I notice that math teachers often draw a dichotomy between rich, open tasks and drill-oriented practice. I wonder if it would be helpful to try and articulate some of that middle ground, the rich tasks that also act as practice, practice that one can look back on and draw new connections, and any number of other places to bridge the gap and help teachers move more fluidly between open tasks and practice.

I still agree with my comment, but I’ve had trouble with what that articulation might look like. Here’s an attempt.

First, what is a rich task? I don’t think any one definition can capture the subtlety I find here, but a rich task has some (though rarely all) of the following qualities:

- Lends itself to multiple strategies
- Has a low floor for entry, whether through solving intermediate problems, making estimates, visualizing, or other places for students to recognize what they already know early in the problem
- Has a high ceiling, naturally leading to extensions or additional tasks
- Allows multiple representations, in particular visual representations
- Has an element of perplexity, provoking students’ curiosity
- Allows some experimentation or trial and error, and meaningful reflection on that work
- Lends itself to intuition
- Starts humble but leads to multiple useful mathematical ideas
- Values concepts and connections over procedures
- Gives students something to argue and collaborate about
- Involves ambiguity and requires making sense of mathematical ideas

Most of all, a rich task captures a slice of the richness of the discipline of mathematics. Rich tasks are hard for students; they involve new norms in math class, often require a positive disposition toward learning math, and can overwhelm students to the point where they aren’t learning. I think they should be used judiciously. But a large part of their value comes in exposing students to the beauty and complexity of mathematics.

Next, what is drill? I don’t like the word drill because of the connotations it brings in, but I do value practice. At a basic level, practice means retrieving ideas from long-term memory to strengthen connections, and often to make new connections as practice tasks increase in complexity.

I see these as two different purposes of math class, and purposes that aren’t necessarily in tension. While some folks might characterize one side as good and the other as bad, I think both rich tasks and practice have important places in math class, and useful opportunities for synergy.

A rich task can be used to introduce a topic by creating intellectual need for an idea, help students learn something new by taking what they already know and extending it a step further, or to give students an opportunity to apply what they know at the end of a unit. Those are very different purposes, and each purpose relies on choosing tasks thoughtfully, facilitating with clear goals, and supporting students to find success.

At the same time, a rich task can be practice. Ben Orlin’s Give Me and Open Middle are great examples. Practice can lead to a rich task, where students practice a skill they already know, then step back to look at patterns in their work and learn something new. Practice can incorporate elements of a rich task, and rich tasks can be interspersed with practice. Studying worked examples is a great bridge between rich tasks and practice that gets students thinking, while also focusing their thinking on specific ideas.

Rather than thinking of these ideas in opposition, I think of them on perpendicular axes. I start planning with a goal for a lesson, and based on that goal I think about what will help my students reach it. I want to offer richness, and I want to offer practice, and I want to find as many opportunities as I can to do both in ways that build off of each other.

]]>To illustrate an early lesson in white racial framing, imagine that a white mother and her child are in the grocery store. The child sees a black man and shouts out, “Mommy, that man’s skin is black!” Several people, including the black man, turn to look. How do you imagine the mother would respond? Most people would immediately put their finger to their mouth and say, “Shush!” When white people are asked what the mother might be feeling, most agree that she is likely to feel anxiety, tension, and embarrassment. Indeed, many of us have had similar experiences wherein the message was clear: we should not talk openly about race.

-Robin DiAngelo in “White Fragility” p. 37

“Race is just a social construction,” is a common refrain in some circles. But what does that actually mean?

Robin DiAngelo’s example in White Fragility illustrates one of the many ways that race is socially constructed. In her anecdote, a child learns that race is not to be talked about in public. The child might also learn that being black is something negative or to be embarrassed of — the mother acts the same as she might if the child pointed out someone was overweight or disfigured, rather than particularly good-looking or well-dressed. Lessons about race become part of the fabric of society because of these everyday interactions. Our language, choices, and responses shape our perspectives and the perspectives of those around us.

The phrase, “Well, race is just a social construction,” is interesting in its use of the passive voice. Race is socially constructed, but who constructed it? Well, all of us, every day. And if it has been made, it can be remade. Mathematics is the same, as are race, gender, and more in the context of the mathematics classroom. Mathematics is what it is because of people, and as Rochelle Gutierrez says, mathematics needs people as much as people need mathematics. The learning of mathematics has changed dramatically over time, more than most realize. It will continue to change. What are some questions one might ask to reconstruct mathematics in a way that better humanizes and values all students?

