I don’t think that being National Board Certified makes me a great teacher. I can give you dozens of ways I’m not. I do think it reflects that I care about the teaching profession, and that I’m working to get better. Board Certification is premised on five core propositions, and I think that these came through in my portfolio — but notice that these speak more to teachers’ growth than their expertise.

- Teachers are committed to students and their learning.
- Teachers know the subjects they teacher and how to teach those subjects to students.
- Teachers are responsible for managing and monitoring student learning.
- Teachers think systematically about their practice and learn from experience.
- Teachers are members of learning communities.

Some things were frustrating about the certification process. The feedback on my portfolio was hard to understand and not very helpful. The guidelines and rubrics were complicated and took forever to sort through. For Component 1, I had to drive two hours to the nearest testing center in Denver and sit in a cubicle staring at math on a computer for three hours; not fun. For Component 2, I had to figure out how to assess student learning at the beginning of a unit, use that assessment to differentiate and give feedback effectively within some uncomfortably prescriptive guidelines, and assess progress again at the end of the unit. For Component 3, I had to struggle to get intelligible audio and video of my teaching, throw out lots of bad clips, and then write something articulate about my teaching. For Component 4, I had to gather information from colleagues and students’ families about their learning, show evidence of how I design assessment systems based on student needs, and demonstrate that I’m learning outside of my school to meet those needs. This last one was a mess; it was hard to sort through exactly what I needed to do for each step and how the different pieces fit together. For the three portfolio components, I had to do a ton of pre-work planning when and where I was going to gather evidence and be prepared when things didn’t work out the first time. Then, I had to piece together what I wanted to communicate in my portfolio, and then actually write the thing. Luckily I like writing about teaching, but it was exhausting.

I think part of the value of Board Certification is that it is a ton of work. It takes time, it costs money, it’s complicated. Teachers can’t start the certification process until they’ve been in the classroom for at least three years, and the credential isn’t worth much outside of schools. It’s not something people are likely to do if they’re on their way out of the profession. And all of the work is embedded in teaching; it’s not like writing papers for a master’s degree because so much of what I did was analyzing my actual teaching practice and talking about where I was working to improve.

Some folks say that the certification process is one of the best professional learning opportunities for practicing teachers. I don’t think this is true, but I feel incredibly lucky to have the MTBoS as a space to share ideas on teaching, hear from others, and push my thinking forward. The NBCT community won’t replace that. But the MTBoS community is different. For one, it’s not all teachers. Lots of people I connect with work in curriculum, technology, instructional leadership, PD, and more. And that’s great! I went to NCTM in Seattle two weeks ago, and those were lots of the people I was hanging out with, and lots of the folks who read this blog. Hi! I appreciate you. The MTBoS is the best place I’ve found for engaging intellectually with teaching math, and I wouldn’t be the teacher I am today without it. National Board Certification dug into the practicalities of classroom teaching in a different way. It was messy and imperfect, but so is the reality of schools and teaching. I have no illusion that being Board Certified will influence my career the way the MTBoS has. But it serves as a symbol of my commitment to the classroom, and my commitment to improving my teaching in the classroom.

My advice to other teachers: if you’re committed to teaching and your school or district is willing to support you financially, take a look at Board Certification. Be careful taking on too many components at a time. Learn to love writing. Know that the first lesson you want to videotape or assessment you want to use work samples from might not work out. Plan the logistics early. Know that it will be frustrating, the rubrics and criteria will be obtuse, and the portfolio will feel like a mountain of paperwork at times. Find someone you trust to look over your work. It’s less about being a brilliant teacher than it is showing off what you already do well, and being willing to reflect on where you want to improve. And you might fail — I didn’t pass by much, and better teachers than me have failed but pushed through.

Board Certification doesn’t decide what good teaching looks like. But it does serve as a marker of commitment to the hard intellectual work and richness of teaching, and a step toward a profession that defines and regulates excellence and receives the respect it deserves.

]]>**Mindset & Competence**

Growth mindset is a hot topic in education right now, and teachers are often told to praise students for their effort rather than their ability. The catch is, in more recent research, changing the way we praise students doesn’t seem to actually influence many students’ mindsets. Carol Dweck has written about how growth mindset has been oversimplified and misused; lots of studies haven’t replicated the optimism of early research on growth mindset, and it seems like praising students a certain way or telling them to have a growth mindset is insufficient for actually changing their attitudes.

