Implementing a Thinking Classroom

I’ve spent some time in the last year experimenting with principles of a Thinking Classroom. Laura Wheeler’s sketchnote seems to have become the go-to summary of the Thinking Classroom framework:

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.

Disrupt Math

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.

1. 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.                  
2. 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.)
3. 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.

Two Perspectives on Classroom Culture

Turkey Baster

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.

What Is Mathematics?

On Friday at PCMI, Jonathan Mattingly gave an afternoon lecture entitled “Math and Gerrymandering.” You can see the talk here, and explore Jonathan and his team’s great work here. It was a fascinating talk, and I learned a ton. Jonathan spoke about Supreme Court cases on gerrymandering that have been recently decided or could be decided soon, looked at the statistics behind how his team models gerrymandering, and explored some really cool representations of gerrymandering to help understand what it means from a less technical perspective. One interesting point is that gerrymandering is not just about oddly-shaped districts. The two maps below are equally biased toward Republicans.

While oddly shaped districts may be a symptom, they could also be bringing together an interest group that wouldn’t have representation otherwise. Mattingly points to something else as what he calls the “signature of gerrymandering,” shown in the image below and which he writes about here:

See how the orange and purple lines jump between 10 and 11 districts, but the yellow and green ones increase more uniformly? The jump means that, in gerrymandered maps, Democrats could get anywhere from 50%-65% of the vote, and their representation in the House of Representatives wouldn’t change. As a contrast, the green data points are a hypothetical map that a non-partisan commission drew but did not implement. In that map, more votes for one side correlate strongly with more seats in the House, and there’s no large jump indicating an advantage for one side

It was a great talk, but one piece of it bugged me. Both Mattingly and Rafe Mazzeo, the director of PCMI who introduced the talk, alluded to the idea that his project is “social” work and not actually mathematics. The first part of the talk I mostly understood, but toward the end he put up a slide that said “Where’s the Mathematics?” (implying that communicating about gerrymandering isn’t mathematics) and then launched into something I didn’t understand and that also seemed much less relevant to understanding and communicating about gerrymandering to the public.

I don’t mean to attack Jonathan, but I see a missed opportunity for expanding the meaning of mathematics here. This bugged me, and I asked a question at the end of the talk trying to push on this. Here is a more articulate version of that question:

In your talk, you alluded several times to the idea that studying gerrymandering is not “real mathematics.” I would conjecture that many who are in or consider entering the field of mathematics would agree — we often see mathematics as Lie groups, Fourier series, eight-dimensional topology, and other stuff I don’t understand. But many others would argue that studying gerrymandering, and applying mathematics to social and political questions more broadly, is mathematics. More than that, social and political work has the potential to change what we perceive as mathematics, and transform our discipline into one that can more concretely improve the condition of humanity. What would you say to a high school student, undergraduate, or graduate student who sees your work as “not mathematics,” and how can we work to create a more inclusive mathematics in the future?

I’d like to go a step further. Most of my students don’t see themselves as mathematicians because they can’t see pathways for mathematics to positively influence their lives. What if, as one small step toward creating richer perceptions of what mathematics is and creating a discipline that has a more positive influence on humans, we chose to center “mathematics for social good” as a core part of what we see as math?

What if university math departments designed undergraduate courses on the mathematics of, say, gerrymandering, income inequality, and applications of big data. What if those departments required math majors to take at least one of these courses, under the umbrella of “mathematics for social good” as part of a math major?

What if high schools designed a pathway focusing on similar themes as an alternative to the current race to calculus?

What if professionals prioritized communication with the public about mathematics as equal in stature to proving theorems?

Michael Pershan wrote recently about how mathematics was taught several hundred years ago — to very briefly summarize, undergraduates read texts like Euclid’s Elements and had to be able to state proofs or theorems from the text or solve simple problems orally. There was very little written mathematics, and very little of what we might call problem solving. And, to them, that was the discipline of mathematics. Universities didn’t change until the middle of the 19th century. That blows my mind. The way we look at what is important in mathematics education has radically changed in the last two hundred years. Why can’t it change again?

I’m interested in conceptualizing “mathematics for social good” because it’s on my mind at the moment, but more broadly, I’m curious what we might change if we had the opportunity to completely reimagine mathematics education. What essential things do we want people to know? What experiences do we want people to have? How do we want the public to understand mathematics? I think those questions might cause us to build something that looks very different than the mathematics we have today.