Who practices mathematics?

Where did mathematics come from?

We spend countless hours worrying about kids understanding fractions — to this day, I am still completely flummoxed by that — and close to no time folding in math history. Somehow ensuring kids can add fractions with denominators nobody cares about is more important than humanizing math education with the hundreds of artists — spanning every culture/civilization on the planet — that have contributed to its creation?

How was mathematics created?

Both Thales, the legendary founder of Greek mathematics, and Pythagoras, one of the earliest and greatest Greek mathematicians, were reported to have travelled widely in Egypt and Babylonia and learnt much of their mathematics from these areas. Some sources even credit Pythagoras with having travelled as far as India in search of knowledge, which may explain some of the close parallels between Indian and Pythagorean philosophy and geometry.

-George Ghevarughese Joseph, “Foundations of Eurocentrism in Mathematics”, see also Beatrice Lumpkin, “African and African-American Contributions to Mathematics”

Why is mathematics worth learning?

And for a lot of students it feels like “just pretend.” Just pretend this is real world. Even though students might feel like “this doesn’t look like anything that’s in my real world.” And that’s where we get that question. “When are we ever gonna use this?” Now the question of “When are we ever gonna use this?” has already been asked by that person, many times. In their head, they’ve said, “When am I gonna use this?” “When are we gonna use this?” comes up when they’re basically asking everyone else in the room to recognize and to comment on the fact that the emperor isn’t wearing any clothes.

-Rochelle Gutierrez, in “Stand Up For Students”

Is mathematics “truth”?

These are only a few of the questions one might ask. What am I missing?

**Some Things I Believe To Be True **

Acting and not acting are both actions; nothing is neutral.

-Imani Goffney

- Most humans dislike mathematics — and not only dislike mathematics, but believe that they are intrinsically unable to learn or practice mathematics — but I think we can do better.
- A narrow subset of mathematics as it is taught in schools is not the only cause, but it may be one.
- Humans could have constructed a largely different mathematics; the mathematics we have is in many ways an accident of history.
- Speaking as a high school teacher, much of what we teach is not essential for students to learn. While I believe that what I teach helps students learn to think mathematically, it is not the only means to that end.
- Asking hard questions about the nature of mathematics is a worthwhile exercise.

I’m not advocating for a new mathematics tomorrow. Instead, I want to push myself to find the small moments — small moments that, when added together, send important messages — to make small changes. Stopping to talk about a mathematician who doesn’t look like what a student might expect a mathematician to look like. Pausing to acknowledge the rich intellectual history of a topic. Unpacking the ways race and gender play out in math classrooms, and interrogating why things are the way they are. Searching out ambiguity and inconsistency to validate students’ experiences that mathematics is not, to them, the system of pure logic it has been made out to be. Seizing on moments of authentic discovery, and helping students to feel what it might be like to practice mathematics. Questioning why we learn what we learn, opening avenues for dissent, and helping students imagine what else mathematics might be in the future.

Whether I realize it or not, everything I do influences student beliefs about mathematics. I can choose to ignore these questions and entrench the status quo, or start to find ways to communicate new values and new perspectives.

**Coda: On Competence **

In discussing on Twitter some of the ideas that came together as this blog post, I was accused of being a bad teacher because asking questions like these would just confuse students and leave them feeling even more helpless in math class than they did before. I think it’s worth asking hard questions, but what are the trade-offs of complicating a subject so many students already dislike?

Mathematics is made by people. Who will take the opportunity to remake it? I want students to see the richness that mathematics is, and that it might be. But I also have a responsibility to help students be successful within the parameters of the system we have. I think that the most powerful thing I can do for a young person is to help them develop a sense of mathematical competence: to recognize the ways that they are mathematically smart, and to create space for those smartnesses to flourish in my classroom. And, inevitably, most of those smartnesses will reflect mathematics as it is, not mathematics as it might be. I’m not advocating for radical change. Instead, I’m advocating for great everyday teaching that helps students gain the skills they need and recognize the incredible talents they have. At the same time, there are innumerable opportunities to ask hard questions and engage students with the tensions inherent in mathematics education. Those opportunities, taken judiciously and purposefully, can only expand the pool of students who see themselves as potential mathematicians, and expand the discipline that students are learning.

