Becoming an Outdoor Educator: Pemi West

My love of the outdoors really took off in the summer of 2021, so it was a joy and coming-full-circle moment to co-guide a wilderness expedition in the summer of 2023 for a group of teenagers in the Four Corners region near southwest Colorado. My friend Patrick, who’s the Assistant Director at Camp Pemi, put together a great blog post about the trip!

Here’s a full trip report I wrote if you’re curious about what it was like day by day.

Expanding the Overlap: UDL and Progressive Math Pedagogy

I’ve been thinking about the extent to which there is tension between the principles of Universal Design for Learning and progressive, inquiry-based, student-centered math teaching.

I’ve done a lot of math teaching in independent schools that falls within the rightmost section, and some in the sweet spot in the middle. Within the public and charter school system, the challenge has been more about moving from the splash of inaccessible, traditional math teaching into universally designed (but still fairly traditional) teaching.

For a bit of background, UDL is an educational framework that began in 1984 which calls for multiple means of engagement, representation, and action/expression to ensure access for all learners by removing barriers to learning. You may have seen this visual metaphor:

Here are the specific UDL guidelines:

UDL is a rightfully ambitious framework which in practice is aspirational rather than fully realized in any particular lesson, unit, or course. When it comes to access to mathematics, the challenge is in the implementation details of any curriculum or pedagogical approach.

When I think about implementations of a progressive math teaching philosophy, approaches such as Exeter Math (problem sets) and Building Thinking Classrooms come to mind. I’ve used aspects of both of these over the years. Both of them hold student collaboration as a central course goal. They are both opinionated about how students should work (together, often at the board) and that students should learn by solving problems and generalizing from there.

I’ve been wondering lately about the extent to which these progressive, inquiry-based math approaches can be inaccessible to some students based on their learning needs. And if so, are these barriers that can be removed, or are they intrinsic to the particular approach?

I happen to be a huge fan of Exeter math. I love the spiraled problem sets, the fact that students are expected to own the cognitive lift, and the way the curriculum emphasizes connection, sense-making, and novel problem solving. At the same time, I also recognize that it’s not an approach that could be widely adopted at schools across the country. There are practical barriers in terms of the time commitment required outside of class, lack of board space and large class size in many places, and limited opportunities for differentiation. This makes me wonder about what a universally designed, problem-based approach would look like.

To meet the requirements of UDL, it seems the things we can be opinionated about as teachers and curriculum designers have to be limited. For example, we can still insist that students do most of the talking and thinking within the classroom. But we probably have to let get the requirement that students work in groups most of the time since this infringes on student choice. (To me, there is a balancing act to navigate between UDL Checkpoint 8.3 about collaboration and community and UDL Checkpoint 7.1 about individual choice and autonomy.) That’s a painful concession for me to consider because of how strongly I believe in the value of collaboration in math class.

Expanding the Overlap

I’m sure many educators have found ways over the years to take these progressive math approaches and make them accessible to as many students as possible. I wonder if these are some of the both/and beliefs they embody:

  • Student collaboration is a central course goal, and we provide choice for how students collaborate.
    • Maybe the only option isn’t just students working together at the board or at their tables, but also in writing using Google Docs, post-its, etc.
  • Novel problem-solving is required, and we provide literacy scaffolds, concept organizers, on-ramps, explicit instruction about strategy, and student choice of difficulty level
  • Math is treated as a way to have fun with your brain in the same way that puzzles are fun, and direct applications and connections to students’ lives are regular features of our curriculum
  • We keep the complexity of the math we ask students to learn, and provide multiple representations beyond text to support decoding and sense-making
    • This might mean we need to step our visual design game!

Building an Anti-Racist Math Program

My school’s math department has been working hard to articulate a vision for an anti-racist math program and to take action to realize that vision. I’m grateful to our talented communications department for highlighting this ongoing work!


As a study of quantity, structure, space, and change, mathematics might not be top of mind as a discipline in need of an anti-racist lens. However, from the origin of the number system we use today to the mathematicians who are studied and revered, elements of white-washing and exclusion can be found throughout the field. 

Packer’s mathematics department has been exploring ways to address race and equity in both its curriculum and its teaching practices. This is a reflection of the teachers’ own interests, as well as a response to Upper School students’ call this past summer for the re-evaluation of curriculum throughout the division.