Equity and Cognitive Science

Here’s something I’m curious about. There are two different areas of mathematics education that have interested me recently. The first might be called the equity perspective. Here’s Rochelle Gutiérrez in The Sociopolitical Turn in Mathematics Education:

I use the term sociopolitical turn to reference a growing body of researchers and practitioners who seek to foreground the political and to engage in the tensions that surround that work. The sociopolitical turn signals the shift in theoretical perspectives that see knowledge, power, and identity as interwoven and arising from (and constituted within) social discourses. Adopting such a stance means uncovering the taken-for-granted rules and ways of operating that privilege some individuals and exclude others. Those who have taken the sociopolitical turn seek not just to better understand mathematics education in all of its social forms but to transform mathematics education in ways that privilege more socially just practices.

The second is the cognitive science perspective, which has been shared widely by the Learning Scientists and Dan Willingham. I’ve learned a ton about memory, cognition, and learning, and I’ve found cognitive science useful in better understanding the teaching in my classroom and thinking about how I structure activities to be consistent with cognitive research.

Melvin Peralta wrote in the spring about the importance of bringing both of these perspectives into classroom practice. They each have important insights for educators. But beyond learning from both perspectives, I wonder if I can learn more by putting ideas from the cognitive and sociopolitical perspectives into dialogue with each other. Here’s a first attempt at doing that.

I.

Memory is the residue of thought.

–Dan Willingham

What’d you learn in class today?
Don’t walk fast, don’t speak loud, keep your hands to yourself, keep your head down.
Keep your eyes on your own paper, if you don’t know the answer, fill in “C”.
Always wear earbuds when you ride the bus alone,
If you feel like someone’s following you, pretend you’re on the phone.
A teacher never fails, only you do.
Every state in America, the greatest lessons, are the ones you don’t remember learning.

–Brave New Voices, slam poem by the Los Angeles Team

We remember what we think about, and we remember more when that thinking is spaced over time. This cognitive principle is often invoked as a structure for effective studying, or as an argument for spaced practice to review topics that have been previously taught. What if we instead ask the question: What are some ideas that our students are thinking about, day after day and year after year, that might help us better understand what they actually learn in school? The young women who wrote the slam poem above (which I highly recommend watching in its entirety) might argue that the lessons students think about most often and spaced most consistently over time are about obedience, silence, and power. They are thinking, “school is a place where my voice is not important,” and “following rules without questioning is the best way to make it through the day,” and “certain people get to be in charge and that’s just the way it is.” Are these the lessons we want young people to take from school?

II.

Our working memories can only hold so much, and when our working memories are overwhelmed learning is harder. This is John Sweller’s Cognitive Load Theory The theory is often used to argue that inquiry learning cannot work because problem solving overloads the working memory of novices who don’t have enough knowledge about the problems that are meant to lead to learning. Students lose the forest for the trees as they get stuck on the particulars of a specific problem and struggle to step back and see broader connections.

But math problems are not the only thing that can consume working memory. What if we look to understand student identity through this lens?

“I’m always ready for that lady’s class and she gets me suspended because she doesn’t know what she’s doing. She sees what she wants to see.” As we talked more, I mentioned that the teacher said she never had her books with her for class. She responded that a friend shares her books with her and lends her something to write with whenever she needs it. For her, that made it obvious that she was prepared to learn. She then mentioned that she was always on time for class. “I’m always at the door when that bell rings. I’m always there.” The student saw herself as prepared and on time, but the teacher did not see the student the way she saw herself.

The point here is not to debate whether the teacher or the student was right or wrong; there isn’t a clear answer to that question. What’s important to note is that the teacher in this scenario had rendered the student’s self-image as “prepared and on time” invisible.

-Christopher Emdin, For White Folks Who Teach in the Hood…and the Rest of Y’all Too, p. 19

I can’t count the number of times I’ve made a decision in the moment and made a student feel like they are invisible, invalidating their best intentions. What is that student thinking about in class after an interaction that threatens their identity as a learner? How can they learn when their working memory is consumed with thoughts of our interaction and the disconnect between our interpretations of their place in the classroom?

III.

Abstract ideas can be vague and hard to grasp. Moreover, human memory is designed to remember concrete information better than abstract information.

–Yana Weinstein & Megan Smith

I want every student to have equal access to high-level mathematics, and also for every student to feel empowered to pursue future mathematics and feel a sense of agency in their education. But that agency is an abstract idea; if a student doesn’t see images of people who share their identity doing mathematics, do they have access to the concrete examples to feel empowered?

