data collection station. It’s a well oiled machine.

We weren’t sure how kids would feel about us approaching them and asking them to wear the cameras on their heads. It turns out that they love it. They are really happy to share, as are the amazing volunteers, who have been very agreeable to getting captured in the video footage as the children play.

Our research findings will help us identify more ways to make mathematics play a part of instruction. Already, many children are telling us that Math On-A-Stick math is different than what they do in school –– even kids who are inventing and problem solving in impressive and novel ways. We are looking forward to analyzing the hundreds of hours of data we collect and sharing what we learn with all of you.

 

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.

What Does It Mean to Study Teachers’ Learning from a Sociocultural Perspective?

I try to be a plain-talking academic when I engage in the public realm of social media. Sometimes, despite my best efforts, I find myself wanting to use academic jargon. My goal in writing this blog is to have conversations with both educators and researchers, so I think it is okay to have “turns” of conversation that lean a little more on my research voice than my educator voice.

Sociocultural is jargon word that I have wanted to invoke from time to time when talking to my practitioner friends. In particular, the research I do uses sociocultural learning theories as a way of describing both how students and teachers learn.

But what does that mean? In order to understand, you need a little history on how we have come to think about learning the way we do.
In the late 19th and early 20th centuries, U.S. research on learning was dominated by behaviorism. Seeking a rigorous empirical basis for a study of behavior, researchers like E.L. Thorndike and B.F. Skinner sought to explain how learning happened by documenting what they could see empirically.

Out of this theory, we have ideas like operant conditioning, where actions are shaped by stimulus and responses in the environment to ultimately change behavior. Skinner famously made little operant conditioning chambers called “Skinner boxes” that successfully “taught” pigeons to dance. Through the boxes, food was dispensed in response to the pigeon’s movements. If he turned his head to the left –– the stimulus –– he would get a food pellet –– the response. The next time, he had to turn his head a little further to get his food. Eventually, through operant conditioning, the pigeon learned to turn in a full circle –– to “dance” –– to get food.

dancing pigeons

Behaviorism explained some forms of learning, but it couldn’t explain everything. In the 1950s, the cognitive revolution began. Researchers like Jerome Bruner began to critique behaviorism, noting that a sole focus on behavior precluded a study of how people created meaning, a central question in understanding why people do what they do. Researchers realized they could do empirical studies that included a theory of the mind. Using methods like case studies and talk aloud protocols, investigators could examine how people made sense of their activities in the world.

Cognitive science, as it came to be called, led to important insights like schema theory and conceptions. A schema is a general system for understanding how knowledge is represented and how it is used.

Researchers can look for evidence of different schemata (the plural of schema). Like the behaviorists, they observed what people did to understanding learning. However, they augmented this by asking people to explain their thinking through interviews and surveys.

To give an example of a schema, let’s take the word “dog.” When I say “dog” what do you imagine?

You probably think of four-legged animals that bark, are furry, have tails. But how do you know that these are all dogs?


How do you know that these are not?


This is the question that underlies the idea of schemata.

The examination of schemata started to point to the importance of culture. Schemata are closely related to prototypes. So, for example, when I say the word “furniture” what do you think of?

Linguists have found that when you say the word “furniture” to Americans, they think the best examples are chair and sofa.

When you say the word “möbel” to Germans, however, they think the best examples are bed and table. Our schemata and our prototypes –– the building blocks of concepts in the world –– are culturally specific.

By the early 1990s, this increasing recognition of the importance of language, culture, and context shifted our ideas about learning yet again. Language and culture were not just the setting for development and thinking –– some kind of external variable to be controlled for –– they were, in fact, fundamental components of these mental processes. This insight meant that, to explain some learning phenomena, researchers needed to do more than describe mental structures.

This required another broadening of research methods. Using linguistics, anthropology, and sociology, learning researchers wanted to account for how concepts stretched beyond individual minds and into the world. Deeply influenced by Soviet psychologist, Lev Vygotsky, researchers working in this sociocultural tradition examined learning as it happened in interactions in the world, requiring new units of analysis. That is, instead of studying individuals as they learned, researchers sought ways to study individuals in context.

My own research takes up these sociocultural insights to re-think how we study teacher learning. Let me paint a bit of a picture for you about the intellectual traditions that shape my work.

First, when I entered my doctoral program at UC Berkeley in the mid-1990s, debates between cognitive and sociocultural perspectives on learning were quite active in my courses and in research groups. Although most arguments centered on questions of student learning, there was a growing interest in what was often called “out-of-school learning.” Influenced by anthropological researchers like Jean Lave, a small group of scholars studied workplace learning, a particularly pressing topic in our modern information economy, where workers must constantly adapt to a rapidly changing world.

Meanwhile, in educational policy studies, there was a growing recognition that research on school organization, curriculum, and teacher professional development had overlooked a central question: How do teachers’ learn? Since almost all school improvement efforts want to improve instructional quality –– through curricular reform, changes in scheduling or assessment techniques –– they all depend on what happens inside of classrooms. And that, of course, depends on what happens with teachers.

For this reason, educational policy scholars like Judith Warren Little and Mike Knapp were recognizing that teachers’ learning is an underanalyzed component of any efforts at school change or instructional improvement. Yet it was not central to policy designs –– let alone to analyses of their effectiveness.

The moment was ripe for somebody to connect these ideas. My work starts with the policy-based observation that designs for instructional change must consider teacher learning. I then use methods and insights from sociocultural theories of learning to examine how teachers’ learning happens in the school as a workplace. As the sociocultural theorists suggest, what teachers know and learn is not solely a product of what is in their individual heads.

Concepts for teaching draw on culturally specific practices and language in the world. For instance, in the U.S., we often start grouping children by ability levels at a very young age. The concept of a “high ability 6 year old” makes sense for American teachers in a way that it would not to teachers in countries that do not track in the elementary years. There are consequences to that concept having social meaning, as educators make decisions about their schools and classrooms and parents advocate for certain experiences.

By using sociocultural perspectives to explain teachers’ learning, my research is culturally specific and theoretically specific. Although the details of what I find about U.S. teachers may not generalize to other countries, it is my hope that my descriptions of teachers’ learning can be more generalizable.