]]>- Is too much procedural fluency bad for conceptual understanding?
- Is it possible to have lots of procedural fluency without any conceptual understanding? Is it possible to have lots of conceptual understanding without procedural fluency?
- Is conceptual understanding more about what students can
*do*or what they*know?* - Does conceptual understanding support student engagement? Does procedural fluency?
- Adding It Up from the National Academies Press defines five strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Are we missing useful complexity by narrowing our focus to conceptual understanding and procedural fluency?
- Here’s a graph. What does an ideal learning trajectory look like?

- Does that trajectory depend on the content?
- If a student can explain how they solved a problem, do they definitely have conceptual understanding? If a student can’t explain it, do they definitely not have conceptual understanding?

**My Hot Take **

Here are two tentative ideas that I think might contradict each other, but might also both be true.

- It’s easy to overcomplicate conceptual understanding, but really it’s just transfer. Can a student take what they learned in one context and apply it in another? And transfer is, or should be, the primary goal of education.
- Conceptual understanding is actually composed of lots of little pieces, and those pieces depend on the content, the teacher’s goals relative to that content, and the students’ prior knowledge, skills, and dispositions. It’s easy to overgeneralize, but building conceptual understanding is context-specific and there aren’t any one-size-fits-all ways to get there.

Further reading that’s on my mind:

- All of the links in Dan’s blog post
- Kate Nowak on why Illustrative Mathematics’ avoids cross-multiplication
- Dan Willingham on Inflexible Knowledge
- Are Cognitive Skills Context-Bound? by Perkins & Salomon

But despite the attempted removal of the weeds, this hope of a community never formed. “I just couldn’t build the community I am usually able to build,” Emily lamented. Her disappointment was palpable.

One reason for this is that exclusion does not build community–it destroys it. The problem with weeds is that when you pull up one, many more sprout with a vengeance. It isn’t the behavior of the children that threatens community; it is the response to that behavior, the use of exclusion, that threatens community.

When a child is excluded, it teaches the other children that belonging to the classroom community is conditional, not absolute, contingent upon their willingness and ability to be a certain kind of person. In this paradigm, belonging is a privilege to be earned by docility, not a basic human right that is ensured for every child.

-Carla Shalaby, *Troublemakers, *p. 162

*Troublemakers, *by Carla Shalaby, might be one of the most impactful books I have read on education. She makes a compelling argument for education as a place to “be love and practice freedom,” and looks at all students, and especially the “troublemakers,” with empathy and an authentic desire to understand, rather than to control and coerce. She follows four of these “troublemakers” to their first- and second-grade classrooms, and the portraits she paints are both tragic and moving.

First, thanks to Grace for first writing about the book, Becky for lots of thought-provoking discussion about it, and Val for leading the #ClearTheAir discussions exploring further. These reflections have me thinking about the role of teachers in educating students to be thoughtful citizens — interpreting “citizens” broadly, not necessarily as citizens of this country, but as young people who can and will inform the future of democratic government.

I see education for thoughtful citizenship differently today than when I started teaching; I want to be an educator who, in ways small and large, prepares students for a world where citizenship includes questioning authority, insisting on respect and dignity, and protesting effectively. I believe — and Shalaby articulates — that these skills are taught, or untaught, in schools. I recognize that many would call these *political *values, and teachers aren’t supposed to be political. But our present moment is a reminder that silence is a choice, and the status quo is a political value that we perpetuate by closing schools off from political perspectives.

In trying to understand what Shalaby’s values will look like for me, it was helpful to put into my own words the values I want for students:

**Dissent**. I want students to be able to question authority, ask “why?”, and “what if?”, imagine a better world, and cultiavte the tools to work toward it.**Compassion**. I want to look for the best in students, believe that they are growing and learning as humans, to honor their dignity, and to teach students to do the same for those around them.**Freedom**. I want students to recognize their agency in valuing what they want to value, doing what they want to do, and being who they want to be.

These are ideals. I’m not sure what they look like in practice, but I do know they can inform the small, everyday interactions that shape students’ experiences in school. I also know that the way that the institution of school is organized is antithetical to these values in many ways. Still, I want to shape those interactions with Shalaby’s perspective on what it means for students to be “indigenous” to classrooms:

]]>Duncan-Andrade reminds us that, in the words of one educator, the students are all “indigenous” to the classroom and therefore “there are no weeds in my classroom.” The young people are indigenous because they are the natural part of the school community. They are indigenous to the neighborhood to which the school belongs, and they are indigenous to the culture of childhood that dominates the classroom.