“There’s a hunger for how math can be used to make the world a more equitable space, how it can be used to understand racism and other dynamics of oppression and how it can be used…as a tool for social justice,” said Interim Upper School Math Department Head Tom James. 

Last spring, the math department held an anti-racist math teaching forum, where they identified issues of racial inequity and pulled together a collection of anti-racist resources. The group has been working to articulate a vision for what anti-bias and anti-racism looks like in their department. While the group has addressed gender bias in the past, Tom said, “it’s important to focus anti-bias work on race specifically because that’s an area of growth for us and something that needs to be named explicitly.”

The teachers’ work, coupled with the feedback shared by the student-led Change Committee, has brought noticeable changes to what is taught in Upper School math classes and how. The department employs a “journey partners” model to support educators in this work. In pairs or trios, math teachers work together to develop and implement anti-racist curriculum changes in their respective classes. Through regular check-ins, the partners serve as sounding boards for new ideas and as accountability buddies in the process of unlearning racism and bias. …

One noteworthy focus is the way in which teachers are attempting to shape classroom culture. While collaboration is common in other disciplines, Tom said students sometimes approach math with a strong focus on personal rather than collective understanding—a mindset that is prevalent in dominant culture. …

“A lot of teachers in our department are very deliberate about emphasizing really strong collaboration, teaching those skills explicitly,” said Tom. “[They’re] sending the message very directly that [math] class is not just about you being in it for yourself and your own understanding, but actually about building collective understanding.” …

In Tom’s Advanced Algebra II and AT Symmetry and Transformations courses, he is working to humanize mathematics by having students read stories about the professional journeys of mathematicians across identifiers. “It’s important to us that we’re not tokenizing mathematicians of color, so we’ve looked at female mathematicians, LBGTQ+ mathematicians, mathematicians of color, of a number of different backgrounds,” said Tom. Classroom discussions explore the positive and negative forces these professionals have faced and how those same forces show up in Packer’s classrooms. 

These examples represent just some of the immediate ways in which the mathematics department has answered the call to become anti-racist. On-going efforts will include additional shaping of the curriculum as well as further exploration about how math is taught, how work is graded, and how to achieve proportional representation across course levels. The goal of this work is to create an inclusive and supportive learning environment so that all students will be equally engaged in the study of mathematics.

“We really want students to understand that acquiring mathematical knowledge and expertise is in itself an act of personal and collective liberation,” said Tom. “These are the tools you need in order to both understand and come to terms with [the idea that] society is not the way we want it to be and to be able to make a plan toward building a more equitable and just society.”

Takeaways from my First Summer at the Klingenstein Center

(I wrote most of this post before the school year started but didn’t get around to publishing—using the end of winter break to finish it off!)

I’ve been teaching pretty much non-stop since I started in 2012. Every summer I’ve been an instructor at a couple different summer academic programs (a.k.a. nerd camps). But this summer, I took a break from teaching to work on a master’s program in educational leadership. This meant taking five classes in six weeks:

  1. Negotiations
  2. Ethics & Education
  3. Private School Leadership
  4. Instructional Leadership
  5. Research Methods in Education

Key learnings


Pearls of wisdom

  • You can build or lose trust with very, very few words. Pay careful attention to the messages you signal to others by the details you choose to share about yourself. What do you emphasize, and how is that received?
  • An excellent negotiator creates value for everyone while protecting their own interests.
  • Attend to process—every negotiation has one, regardless of whether it is stated explicitly.
  • Time is one of the most critical aspects of negotiation. Life is timed and open-book. Time pressure can be a major enemy to a successful negotiation, so a good first move is to pause and consider process.
  • Too often, we focus on our positions rather than our interests. Always look for ways to reframe the problem.
  • It is important to prepare for negotiations and difficult conversations.
  • There are many techniques for blocking questions, such as changing the subject, over- or under-answering, answering only the helpful part, deflecting with humor, responding with a probing question, or ruling the question out of bounds. To get an answer, ask four questions about the same thing, since most people can only block once or twice.