But why do students come to math class with fixed mindsets in the first place? They develop these attitudes toward math over years (for my students, a decade) in math classes that send narrow messages about what it means to be good at math. Lani Horn writes that “Schooling favors one type of mathematical competence: quick and accurate calculation” (Motivated, p. 61). Horn argues that we can value broader mathematical competencies — making astute connections, seeing and describing patterns, developing clear representations, being systematic, extending ideas, and more. Instead of trying to convince students to have a growth mindset, we can give students experiences in which they can recognize the different ways they are mathematically competent. As students see the value they bring to math class, they can start to develop a more positive identity as a math learner.

**Routines **

About one-third of our students are likely to experience some type of anxiety disorder during adolescence.

Anxiety is a pretty rational response to the stresses of adolescence, both within and outside of school. While we may not be able to address many of the root causes, we can create classrooms where students experiencing anxiety, as well as the rest of our students, recognize their competencies. Routines are an opportunity for students to feel safe, to worry less about what’s happening next, and think more about the math. Within a routine, students can become more comfortable taking risks and sharing ideas. There are also a ton of routines out there. In our session we used Stronger and Clearer Each Time, Number Talks, Visual Patterns, Five Practices, and Stand & Talks, but these are just a few examples we are partial to, and other routines would work better in different contexts. Lisa’s blog and the Stanford GSE have plenty more examples.

Routines have value in creating spaces where students can take risks and feel comfortable thinking mathematically, but they also add value for teachers. As I use more routines, I become more comfortable with the structure of the routines, thinking less about what comes next in my lesson, and thinking more about how students understand mathematical ideas and finding more valuable conceptions that I can build off of.

**Routines & Competence **

What’s the connection? Here’s our premise: routines are a valuable teaching tool, and every teacher already has routines in the ways we set up our classrooms and lessons, even if we don’t make them explicit. How do we start class? How do we launch problems? How do we have students practice? These routines send a message about our values. If our routines value a narrow vision of mathematics that causes students to focus on their deficits rather than their strengths, then negative feelings, negative mindsets, and anxious behaviors become entrenched. If our routines create rich and varied opportunities for students to recognize the ways they can be successful in math class, and to recognize those successes in lots of different ways, students who have felt alienated in the past can start, slowly, to change their perspectives. Stronger and Clearer Each Time values revision and improving ideas. Visual Patterns value different perspectives and unusual interpretations of a problem. Five Practices helps us to highlight every student’s thinking, rather than just loud students.

This is slow, humble work. Students may come to class having been convinced for years that they aren’t math people and that math class is not a place that values their thinking. We can’t change that overnight with a flashy lesson or a quick pep talk. And routines aren’t the only way to create that change. But routines are one practical place that classrooms can find ways to value every student and help all students to see themselves as mathematicians.

What messages do your routines send about your values as a teacher? What is one opportunity to incorporate a new routine that broadens student conceptions of what it means to be smart in math class?

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Meanings are acquired from experiencing differences across a background of sameness, rather than from experiencing sameness against a background of difference.

One common thing I do in class is have students practice something. Some students get bored quickly, some work happily along, others struggle. This post is an attempt to design practice in a way that supports the learning of all of these students.

Students have been introduced to arithmetic series and need to practice. Here are two sets of problems:

Which sequence of problems better helps all students?

I’m going to argue the first. Three reasons:

- First, each problem only varies in small ways from the previous problem. Students’ attention is then focused on these small changes, and they are more likely to make sense of the components of an arithmetic series problem, rather than having to start from scratch for each problem. When the problem changes in only one way, students can better understand the impact of that change on the mathematics.
- There is more potential for extension. The structure of the problems means that students can find shortcuts, using one answer to more easily solve another. Then, we can return to those ideas to review as a class, providing more opportunities for discussion than typical practice.
- There is more opportunity to scaffold success. A student who is struggling might have trouble at first, but varying only one element of the next problem makes it more likely that they can use what they figured out right away and better consolidate their understanding.

This idea comes from Variation Theory, which Craig Barton talks about in How I Wish I’d Taught Maths. He writes:

By working through carefully chosen sequences of questions, students have to carry out procedural operations, thus engaging in vital practice. But through connected calculations, they also have the opportunity to consider the deeper structure. Such variation allows students to anticipate, notice and then generalise, instead of permanently playing catch-up (249).

I think there is more potential in these sequences of problems both for students who already have strong skills and have the opportunity to notice new connections, and to students who are struggling with the concept and can benefit from only focusing on the essential differences between problems. But the problems above are only one very narrow type of question. What about when students need to distinguish between similar problems?