Given the realities of school segregation and the demographics of the teaching profession, young people have much more in common with one another–culturally, socioeconomically, linguistically, developmentally–than they do with thier teachers. The young people comprise the community. The teachers are the interlopers, the oustiders, the ones who come and go, the ones who don’t fundamentally belong. The children are a community garden long before the teacher arrives on the scene with her own outsider tools, so when she pulls a “weed” she disrupts the balance of community by creating the threat that any child, at any time, can be excluded at will. She leverages power and authority to show that she is the ultimate arbiter of community belonging.

pp. 162-163

Here’s a thought experiment:

I wonder what math class might look like if our most important goal was to help young people love solving problems.

Literacy teachers have lots of goals, but I would wager most would tell you that above all they want their students to love reading. English classrooms are often filled with books, teachers are knowledgeable about the interests of their students and suggest books appropriately, and teachers work to build a love of reading in every student.

Is math just different?

There are tons of problems out there. Free sites like Alcumus, Brilliant (especially their 100 day challenge), and Play With Your Math. Julie Wright has a great collection of puzzles and games.

But problems don’t seem to be our paradigm for a successful math class. If a student or group of students does well, we’re more likely to have them start learning the next year’s math than embrace the depth and complexity of non-curricular problems that student might enjoy exploring. Imagine if a student was a great reader and someone said, “hey, you’re going to do *To Kill a Mockingbird *in English class next year, why don’t you just get ahead and read it now,” rather than prompting the student to explore books that they’re interested in.

Sam Shah’s prompt for this conference was:

How does your class move the needle on what your kids think about the doing of math, or what counts as math, or what math feels like, or who can do math?

I want to move the needle on my students’ love of problems. This piece is more aspirational than anything — I don’t know that I do a particularly great job of fostering a love of problems in my class. But it’s something I care about, and something I am working to get better at. Here are some questions I have about helping students to love solving problems:

- I’ve observed that students are much more likely to enjoy solving problems when I find that “just right” task. How can I better do that for all students, while still valuing a social and collaborative classroom?
- The resources I referenced above are pretty abstract and logic-oriented, in the vein of many publications on problems and puzzles. How can I broaden my conception of “problem” to include problems about solving practical challenges that humans face and help math feel relevant to more students?
- A human can become pretty literate (after an initial period of learning to read) by just reading lots of books. Is something similar possible in learning math — could someone learn by just solving lots problems?
- To what extent do the goals of helping students to love solving problems, and helping students to learn required content, work in opposition or in parallel?

It’s simple. On any assessment that was previously individual, students are assigned to groups. Randomly at first, though it can be more strategic later. Assessments can be the same, though group assessments also create an opportunity to pose harder problems.

I imagine many folks are saying, “well I can’t do that.” Maybe. But maybe that’s just a norm that we’ve created for ourselves. Why not? What’s wrong with making every assessment a group assessment?

*But the grades won’t be valid! They won’t actually communicate what each individual student knows and doesn’t know.*

Why is that the point of grades? Why are we so obsessed with putting young people into silos and ranking and sorting them so that they can have access to different opportunities in the future? Our entire education system is premised on educating individuals, but humans learn best in groups, and practice mathematics with support of collaborators. Do we really have to have a system where we watch students struggle, silent and alone, to figure out what they have learned?

*But you have to give a letter grade at the end of the year!*

Our education system is deluding itself that grades actually say anything substantive about what a student knows or doesn’t know. Why not just end that pretense? Grades signal what we value. I value collaboration. Why shouldn’t students collaborate on assessments?

*But students will fall through the cracks, depending on each other and never taking responsibility for their own learning.*

I can still ask students to answer an exit ticket or similar formative assessment on their own — though not for a grade. And I can use that information to respond to what students know and don’t know. But I think that, when the stakes are high and an assignment is going into the gradebook, asking students to complete assessments alone is fundamentally dehumanizing. Think about the enormous percentage of adults who hate math and spend their lives terrified of it. What if we could change that?

*Ok so there’s this thing called standards-based grading. You describe all the different skills that you want students to learn, and you report the results of assessments based on those specific skills so students know what they need to work on and grades are actually meaningful.*

Eh. Not impressed. Sounds like a lot of work, and really just puts a new veneer on top of assessment without changing the student experience. I think our system is fundamentally broken, and standards-based grading seems like a change in style rather than substance.