Tensions in a negotiation

  • Creating vs. distributing value
    • There is a pervasive cultural assumption that negotiations are win-lose, with an adversarial relationship between parties.
  • Empathy vs. assertiveness
    • Our culture frequently holds these as contradictory, but they are better thought of as orthogonal. We should strive to both advocate for our own interests while seeking to understand the other person’s interests. Making your needs known is key—this invites the other person to problem-solve with you.
  • Principals vs. agents
    • In a school setting, many different actors can try to claim to be the agent acting on behalf of the principal (the student).
  • Being yourself vs. playing an institutional role

Difficult conversations

  • You don’t have to agree with another person’s story, but you do have to understand it.
  • Attend to your own feelings and decide if you want to share them. Make sure to anticipate others’ feelings and which might be hard to hear. (For example, hearing disappointment can be very challenging.)
  • Intentions can be positive, negative, or neutral. When you’re only perceiving negative intentions, brainstorm neutral or positive intentions that could be underlying the other person’s actions.
  • It’s critical not to conflate your work identity with your personal identity.

Ethics & Education

Pearls of wisdom

  • Case studies are an immensely powerful tool for identifying our values and drawing out our hidden assumptions as educators.
  • Studying ethics helps us understand why there is conflict and misunderstanding around decision-making in schools.
  • There is a key difference between reading a text to recover meaning (the “hermeneutics of recovery”) versus comparing it against an existing theory (the “hermeneutics of suspicion”)
  • The more we understand the nature of social power, the more we can understand human practices and thus change our actions, and moral agency becomes increasingly possible.

Private School Leadership

Pearls of wisdom

  • There’s an inherent tension in the model of “leader as learner”: if I still have so much to learn, how could I possibly think I’m ready to lead? This is normal.
  • Personal transformation is part of change leadership. If we can’t change, our schools can’t change!
  • Culture eats strategy for breakfast.
  • A useful metaphor for learning and growth is thinking about “maps and compasses.” When we operate using a map, we have clearly defined goals and the process feels comfortable. With a compass, we can go off the beaten path but there is more discomfort with the process.
  • The shift to thinking about adults as learners is fairly new in schools (~last 15 years), and especially independent schools.
  • It’s important to explore what it means to be a listening leader. Consider there ways in which listening is both a situated (i.e., context matters) and a fully embodied practice.
  • In decision-making, schools seek coherence, not consensus.
  • A soundwalk can be a valuable tool for understanding the life of the school in a new way by allowing you to pick up on what often goes unnoticed. Learning walks are a related tool to keep the pulse on the day-to-day. You can ask yourself:
    • How did I hear the mission of the school? What did I hope to hear?
    • What is the heard culture? The hidden culture?
    • What affirms or contradicts school values?
  • When analyzing a text, ask yourself:
    • What assumptions is the author making?
    • What do I agree with?
    • What do I find myself wanting to argue with?
    • What could I aspire to?
  • An excellent question to ask of any group of people is, “What went well?”
  • When introducing a new tool (such as the creation of an asset map) and folks are asking, “Is this right?”, it can be useful to offer the reframe, “Was this useful to you?” to emphasize that it’s not about following a formula, but about identifying and helpful practices and personalizing them.
  • There’s a difference between task leadership and social-emotional leadership. Both are critical components of change leadership, because there’s a gap between wanting change and actually going through with it.
  • There’s value in unfinished work—see the body of scholarship studying unfinished artwork.
  • When building out new structures, you need to establish rapport, which comes from presence and consistency.
  • Projects and core organizational change are managed very differently. But both need a vision and an elevator pitch others can echo.
  • As a leader, every word you use matters. For example, start with “anti-bias” if folks can’t hear “anti-racist.” It’s not cowardice, it’s a plan for moving to the next level of good.
  • When walking into a difficult meeting, pause to consider your purpose and how you intend to conduct yourself.