–Mun Ling Lo

If you do not know what English is and you hear 100 people speaking English, you will have no better idea of the meaning of “a language”. If you do not know what “a lively style of writing” is, and you read 100 articles, all of them written in the same lively style, you will still not know what “a lively style of writing” means.

Let’s say I want to help students distinguish between arithmetic and geometric series, and as a secondary goal practice identifying the common ratio of a geometric series. The above sequence of problems strips away any unnecessary ideas, and gives me a great chance to see exactly where student thinking breaks down, and to address those breakdowns. I don’t think all practice should be structured as variations of one problem; after these six problems, I might start with a new sequence focused on different ideas and asked in a different way. But by only varying a single element of a problem at a time, I get more precise information about what students know and don’t know, and can facilitate a more fruitful discussion of the problems.

I think there is one more possible use for this type of minimally different problems. Let’s say I want to introduce students to sigma notation. I often struggle to explain sigma notation concisely, and a few examples can go a long way. I might give students a few examples of sigma notation to notice and wonder about. But with too much variation, it just looks like Greek alphabet soup. By only minimally varying problems, I give students more to latch onto, and make it more likely they notice what I would like them to notice:

I really like these sequences of minimally different problems. They still serve goals I had before. But now, students’ attention is more focused on the essential ideas of a topic, sequences of problems scaffold success for more students, and I open up natural opportunities for differentiation as students can make new connections and generalizations. While I’m only starting to experiment with minimally different problems, I also think that over time these problems could help students to see that math can make sense and isn’t just a collection of disconnected ideas. As students see more sequences of problems like these, they might start to believe that they can find shortcuts and new strategies for problems, and develop a disposition to look for patterns where they might not have before.

]]>**Stories**

I became interested in learning about groups after reading Patrick Honner’s October article, The (Imaginary) Numbers at the Edge of Reality in Quanta Magazine. It’s a great read, and it positions group theory as part of a larger story, framing different number systems in terms of their connections to physical problems and sharing the stories of the mathematicians who first worked with them. I became fascinated by quaternions, and I’m lucky that the text I’m learning from uses quaternions as an example in different contexts and keeps me connected to a narrative beyond the math itself.

How often does this happen in math classes? Not very often in mine. Now I’m thinking about how I can find ways to position the math that we’re learning as part of a larger story. I don’t think this needs to be a radical change; it can be a quick addendum of historical context, a narrative about a relevant mathematician or mathematicians, an interesting application of a topic, or just taking a moment to share how different concepts are related, framing where students have been and where they’re going. But humans learn from stories, and are motivated to learn from stories, and I think this is something that is underused in my classes.

**Examples**

An example is worth a thousand definitions. You can define “ideal” as carefully as you like and I’m still going to be confused the first time I learn about it. Share a handful of well-chosen examples and non-examples and all of a sudden it makes sense. In math we love definitions. I find many definitions elegant and beautiful. I spend lots of time thinking about how to explain concepts in ways that will make sense to students. These things are important, but it’s possible to overestimate their importance. Examples work with explanations to create students’ mental models of concepts. Examples give something concrete to latch onto, and they can illuminate boundary cases and subtleties that might not make sense in an explanation or be clear from a definition. And as I see more examples, I start to create new generalizations and come up with explanations that make sense to me. When I first read about ideals, they were nonsense. Now I think about them like a magnet — they’re this set of objects that pull other objects in, and once you’re in, you can’t get out. For instance, if I’m working with integers, once something becomes a multiple of 3, no matter what you multiply it by, it stays a multiple of 3. That might not make sense to you, but it makes sense to me. And the more I see new examples and incorporate them into my mental models, the better I can apply that knowledge. Examples give me a chance to test my understanding and see whether my ideas make sense in a new context.

I don’t think I do this very well with students. Student understanding often happens within the paradigm of my explanations and my ways of looking at mathematical ideas. There’s a place for that, especially to minimize confusion and misconceptions. But there’s also an place to give students lots of examples to work with, to ask them to come up with explanations that make sense to them, and to embrace their ideas and perspectives. I can explain the end behavior of rational functions until I’m blue in the face talking about top-heavy and bottom-heavy functions, and students are often still confused. Offering a set of well-chosen examples and letting students come up with language and analogies that make sense with their experience could be a much less painful way to do it.