In all seriousness, I don’t know that I will go all-in on group assessments, but I really am intrigued. Do we as teachers insist on individual assessments because they are best for student learning, or do we do it because it’s an institutional norm that is baked into some unfortunate ways we think about education? Could group assessments transform the evaluation of learning into a humanizing and affirming process, rather than a process that instills anxiety and fear? I’ve experimented with lots of assessment systems the last few years, but every time I’ve felt like I’m just tinkering around the edges, that despite grand ambitions my changes haven’t actually influenced how students experience assessment. Maybe the answer is to change something more fundamental.

]]>As I’ve become more involved in the professional math teacher world, I’ve seen a lot of conversation and action around equity and specifically around addressing anti-black racism in and beyond the math classroom, and I’ve noticed many white teachers under-prepared to participate.

As a white person, I spent much of my life shielded from racial awareness. Once I started to tune in, I felt intimidated by my own ignorance. I have found that educating myself through a lot of reading and listening has been (and always will be) a vital component of my equity work, so that my friends and colleagues of color don’t bear the burden of teaching me the basics and I can build up the analysis guiding my action.

Something that I love about the #mtbos is our shared passion for continuous learning and growth. Here’s a very non-exhaustive list of some of my favorite resources on anti-black racism and anti-racist action. They are each good starting points or good deepening points, and I find myself referencing each of them often.

**“The Case for Reparations”**** by Ta-Nehisi Coates
**In this article, Coates masterfully lays out our history of systemic property-based discrimination against black people, and how that dictates our present reality. Absolutely central to my current understanding of institutional racism and wealth disparities.

**@prisonculture**** Mariame Kaba’s Twitter feed
**By far my #1 most favorite Twitter follow, from whom I am constantly learning. I had picked out one of her blog posts to share but really I just want you to follow her. I learn a ton of activist history, theory, and strategy from Kaba, especially around prisons and policing. One idea she tweets about regularly: hope as a discipline.

**“Choosing a School for My Daughter in a Segregated City”**** by ****Nikole Hannah-Jones**

A reporter on segregation in education reports on her own decision and experience as the parent of a black child. Through this article and her consistent voice on Twitter, Hannah-Jones pushes me to question, “If I wouldn’t accept this for my child, why is it acceptable for other children?”

*Why Are All the Black Kids Sitting Together in the Cafeteria*** by Beverly Daniel Tatum
**Dr. Tatum patiently and clearly explains developmental stages of racial identity formation in ways that have helped me understand both my own past experiences and what is happening for my students.

**“Greening the Ghetto”**** Ted Talk by Majora Carter
**Carter explains environmental justice and some of the disparate impacts of environmental degradation and city planning alongside her personal story and vision for change. I have an extra soft spot for her because after my students in New Orleans wrote her letters about their related experiences she flew out to meet them (!!!).

*Meridian*** by Alice Walker
**I am constantly trying to talk about this novel with people who turn out not to have read it, so if you do read it please hit me up to discuss! Warning that it includes sexual and racial violence.

“**No-Man’s-Land****” by Eula Biss
**Biss, a white woman, has shaped the way I think about whiteness and the meaning of fear through this essay.

If you give one or many of these a try and have thoughts or questions or criticisms or epiphanies, I would love to discuss them here or on Twitter. These are just a few of my favorite teachers-from-afar and I LOVE giving personalized reading and listening recommendations, so feel free to reach out for that too. What are some of your favorite resources for self-education about race?

(While it’s the focus of this post, I want to note that of course race and racism in the United States are not just black and white, and I hope that we’ll also have a lot of learning, conversation and action around other racial identities and oppressions.)

]]>Peter Liljedahl, the researcher behind these ideas, has a short piece on it in Edutopia here, and a longer paper here. I had the chance to meet him and experience the framework at PCMI this summer. I had two big takeaways:

First, Thinking Classrooms are often oversimplified. On the surface, you put students in random groups, send them to vertical whiteboards, and give them problems. In reality, there are a ton of micro-moves much more subtle than having students do problems at whiteboards in random groups that ensure every student is learning and that the class can reach meaningful mathematical goals. Second, each element of the framework has a purpose — it’s not some magic system that causes learning by itself; it requires constant monitoring and feedback to make sure the intent of the Thinking Classroom comes through. As a teacher, my role is to look for specific things, and make adjustments as necessary.