At the Equity & Math Education panel at PCMI last week, KiMi Wilson argued that we have a responsibility to find examples of professionals who use mathematics and represent the identities of marginalized students to come into math classrooms and talk to students. Short of that, Annie Perkins has assembled information on dozens of mathematicians from a range of identities and ideas on how to share diverse role models with students in empowering ways. Telling students that they can learn mathematics is unlikely to be successful on its own for many students; sharing concrete examples of mathematicians who share their identities is much more likely to help students understand that they can be mathematicians as well.

IV.

I wonder if pursuing connections between these bodies of work is worthwhile. I’ve learned a ton from cognitive science, but I think it can also seem like a discipline that is distant from the realities of classrooms and students, providing recommendations that treat all students and all contexts the same. Can applying cognitive science to questions of equity help to bring more nuance to applications of cognitive science research? At the same time, perspectives on equity are often dismissed as “soft” or as distractions from the most important parts of teaching and learning. Can a dialogue between equity and cognitive science help to surface the importance of multiple perspectives in education research?

Positioning Students as Competent

Two thoughts on competence, reflecting on time at PCMI. First:

We’ve been thinking about Lani Horn’s book Motivated, and talking about the idea of competence. I had been thinking of competence as a student’s need to feel successful in doing mathematics. If students don’t feel like they are successful in doing math they are unlikely to engage in unfamiliar or challenging problems, take risks by sharing ideas, or persevere when learning feels hard.

We spent some time with Peg Cagle, and Peg offered a different take. She described competence as a student’s need to recognize their success in doing mathematics. I see this as an important shift. While it’s just language, the word recognize makes an assumption — that all students bring meaningful mathematical ideas and mathematical thinking skills to class. My job as an educator is to create structures and space to help students recognize those competencies. Horn offers a partial list of mathematical competencies that teachers can value beyond what is traditionally valued in math class — fast and accurate computation. Those broader competencies are:

• making astute connections
• seeing and describing patterns
• developing clear representations
• being systematic
• extending ideas

Peg shared this image from her classroom as well:

I’m sure other educators could expand on these. The point is that there are lots of ways for students to be mathematically competent. If my goal is for them to feel  competent, I might work to prop up their confidence or self-esteem without changing any structures in my classroom. If, instead, my goal is to help students recognize the ways they are mathematically competent, I am obligated to find ways to surface and highlight broader competencies that create avenues for every student to recognize their successes. Those are two very different classrooms; the second is much more responsive to the needs of particular students, and develops a much richer idea of what it means to do mathematics.

Second idea. In talking about competence yesterday, we also talked about the challenges of grades. What does it mean to value broader competencies like extending ideas when we are obligated to put a letter grade on a transcript at the end of the term? Those grades seem to be all many students care about. Do grades erase any work we do to assign competence in broader ways? I can’t realistically improve at more than one or two areas of my practice at a time. In the context of schools unwilling to change grading policies, would I be smart to put my effort somewhere else?

A Thought Experiment on Tracking

Lots of talk on tracking in the math world right now, and I’ve enjoyed following along with the comments on this recent NCTM President’s Message from Robert Berry and this blog post by Michael Pershan, as well as the conversation on Twitter (for instance, here).

In the comments linked above, many people make an argument along the lines of, “but what about the gifted/smart/advanced students?” Tracking is the status quo, and these are the students who many perceive would be affected by a change. They argue that ending tracking will reduce opportunity for a certain group of students, labeled as more able than others. They argue that these students will be less challenged, will love math less, and will struggle to be engaged in the typical heterogeneous class. For more detail, check out the comments and conversation in the links above — there are many compelling arguments, from the perspectives of educators, parents, and former students.

Here’s a thought experiment. What if, instead of a world where tracking is the norm and NCTM is advocating to end it, we imagine a world where there is no tracking, and someone is advocating to institute grouping students into classes by perceived ability? What might people say if heterogeneous classes were the status quo, and we argued to change that?

Here’s an argument I might make:

What about the students who won’t be selected for a higher track? They’ll be pushed into low-level classes taught by less qualified teachers, they’ll interpret tracking as a message that they are less “smart” (likely based on standardized tests), and they’ll be segregated based on those messages, undermining community as entire classes come to believe they’re not mathematically capable. Given the way that academic ability is currently assessed, tracking will create a system of de facto segregation based on race and class, exacerbating differential access to the type of education that young people need to be full citizens in their country. And as tracking spreads, opportunity will be hoarded by families who know how to manipulate the system and buy advantages for their children, whether through private tutoring or pressure on their schools. All of this will limit the mathematical trajectories of many students, often before they’ve hit puberty.

In this alternate world, where tracking is a change to the status quo, what would you advocate for to better support all students?