Instructional Leadership

Pearls of wisdom

  • It’s crucial to distinguish between “effort” and “effective effort.” We have to explicitly teach what the latter looks like inside and outside of class.
  • Most schools are not set up to promote long-term learning.
  • When asked to reflect, most people just describe or narrate what happened. We have to explicitly teach what meaningful reflection looks like. People learn from reflection on an experience, not the experience itself. This is because memory is the residue of thought (see Daniel Willingham’s writing).
  • Feedback given to students is only useful insofar as students engage with it to increase their level of understanding.
  • The reason looking over notes is ineffective is because you’re recognizing, not retrieving. We can think about forgetting as retrieval failure.
  • Struggle is key to deep, enduring learning. However, struggle brings up emotion, and we have to attend to it. First, we need to build up a culture of trust and intentionality.
  • Forgetting is essential to learning. We need to remind students that this is a normal part of the process. Re-learning goes much more quickly and reinforces those connections.
  • A great interview question is, “If you ran the new teachers program, what three principles of learning would you make sure everyone grasped?”
  • Put the big ideas of a course out early, then provide spaced practice and interleave them.
  • Expertise is not gained simply through hours spent—it’s about deliberate, effective effort.
  • When planning for class, practice taking each individual student’s perspective rather than the students’ perspective as a whole.
  • In terms of feedback, we should be giving baseball cards (with have multiple measures), not grades!
  • Design lessons to require students to construct meaning for themselves. (See the memory demonstration described here.)
  • Effective feedback doesn’t change the underlying product (external). It changes the learner’s understanding (internal).

Questions to ask students

  • What did we learn yesterday?
  • What are the most important ideas you will be tested on?
  • What for you is worth remembering?
  • What would I ask you if you were stuck in class?
  • What would I ask you if I were there with you? (reflective question for students to consider while working on homework)

Metacognitive questions

  • What do I do when I’m stuck?
  • How does this connect to prior knowledge or past experience?
  • What alternative approaches are available?
  • How do I know that I understand?
  • What am I noticing? What am I wondering?
  • If I were to do this again, what would I do differently?
  • How effective was my planning and my effort?
  • When did I pause to evaluate how I was doing?

Research Methods in Education

Pearls of wisdom

  • In general, we are not sufficiently critical of summaries of educational research.
  • (sorry, ran out of time!)

Readings to revisit

Beyond Winning. The chapters specific to lawyering can be skipped.

Source: Mnookin, Robert H. Beyond Winning Negotiating to Create Value in Deals and Disputes. Harvard: Harvard University Press, 2004.

Difficult Conversations. This is worth rereading a chapter or two every month.

Source: Stone, Douglas, Bruce Patton, and Sheila Heen. Difficult Conversations: How to Discuss What Matters Most. Yuan Liou Publishing, 2014.

Epistemic Injustice: Power and the Ethics of Knowing. The concept of “testimonial injustice” is useful for naming systems in which marginalized individuals’ voices are met with doubt and disbelief. The first chapter of Fricker’s book very carefully builds up definitions along with examples and non-examples. I’d like to read the rest of this book where she develops the idea of testimonial justice.

Source: Fricker, M. (2011). Epistemic injustice: Power and the ethics of knowing. Oxford: Oxford University Press.

Caring: A Feminine Approach to Ethics & Moral Education. In the first chapter of this book (“Why Care About Caring?”), Noddings develops the concept of “care ethics.” Whenever I get bogged down in the details of math pedagogy, this chapter will be an important resource for humanizing the process of education and our students’ needs as human beings.

Source: Noddings, N. (2003). Caring: A feminine approach to ethics & moral education. Berkeley: University of California Press.

Democracy and Education. In the chapter “Education as Growth,” Dewey names a mechanism through which much of the deficit thinking about children and adolescents arises. When we think about students as unfinished adults, we necessarily close ourselves off to the powers and strengths unique to young people. This chapter is worth rereading every year.

Source: Dewey, J., & Boydston, J. A. (2008). The middle works, 1899–1924. Carbondale: Southern Ill. Univ. Press.

Educator Competencies for Personalized, Learner-Centered Teaching. This resource enumerates and unpacks the skills, understandings, and dispositions teachers need to enact student-centered practices. I’ve previously focused on making sense of the instructional and cognitive competencies, and this report also dives into the interpersonal and intrapersonal dimensions of student-centered teaching. I could see myself using this to ground myself as I analyze the competencies that are present in myself and the educators I’m working with.

Source: Jobs for the Future & the Council of Chief State School Officers. 2015. Educator Competencies for Personalized, Learner-Centered Teaching. Boston, MA: Jobs for the Future.