I’m not totally sold on the Thinking Classroom framework. But something I realized this summer is that last year, experimenting with different elements piecemeal, I never really understood how they all fit together. Seeing these connections has helped me to understand why I often felt frustrated with the results, and has me reconsidering my approach for this year.

These don’t feel like groundbreaking ideas, but it’s been helpful for me to step back and think about how the different elements of the framework fit together. Liljedahl advocates for implementing the framework one “level” at a time, and each level focuses on different ideas. Here’s where my thinking is right now, summarizing the big ideas of a Thinking Classroom and what I would need to look for to make sure it’s working effectively.

**Renegotiating Norms **

The first three elements of the framework are all about setting norms for thinking. Students are used to certain norms in schools, in particular in math class. They come in, sit down, face front. They copy things down that are written on the board. They solve simple problems after being shown how to do so. Liljedahl’s hypothesis is that these norms actually prevent thinking, and in some ways assume that students can’t think, by doing so much mathematical thinking for them. The first three elements of the framework try to reset those norms in ways that encourage thinking, to create a baseline of engagement in the classroom higher than what students are used to.

1. Begin with problems. Students are used to coming to class and being shown how to do a problem before solving it. Beginning class with a problem gets students thinking from the beginning, and uses student thinking to launch any instruction. At first these are just engaging problems; later they become curricular. Liljedahl’s site has some good problems on it, and Jo Boaler’s resources are useful as well. I’m looking for students to become more willing to try a problem that they haven’t seen before, rather than giving up at the first sign of difficulty.

2. Use visibly random groupings. Students are used to picking their own partners or being assigned partners or groups. In those spaces, students fall into predictable roles; some students are the “smart kids” who explain things, some rely on copying others’ work. Visibly random groups disrupt those roles, helping students become willing to work with anyone in the room and share ideas openly. I’m looking for students to move between groups flexibly, and for students to fill multiple roles in the group, sharing ideas, recording thinking, and generating new approaches.

3. Work at vertical non-permanent surfaces. Working at desks or tables makes it much easier for students to hide their thinking. Getting them standing and working vertically creates opportunities for collaboration, using erasable surfaces facilitates risk-taking, and orienting the work vertically allows groups to share ideas more easily. Each group should have only one pen to facilitate conversation, rather than a few students working in parallel. I’m looking for students to attempt problems right away, share ideas between groups when they get stuck, and look to each other for help when necessary.

**Helping Students Use Each Other As Resources**

The next five elements of the framework all focus on creating an environment where students look to each other for help, rather than exclusively to the teacher. Liljedahl spoke a lot about learned helplessness — one of the biggest things students learn in school is that they cannot rely on themselves to reason through challenges. These five strategies work together to create a culture of collaboration that empowers every student to own their learning.

4. Oral instructions. This one feels controversial to me — what if a student has trouble understanding ideas verbally? But the purpose isn’t to make sure every student understands every part of a problem instantly. Contrast giving directions verbally with handing each group or student the problem on a piece of paper. Giving the problem verbally encourages groups to make sense of it together, starting conversations and clarifications that lead to more thinking. Giving the problem on paper encourages silent reading and slows the collaboration of the group. I’m looking for groups to try to answer questions about a problem themselves before coming to me, and asking each other questions right after receiving a problem.

5. Defront the room. Students are habituated to look to the front of the room for knowledge and answers. Defronting the room — orienting desks in different directions, standing at random locations around the room, and using the whole room flexibly — breaks that habit. The less students look to the front of the room for answers, the more they rely on each other. I’m looking for students to become comfortable seeing their peers’ work as the most valuable resource for their learning.

6. Only answer “keep thinking” questions. When a student asks a question, it can serve one of two purposes. A “stop thinking” question might be, “is this right?” or “how do I do this?” and might only be asked because the teacher is close by. These questions short-circuit potential learning. On the other hand, a “keep thinking” question is one that allows a student who is stuck to keep working — maybe I tell them the formula for something they’ve seen before that will let them access a more complex problem, or I clarify something ambiguous in the problem to point them in a useful direction. I’m looking for students to persist in asking each other questions and using peers as resources when they get stuck, rather than trying to get the answer from the teacher.