There’s nothing groundbreaking about either of these ideas, but as someone who knows a lot of math and isn’t often in the position of learning new math, they’re easy to forget. A constant challenge of teaching is the curse of my own knowledge, and learning something new, even when it’s hard, is a great way for me to see learning from a new perspective and push myself to teach in ways that are accessible and engaging for all students.

]]>“I just can’t do math.”

-Lots of students since forever

When I started teaching, my typical response was to blame the student. Of course they can do math. Anyone can, with the right support and patience. Why can’t they just have a growth mindset?

Here’s the question I’ve started asking: why does that student have a fixed mindset in the first place? We work in an education system that excels at communicating to some students that they have intellectual promise, to others that they don’t, and that there’s nothing anyone can do to change it. This message is implicit in the ways students are sorted into tracks from an early age, to the way we talk about professionals in our fields and promote or discourage representation, to the grades we stamp on students’ assessments. And all of those messages accumulate over time to exactly the opposite mindset that we might hope students develop.

Growth mindset suffered from the mistaken idea that someone’s mindset can change just by being praised for their effort rather than their ability. In reality it’s a little tougher than that, and researchers have struggled to design interventions that consistently change mindsets. But understanding where mindsets come from helps in having some humility about changing a student’s mindset. It’s definitely possible, but it’s also definitely harder than we might like to think.

My job is to find ways for students to recognize the ways that they are mathematically competent. One way we’ve convinced students of their lack of mathematical ability is by valuing a narrow vision of mathematics that emphasizes computational speed and accuracy. If I’m deliberate I can send new messages that, over time, broaden that student’s idea of what it means to learn and practice mathematics, and help them to recognize their mathematical competence. Doing so helps to change mindsets, and to create classrooms that communicate the values of mathematics and help every student to see themselves as a potential knower and doer of math.

I try to do this through routines that value different competencies. Routines, repeated over time, help students to become comfortable practicing mathematics in new ways and give them opportunities to practice and recognize their own brilliance in new ways. I come in with the perspective that every student already has great mathematical ideas; my job is to create space for students to share and recognize those ideas. Routines are the healthiest space for those ideas to grow. And routines help to build enduring messages that communicate new ideas about students’ mathematical potential.

One important result of this shift in perspective is that it helps me to be patient. Instead of becoming frustrated that my students aren’t ideal math learners, I try to understand why they feel the way they do, and to feel like I have some concrete tools to help them shift their perspective. I know that nothing will change overnight, but I also know that I’m playing a longer game that can help students to see themselves in a new light, based on their successes in class each day.

I’ll be presenting on this topic with Lisa Bejarano at NCTM Regionals where we will practice a few routines that allow students to see their own brilliance and discuss many more while considering how they can be adapted to fit your own classroom.

Come join the discussion in our session at NCTM Regionals in Seattle:

**Anxiety, Mindset, and Motivation: Bridging from Research to Action***November 30, 2018 | 9:45-11:00 a.m. in Washington State Convention Center, 606*

Developing a supportive class culture and growth mindset can reduce students’ anxiety, allowing learners to engage thoughtfully with each other around mathematics. Participants will discuss the challenges of shifting mindsets, experience routines as learners and leave with resources and ideas to implement these structures in their classroom.

I struggle to define “problem” despite always having tons of them rattling around in my brain. It seems like a decent way to define it is to offer a bunch of examples. Here are some favorites:

**Split 25
**

(Play With Your Math)

**Self-Aware**

(Play With Your Math)

**Cows in Fields**

(Singh)

**To Cross the Bridge**

(Singh)

**The Census Taker**

During a recent census, a man told the census taker that he had three children. The census taker said that he needed to know their ages, and the man replied that the product of their ages was 36. The census taker, slightly miffed, said he needed to know each of their ages. The man said, “Well the sum of their ages is the same as my house number.” The census taker looked at the house number and complained, “I still can’t tell their ages.” The man said, “Oh, that’s right, the oldest one taught the younger ones to play chess.” The census taker promptly wrote down the ages of the three children. How did he know, and what were the ages?

(Batchelder & Alexander)

**Long Division**

The following long division problem has a unique solution, despite providing just one digit. The Xs can represent any digit, and the problem is an 8-digit number divided by a 3-digit number producing a 5-digit number and dividing evenly.

(Gardner)

**105**

In how many ways can 105 be expressed as the sum of at least two consecutive integers?

(Liljedahl)

**Circle in a Parabola**

There are many circles that will “fit” inside a given parabola. What is the largest circle that will do so? Why?