Leadership for Increasingly Diverse Schools. An equity audit can be used to identify and address issues concerning proportional representation in various areas of the school. The suggested timeline for accomplishing goals is about three years, and it’s crucial to link these goals to student learning. This tool provides a clear structure for meeting a particular type of school-based inequity head-on.

Source: Theoharis, George, and Martin K. Scanlan. Leadership for Increasingly Diverse Schools. Routledge, 2015.

Organizational Culture and Leadership. The opening chapters from this resource pushed my understanding of organizational culture forward by laying out three levels of culture. I particularly appreciated the explanatory power of invisible but deeply influential underlying assumptions when we encounter structures and behaviors that don’t align with our publicly stated beliefs. I could see myself appealing to the lily pond metaphor to explain this idea to others who are invested in change. I’d like to read more out of this.

Source: Schein, Edgar H. Organizational Culture and Leadership. 5th ed., Wiley, 2017.

Dilemmas of Educational Ethics: Cases and Commentaries. Our ethics professor suggested we read this. Reading and thinking through case studies was one of my favorite activities of the class, so I’m looking forward to reading this.

Source: Levinson, Meira, and Jacob Fay. Dilemmas of Educational Ethics: Cases and Commentaries. Cambridge, MA: Harvard Education Press, 2016.

Practices to implement

Visible random groupings.

Retrieval practice.

Spaced practice.


Connecting to prior knowledge.

Coaching. By offering help to my advisees in a non-directive way, I can honor their autonomy and ability to solve their own problems. There’s a toolkit I can use that focuses on the now and the future and spends more time on the values, wants, passions, and motivations of the person I’m working with. This is more effective than jumping right into the logical and logistical domains. I can also practice asking more “what” than “why” questions to keep the focus on coaching the person, not the problem.

Sharing hopes/fears anonymously.

Listening walks. Slow down in the hallways.

Teaching metacognitition explicitly.

Teaching executive functioning skills explicitly. For example, have everyone write on an index card the “movie script” of how they did their homework. Where were they physically? What time was it? Where was their phone? What resources did they use? Collect, shuffle, and redistribute the cards. In groups, compare the approaches. Share out the ones that sounded most effective.

Why I “Leaked” My Logarithms Test

In the somewhat rushed lead-up to our unit test on logarithms in my Advanced Algebra II class, I realized I was asking students to study from an enormous corpus of practice material I’d provided. I could tell that this was overwhelming, so I got the idea to give the test to my students in advance.

Not the full test—just a redacted version:

I had dropped a hard copy in one student’s open backpack during a break, and then I made a big show of asking him to pull out the paper marked “top secret.” Then I was all, “ohmygoodness what’s that?!” and “Did I accidentally post a copy of the test to Google Classroom?” My students were thrilled. (I did make it clear that this was a one-time thing!)

Even though I had already provided a list of learning targets, I had two main goals by “leaking” the test itself:

  1. Reduce student anxiety by providing the parameters of the test (number of problems, layout, types of question prompts, etc.) very explicitly
  2. Focus students’ studying on the ideas I considered most important

This was a spur-of-the-moment idea, but definitely one I would use again under the right circumstances!

Understanding Pushback on Inquiry-Based Math Learning

I teach in a progressive math department committed to inquiry-based learning, and I’ve been working on sharpening how I talk about why I believe in this approach.

A conversation will often start when I hear a student or a parent say one of these:

  • “I learn better with a traditional approach where we learn something and then practice it a bunch.”
  • “My kid needs a teacher who actually teaches.”
  • “I get that some students can figure out how to do math without being told how, but my kid just needs you to show them the steps to follow.”
  • “My teacher made us figure most of the material out on our own last year, which was frustrating. Instead of telling us the answer, they would just ask us more questions!”