7. Meaningful notes. Rather than having students copy things I write on the board, after a problem and debrief, I ask them, “what do you want to make sure you remember from this problem?” Students need to figure out, themselves or with peers, what the important learning was, and record it in ways that are useful to them. I’m looking for students to step back and consider the connections between ideas and problems and their implications, rather than relying on me for all of their knowledge.

8. Build autonomy. I want students to be able to visit other groups when they are stuck and create their own extensions to continue their thinking. I need to narrate and give feedback on these behaviors to make them a part of the class culture. I’m looking for students to interact less and less with me and more and more with each other as the year goes on.

**Meeting Mathematical Goals**

Elements nine to eleven of the framework are where the learning really happens. The first two conditions could be satisfied — setting new norms where students are doing more thinking, and creating a culture where students look to each other as resources –without anyone actually learning math. These teacher moves focus thinking on specific mathematical goals, and hold students accountable for those goals.

9. Give hints and extensions to manage flow. First, I always want students to be in the sweet spot between frustration (too hard) and boredom (too easy). Hints and extensions help to keep them in that zone. But second, hints and extensions focus students on specific goals. I don’t want to give extensions at random; as students solve problems, I want to keep moving them toward my goals for the lesson. Hints can help to manage the complexities of many groups working simultaneously; if there’s something I want everyone to get to, I can use hints to move slower groups in that direction. I’m looking for students to be constantly engaged rather than frustrated or bored, and for student thinking to zero in on larger goals of a class.

10. Level to the bottom. After a problem or series of problems, we step back to debrief and consolidate understanding. My goal in these conversations is that every student has engaged with the big ideas we’re summarizing (the bottom). If they haven’t, they might as well not have worked on the problem. So I time my debriefs after every group has reached a minimum threshold, and I design tasks, hints, and extensions to help every group get there. I’m looking for students to be able to actively make sense during debriefs, rather than receiving knowledge that feels unfamiliar, confusing, and disconnected from what they were just working on.

11. Assign check for understanding questions. After a task, I want to know what students learned. Check for understanding questions give me information about where to go next, and help students to monitor their own learning. Students can choose to do these problems alone or in a group, on whatever space they like. I’m looking for students to take ownership of their learning and advocate for themselves when they don’t understand something.

**Assessment & Reporting **

Liljedahl minimized the value of assessment — while assessment is a necessary evil in schools, it’s not where much of the learning happens. That said, some practices are better than others to promote thinking and learning. Very quickly:

12. Tell students where they are and where they are going. Formative assessment should help students understand what they know and what they don’t know, and position learning as part of a larger trajectory.

13. Evaluate what you value. Summative assessment communicates to students what is important in class. If collaboration and process skills are important, then they should play a role in summative assessment.

14. Report out based on data, not points. What does an 87 mean? What does 46/54 mean? Reporting should emphasize what students can and can’t do, rather than an aggregate score without meaning.

**Classrooms as Systems **

One interesting argument Liljedahl made was that small changes are often ineffective. His argument is that classrooms are like systems at equilibrium; students find a space where they are compliant but don’t have to think too hard, and teachers enforce norms to keep class organized and moving through curriculum. Small changes are likely to revert back to the status quo. Only large changes will disrupt the system enough that it finds a new equilibrium, one that pushes students toward higher levels of engagement.

It’s fascinating food for thought. Big change vs small change. Disruption vs evolution. I’m still not sure where I stand, but I’ve really enjoyed coming to understand the framework and its application better.

]]>Look beyond the numbers. Look around and through them. Answer questions we don’t even know how to ask. The math that doesn’t exist.

-Hidden Figures

I’m fascinated by the English teacher community around #DisruptTexts. The community hosts Twitter chats during the school year as well as an ongoing conversation around disrupting the white male literary canon. One goal is to replace texts, making space for new works that help more students see themselves in the literature they read and reflect perspectives that have been excluded from the canon. A second goal is to apply a critical lens to texts that remain, looking at literary classics from new perspectives and challenging narratives by questioning the ways that groups are centered or marginalized. See this great post by Tricia Ebarvia for a deeper dive into what disrupting texts looks like.