(Lockhart)

**7.11**

A guy walks into a 7-11 store and selects four items to buy. The clerk at the counter informs the gentleman that the total cost of the four items is $7.11. He was completely surprised that the cost was the same as the name of the store. The clerk informed the man that he simply multiplied the cost of each item and arrived at the total. The customer calmly informed the clerk that the items should be added and not multiplied. The clerk then added the items together and informed the customer that the total was still exactly $7.11.

What are the exact costs of each item? (Assume that they multiply to 7.11 exactly, with no rounding.)

(Brumbaugh)

**No Trigonometry Required!**

(Brilliant)

**November Nonagon
**The figure below shows a square within a regular nonagon. What is the measure of the indicated angle?

**Why Problems? **

I think math is worth learning for lots of reasons. I want students to be quantitatively literate in a world that increasingly requires mathematical knowledge to be an informed citizen. I want students to understand math to open doors for them in the future, as mathematicians or in any number of other disciplines that rely on mathematics. I want students to cultivate skills of abstract reasoning, recognition and generalization of patterns, critical argument, precision, and structure. I want students to see math as a subject full of challenges that they are capable of overcoming, and for math to help them recognize their intellectual potential.

But from my perspective, the most important piece is for students to get a sense of the beauty and joy of mathematics, and to experience the “a-ha moments” that characterize our discipline. English has great literature. Science has the mysteries and wonders of the natural world. History has the gripping narratives of the past. Math has problems.

**Two Things**

I want students to experience the a-ha moments of problem solving as a catalyst to help them understand the discipline of mathematics and their potential as mathematicians. But not all problems are equally useful for creating these moments. I’d like to hypothesize two elements that allow a problem to facilitate students’ love of problem solving.

**Insight vs Experimentation **

On one end of a spectrum are insight problems, like “November Nonagon,” “To Cross the Bridge,” and “No Trigonometry Required!” These problems lend themselves to certain representations and strategies, but the approaches one takes at first are unlikely to be successful. Solving the problem relies largely on an insight: a change of perspective that illuminates a path to a solution. A solver might end up staring at the problem, making no progress, for some time. With some luck, the insight will whisper itself at an opportune moment, and the problem will be solved. On the other end are experimentation problems, like “Circle in a Parabola,” “The Census Taker,” and “Split 25.” These problems lend themselves to trial and error and don’t require any large leaps of logic or intuition. A solver can try a number of different approaches, stepping back to look for patterns as necessary, on a much more well-defined path to a solution. That’s not to say that these problems are easy, just that they are more likely to suggest plausible pathways than dead ends.

**Success**

A second spectrum is how quickly a solver is likely to experience a feeling of success — whether or not they solve the problem, can they make some concrete progress early on? The problems “Cows in Fields,” “Circle in a Parabola,” and “105” allow a solver a quick taste of success, where one or several examples are readily available, although finding all of them still requires a great deal of persistence and ingenuity. These successes can act as springboards to the rest of the problem, rather than experiences of frustration from the beginning. Alternatively, problems “Self-Aware” and “7.11” resist easy wins. One could try a few ideas, but they don’t lend themselves to quick strategies, and a successful solver will likely have to muddle through a significant amount of failure, trying unsuccessful ideas, to get to a solution.

**What Makes a Problem Useful?**

I think that the best problems to teach students a love of problem solving allow for experimentation facilitate early success. Experimentation allows multiple access points, gives students half-formed and informal ideas to share and argue about, and gives a sense that, while the journey may not be easy, it is at least possible. Early success builds motivation; feelings of success help students understand that problems exist for the pleasure of solving, rather than to frustrate and bore them.

These aren’t necessarily static properties of problems. A teacher could facilitate experimentation in “November Nonagon” with the suggestion that a solver try adding auxiliary lines, or in “No Trigonometry Required” with the hint that the angles can be rearranged (without changing their size) to try to make a useful shape. Similarly, “Self-Aware” could be modified to make early success more likely by prompting students for 5- and 7-digit self-aware numbers in addition to 10-digit ones. These small changes, combined with choosing problems thoughtfully given students’ knowledge and motivation, can make a big difference.