It’s hard to imagine what an inquiry-based math class looks like if you’ve never been in one. I suspect that if you’re a parent and your child is experiencing math-related stress or frustration, there are a few reasons you might order if this way of learning is best for your child. These are some of the images you might conjure up:

  • Students working on a hard problem they’ve never seen before as the teacher watches silently from the side of the classroom
  • Students “learning” an incorrect approach and leaving the class with serious misconceptions
  • A teacher who de-emphasizes procedural fluency to the point that students while students can explain a concept or idea in broad strokes, they can’t actually solve any math problems
  • Students who are stuck, frustrated, and angry

When I think about an inquiry-based math class looks like, I see something very different:

  • Students engaged in problem-solving requiring them to notice patterns and wonder why they occur
  • Students having rich conversations with one another about math under the guidance of a thoughtful, attentive teacher
  • Students working hard to satisfy their own curiosity while being simultaneously “egged on” and supported by a caring teacher

Ultimately, every parent comes to these conversations about math education with the goal of seeing their child successful and happy. Keeping this in mind is critical for addressing their concerns.

Over at the IBL Blog, Stan Yoshinobu writes about the feelings can arise when students haven’t bought into the value of productive struggle:

Mindsets are at the core causes student buy-in issues. When students don’t buy it, it’s often because they don’t like being stuck or that being stuck implies there is something wrong with the problem, them, or the teacher or all of the above and more.

I was digging more into the IBL Blog and discovered this lovely metaphor of “I don’t learn this way” as the tip of the iceberg:

Iceberg 1

This imagery reminds us to look beneath the surface to find the source of resistance to inquiry-based learning. Only then can I speak to my beliefs and vision for my students.

I believe in inquiry-based learning because I think students learn math best when…

  • … they have a chance to explore ideas on their own before being told what the “best” strategy is for solving a problem.
  • … their mathematical ideas are affirmed and valued, even when they’re not fully clarified or correct yet.
  • … they are given opportunities to develop intuition before technical vocabulary and formalism are introduced.
  • … they are invited (and expected) to look for patterns and are regularly asked, “What do you notice?”
  • … teachers explicitly and implicitly communicate to students that mathematical knowledge is not isolated to select “experts” (like math teachers) who then dispense it to others, but rather that mathematical creativity is broadly accessible.
  • … they are invited (and expected) to pose questions of their own and are regularly asked, “What do you wonder?”
  • … teachers explicitly and implicitly communicate to students that the teacher’s mathematical questions are not the only interesting ones, but rather that the ability to ask a rich, thought-provoking question about math is broadly accessible.
  • … they see their peers employ a variety of successful strategies to solve a problem and are encouraged to understand multiple approaches.
  • … they are given opportunities to communicate their understanding to their classmates and receive guidance on how to improve their oral and written communication.
  • … there are structures to support collaboration with their peers.
  • … they spend most of their time in the sweet spot of productive struggle.
  • … they are given opportunities to apply the fruits of their intellectual labor during focused practice, building mastery and supporting long-term retention.

I’ll conclude with some prose from Joshua Bowman, who recently shared a preface he wrote to an IBL course. Here’s an excerpt:

… [T]he success of the class will depend on the pursuit of both individual excellence and collective achievement. Like a musician in an orchestra, you should bring your best work and be prepared to blend it with others’ contributions.

… Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

Interesting Numbers v2

This is a follow-up post to my first day math problem from last year.

I’ve only posted once in a while over the past few years, but I’m very glad I spent the time last year to write about how my first day math problem went. After re-reading my post from last year, I was able to rapidly load the problem and how it went back into my memory, and I decided to make some of the changes I had written about.

Here’s the updated version of the problem, which adds scaffolding and will hopefully keep groups moving along who aren’t as comfortable with this type of open-ended exploration.

And here’s the checklist I’m going to use to keep myself organized as I walk around and monitor each group’s progress. (This idea comes from 5 Practices for Orchestrating Productive Mathematics Discussions by by Margaret S. Smith and  Mary Kay Stein.)

Feel free to pass along any feedback or take this a step further!


Math is Like a Pomegranate

This is my contribution to The Virtual Conference of Mathematical Flavors.

“Math is like a pomegranate—intimidating, and kinda scary looking at first, but also incredibly fascinating and vibrant.”

In order to figure out what flavor of math I’ve been serving up in my classrooms over the past six years, I’m going to take a stab at Sam Shah’s idea of working backwards from what students have written about their experiences in my math classes. (Spoiler alert: the answer is apparently pomegranate; who knew?!) Of course, not all of my students have had transformative experiences and others have straight up had a bad time. But right now, I’m going to focus on the students who have been positively impacted in order to articulate what the best implementation of my ideals has felt like.

