Drill vs Rich Tasks

I’m still thinking about Emma Gargroetzi and Dan Meyer’s responses to the August New York Times op-ed on drill-based math teaching. The comments are fascinating. Here’s mine:

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.

The Social Construction of Mathematics

“Socially Constructed”

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?

–xkcd

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?

–Sunil Singh

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”?

Calvin and Hobbes

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.

Questions About Conceptual Understanding

I’ve been fascinated by the conversations about conceptual understanding, happening on Dan Meyer’s blog and elsewhere. I’ve realized I understand way less about “conceptual understanding” than I thought. Here are some questions that have helped me think about this whole thing:

• 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.

1. 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.
2. 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

Troublemakers

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:

1. 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.
2. 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.
3. 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

Problems

This post is part of the Virtual Conference of Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking. 

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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?

Group Assessments

I saw a short talk on group assessments by Kelly MacArthur from the University of Utah at PCMI this summer. I was intrigued.

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.

Learning About Racism

This is a guest post by Becky.

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.)