This isn’t to say that problems without these characteristics are worthless — they can be fantastic fun for students who have already developed some interest and joy in doing math. But to create that a-ha moment that shifts a student onto the path of being a math lover, I think these two features are critical. Staring at a problem with no clear paths forward or ideas to try is likely to result in frustration for many students. And even when there are clear ideas to try, without some positive reinforcement of early success a student is likely to give up before they get to the good stuff. Not all problems fall neatly on one side or the other and no problem is perfect, but I do think these two features make a problem much more useful for all students, rather than just those who already like math.

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Read about the new direction #TMC19 will take to move toward more diversity! https://t.co/x4ciFapL9w pic.twitter.com/NJ5ber3RUH

— TMathC (@TmathC) October 20, 2018

This year we’re making a commitment to racial diversity. At least 20 attendees of #TMC19 will be educators of color. We will reach out to our networks to make sure that people know this conference wants to be more diverse. We will take specific actions to make sure that people know this isn’t a surface level commitment, we are determined that TMC will be a space that welcomes everyone and where educators of color will be specifically included. So far we are planning:

- A time on Tuesday, July 17 for all of the educators of color to gather, get to know one another, and learn about the plan for the week.
- An equity strand of presentations running throughout the conference.
- A safety plan for travel to breakfast, dinner, and evening activities.
- Waived registration fees for all educators of color (speakers and attendees).
- To consider any and all other ideas for ways to make the conference a better environment, specifically for educators of color.

Diversity is in. But why?

I want to help create and sustain diverse communities, both in the math education space and elsewhere. But I feel a tension between two different arguments for diversity that I want to consider in conversations about how to make spaces more diverse.

TMC writes on their blog:

According to this excellent TED Talk, “Ethnically diverse companies perform 33 percent better than the norm.”

One might call this the “diversity makes us stronger” argument. I believe it. I am a better educator because of educators of color who share their perspectives, yet those voices are not often those elevated by folks with the biggest microphones in education spaces. But if the only argument for diversity is to help white folks in largely white spaces, that diversity is fundamentally extractive. People of color and other marginalized folks do not exist to benefit those who already have power. I’m incredibly grateful to Jose Vilson, bell hooks, Mariame Kaba, Jacqueline Keeler, and others for their writing and activism. But they and other people of color have no obligation to seek me out and educate me.

The second argument for diversity is that we should honor the agency and humanity of every individual, acknowledging that our institutions have conspired, past and present, to keep some out. This means creating spaces where every individual can find what they need — in the math education space, that every teacher can grow and see themselves and their learners reflected in their professional development. I see it as an argument about freedom. As Carla Shalaby writes:

A free person retains her power, her right to self-determination, her opportunity to flourish, her ability to love and to be loved, and her capacity for hope.

-Carla Shalaby in Troublemakers (xv)

The second argument for diversity says that we should have spaces where every individual can be free to flourish, to love, and to believe in the potential of education. Folks have no obligation to make spaces diverse for the sake of diversity or the benefit of majority; instead, we should see diversity as the end result of rethinking the ways professional development spaces are organized to value every individual.

Centering the humanity of every individual is a conscious choice, acknowledging our country’s history of oppression. How do we respond to government-supported housing segregation? Inequitably resourced schools? Systematic plunder of wealth? It’s not an accident that some voices are excluded; it is the ongoing legacy of oppression in our country. And bringing folks in with an emphasis on their humanity and freedom helps us to see diversity as an opportunity for more productive collective action, rather than an exercise in pity that reifies existing inequities. Here is Matthew Kay on having meaningful race conversations in the classroom:

If the race conversation is about a hard problem, encourage students to (1) locate their sphere of influence, and (2) explore personal pathways to solutions. If, as argued in the previous chapter, our students deserve to consider the hard problems, they must also be invited to solve them. This balance reminds them of their agency. Without it, the discussion of race controversies is likely to make students feel a bit like punching bags, peppered by jabbing misery narratives that set up a knockout conclusion. We teachers, with all of our culturally sanctioned agency, can be surprisingly blind to this barrage… Imagine the frustration of having various narrative bits dumped on a desk before you and being asked to contemplate them without the opportunity to put them together into a whole.

-Matthew Kay in Not Light, but Fire (121)

Our students need narratives that not only teach about the realities of inequality in the world, but help them feel a sense of agency in making change in the future. In the same way, education spaces need to move beyond token diversity to a paradigm that values every individual and the potential for change when we bring educators together in spaces that are inclusive and empowering. I’m excited that Twitter Math Camp is working to make the conference more diverse. And, I hope that we as an online math education community can continue to work toward diversity as a necessary means to ambitious goals as well as an end in itself.

]]>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