The #DisruptTexts community came up during the Twitter Math Camp morning session “Taking a Knee in the Mathematics Classroom: Moving From Analysis to Action” led by Marian Dingle and Wendy Menard. Each of my last three years at Twitter Math Camp, there has been a lot of interest in conversations about equity. And each of those years, despite initial interest, those conversations have seemed to fizzle. I’m curious what a rich, sustained community around equity would look like in the online math education world. One thing I’ve noticed in past conversations around equity is a focus on sharing resources. Many teachers want to create or draw from a bank of social justice lessons, or share ways to help students see themselves as potential mathematicians, even if they don’t fit our culture’s stereotypes of what a mathematician looks like. And these are awesome conversations! I’ve really enjoyed them. But what I’m looking for, and what I conjecture the community as a whole would benefit from, is a space that centers *learning.* That’s something I love about #DisruptTexts. While it’s a space to share concrete ideas around disrupting the literary canon, it’s also a space for continuing conversations around why, exactly, it’s important to disrupt the canon, what a more inclusive English curriculum might look like, how to start tough conversations in schools, and broadly just asking questions and learning from others.

One group doing this work has been the #MTBoS book club that Annie Perkins has organized. I’ve enjoyed following along with those conversations, and I think they’ve done a lot of this work. But I’m skeptical that a book club is the best way to draw new people in — there’s a significant barrier to entry, and it by necessity moves slowly.

I’ve heard from lots of folks in the math education world that they care about equity, and they want to increase their capacity in doing equity work, but they’re not sure how, and they’re hesitant to engage because of a fear of saying the wrong thing. What would it look like to create a space that draws people in to conversations about equity and focuses on learning, while also supporting concrete change in classrooms? There are lots of great folks already doing this work. What’s missing is the core message and connecting thread to tie it all together. I’m not sure what that looks like. There are lots of pieces of math education we can critically examine. Here are a few I’d love to have conversations and learn more about.

- Who practices mathematics? Sunil Singh recently wrote a great piece advocating for math history to be taught in schools to help students understand those who have contributed to mathematical knowledge but are not highlighted as mathematicians in classrooms. I just got my tote bag, thanks to Chris Nho! And Annie Perkins has done awesome work collecting information on mathematicians who aren’t just white dudes to help students see themselves as potential mathematicians.
- How is mathematical knowledge created? I love Ben Blum-Smith’s piece on the history of calculus. It’s very funny, and it also gets at the uncertainty in what calculus even was, uncertainty that lasted for centuries. The math we teach in schools didn’t pop fully formed out of brilliant and reclusive mathematicians. It was constructed over long periods of time, through debate and disagreement, and many ideas that are staples of our curriculum weren’t well understood until recently. (See also, the concept of a function.)
- What does it mean to practice mathematics? Is mathematics only a study of abstraction? Can it also be used for social good? Can it be used to understand inequality? Can it be used to make better informed political decisions? Is mathematics only about fast and accurate computation, or can it also be about intuition, about negotiating ambiguity, and about joy? How can we better value the ideas and perspectives that our students bring to our classrooms and build from what students already know?

I have two convictions about this work. First, there are a ton of brilliant people already sharing important ideas, but there is potential to share them more widely and create a more vibrant and ongoing conversation that centers equity in math classrooms. Second, I know I have a ton to learn, as does the larger math community, and a space that draws more people in to conversations to reconceptualize how we think of mathematics and mathematics education could have enormous value for our students.

I don’t know what it might look like. I know thoughtful people have worked on this, and I don’t mean to diminish the work of others who have tried to build an equity-oriented math community. But I want to continue conversations to question how we can do a little better, learn a little more, and reach a little farther.

]]>Something I heard at PCMI, credited to Benjamin Walker, is “you don’t build culture with a firehose, you build it with a turkey baster.” Culture isn’t something that’s established all at once or through the brilliance of one great activity, it’s the sum of all the little things that make a class unique. Planning for culture is less about the first day than it is the micro-moves that reinforce classroom norms the second week and halfway through October and the last week in December. Culture is about patience and small choices, day in and day out.

**Breaking the Didactic Contract**

Also at PCMI, Peter Liljedahl spoke about the non-negotiated norms of classrooms everywhere. Students come in, sit down, face front. Students write in their notebooks what is written on the board. Students complete work on pieces of paper put in front of them. In exchange, teachers don’t require students to think very hard or do any math they have not been shown how to do. These norms are so entrenched that they need to be broken in radical ways, beginning at the start of the first class, to create a classroom where students are willing to think.

I see a compelling argument from each perspective, and I’m not sure how to reconcile them.

]]>