But to be honest, it feels way scarier to share the positive things students have written about me over the years than anything critical. When I was younger, I used to brag and show off; I thought that if people knew about all the things I was good at, they would have to like me. Once I figured out that this is not how relationships work, the pendulum swung hard in the other direction for me. I grew increasingly uncomfortable accepting compliments and I minimized my achievements, working to avoid even the appearance of self-promotion. It’s an ongoing struggle to get right-sized, but lately I’ve begun to internalize the idea that being excessively diminutive is its own barrier to connection.

So with that confession out in the open, here are some of my favorite reflections students have written. (The title of this post comes from one of these!)

A few years ago, I started asking students to write advice to next year’s students. And when I remember, I make sure to share this advice once the new group arrives. Here are some examples of what my students have written.

Reading these, some of the core beliefs I bring to my teaching are apparent to students in different courses and at different schools. I hope that my students internalize them as well:

  • Math is something to get excited about. I want my math-skeptical students stay curious about why certain people openly love math, and I want them to find reasons of their own for loving math. I’m not shy about telling them when I get goosebumps when talking about math, and I don’t hesitate to make corny memes—and be super proud of them—to show how highly I think of a mathematical idea or how much their understanding has grown. (See the “extending the definition of sine and cosine” trigonometry meme I made this year.)
  • Math is a playground for creativity. You can ask and answer your own questions. There are games to make up and play, connections to establish, new approaches and representations to develop, and structures to create and explore. Some of my favorite moments come when a student (or even better, a group of students) comes up with a solution pathway I’ve never considered or notices a pattern I’ve never seen before.
  • Engaging with math is an opportunity to build confidence. No matter where you are in your mathematical journey, there are ideas to wrestle with in math that are hard, but not impossible. It’s like an infinite gym for your brain with an endless selection of workouts. Realizing that you can do something you’d previously found scary, intimidating, or intractable is incredibly empowering.
  • Expect and welcome obstacles. In math, there’s nothing wrong with being wrong, and getting stumped is an invitation to push your thinking deeper or to try something else. For this reason, I react like it’s the most normal thing in the world when a student tells me their approach didn’t work or when they don’t know what to do next. Sometimes they get a little peeved when I don’t rescue them right away, and that’s okay!
  • Math is especially enjoyable when shared with others in a caring and trusting community. Despite the cultural trope of the solitary mathematical genius, there is no rule saying that math has to be a solo sport. The process of guiding another person to a mathematical idea you’ve uncovered requires patience, clear thinking, and careful consideration of what the other person is comprehending. Similarly, the practice of asking for and receiving guidance requires humility, self-awareness, and careful articulation of what you’re understanding and where you’re feeling fuzzy. To help with this, I treat the word obvious like a cuss word in math class, and students usually buy in pretty quickly!

As a final thought, it feels liberating to put this out there in a less formal way than I’ve articulated aspects of my educational philosophy in the past. For comparison, this is what I’ve used in previous job searches, and almost all of it was composed in 2014, before any of the student reflections above were written.

I still stand by everything in this document, but I really appreciate the type of unencumbered sharing Sam’s framing of the prompt for this virtual conference has facilitated. In other words, asking “What mathematical flavor are your serving up?” rather than “What’s your theory of mathematics education?” seems more likely to inspire folks to share a healthy multiplicity of approaches instead of competing formal philosophies. And, it gives us an opportunity to celebrate our wins instead of worrying about all of the things we’re not doing.

Launch Problems for Optimization Unit

I’ve had the pleasure of teaching standard-level calculus with Sam Shah this school year, and recently we’ve been working with the students on optimization. Rather than starting with the canonical calculus optimization problems, we decided to jump in with maximizing the area of various shapes under curves:

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I think I first saw a problem like this in a textbook called Advanced Mathematics by Richard G. Brown (page 167, #12):


I’ve always been fascinated by these types of problems because they’re easy to understand and make guesses about but often have unexpected solutions. I wanted to use Desmos to bring this problem to life, so I put together a Desmos activity and companion sheet (.docx version) to look at four of these problems.

Students worked in groups, and they while they all had their own screen, they were expected to move together and come to consensus on the best possible shape before moving on. This fostered lively debate among students as they tried different shapes and improved their guesses by manually calculating areas.

For the “isosceles triangle under a parabola” problem shown above, there were a variety of responses, but there was convergence around the optimal triangle (whose vertex in Quadrant I is (\sqrt{3}, 6):


Students were also asked to identify the constraints of their shapes before moving on to the next challenge. After locking in their guesses for each challenge, students had to really dig in to the second challenge (the “rectangle under a parabola” problem). This time, we provided a slider that would calculate the area of the rectangle for them as they changed the x-coordinate of the vertex in Quadrant I:

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Student were expected to record these data points on the companion sheet to form a sketch of the area function:


Many students initially assumed that this area function was going to form a parabola, but after plotting more points, the class decided that it couldn’t be because of the lack of symmetry. But this function has a peak—how could they find it? This is where the calculus kicked in!

After carefully taking the derivative of the area function, setting it to 0, solving, and determining the dimensions of the best possible rectangle, students were able to finally determine which group came the closest with their initial attempt. They were also ready to tackle the remaining challenges on the second part of the companion sheet (.docx).

Before jumping in, I was also able to recognize groups for getting the closest to the best possible shape while also pointing out that they could do even better!


All in all, I thought this was a super fun way to kick off the optimization unit while keeping engagement high and providing valuable practice with the non-calculus algebra that can trip students up. Most importantly, calculus was positioned as the aspirin for the headache posted by the Desmos activity.

I would love to see these types of optimization problems become more popular!


P.S. See Sam’s post on a lovely lesson he put together called POP! Popcorn Optimization Problem, which is a way more engaging way for students to tackle traditional optimization problems that usually look like this:


First Day Math Problem 2017–18: Interesting Numbers

I decided to re-sequence my start-of-the-year activities and to lead with a low-floor, high-ceiling problem in assigned random groups of three or four students.

Here is the problem, which comes from Phillips Exeter Academy’s Math 1 curriculum:


I told the groups to figure out everything they could about this situation with prompts like, “What do you notice about interesting numbers? What do you wonder about them?”

As I watched twelve groups of students explore this problem over three classes, I began to see students latch onto different aspects of this problem. All of these questions and discoveries are inter-related, so I’m writing them down now so that I can map them out in the future.


  1. Which numbers up through 20 (or so) are interesting?
  2. Why are powers of 2 interesting?
  3. Are powers of 2 the only interesting numbers?
  4. Are there any interesting odd numbers?
  5. What happens when I sum any two consecutive positive integers?
  6. What happens when I sum any three consecutive positive integers?
  7. If is odd, what happens when I sum any n consecutive positive integers?
  8. If is even, what happens when I sum any n consecutive positive integers?
  9. How can I decompose any odd number?
  10. How can I decompose any multiple of 3?
  11. If is odd, how can I decompose any multiple of n?
  12. How can I decompose any even number?
  13. Is there a general algorithm for decomposing any number?
  14. How many ways are there to decompose a given number?


  1. All powers of 2 are interesting.
  2. Only powers of 2 are interesting.
  3. No odd numbers are interesting.
  4. The sum of two consecutive positive integers is odd.
  5. The sum of three consecutive positive integers is a multiple of 3.
  6. If n is odd, the sum of n consecutive positive integers is a multiple of n.
  7. If n is even, the sum of n consecutive positive integers is n/2 more than a multiple of n.
  8. There is an algorithm for decomposing even numbers.
  9. There is exactly one way to decompose a prime number greater than 2.
  10. The powers of 2 are exactly the whole numbers without odd factors.

There was a split between groups that started by trying to answer (the very natural) question #1 (and thus getting to realizations #1 and #2) and those that started by generating and then trying to answer questions #5 and 6 (and thus getting to realizations #4 and #5). There was also one group in one class that decided to explore the sum of the first n consecutive integers (i.e., they wanted to know about the triangular numbers).

I think I will definitely use this problem again, with perhaps a bit more structure and guided mini-explorations along the way as groups arrive at various questions and realizations. It would probably be worth making a checklist for each group to help keep me organized as I keep tabs on each group’s progress.