My 2018 NCTM Annual Meeting Conference Proposal

We’re two weeks removed from the 2017 NCTM Annual conference, which means we’re also two weeks away from the 2018 proposal deadline.

I just submitted mine. I’ll share it here (with some light commentary) in case anyone’s interested in my process or proposal.


Websites, browsers, the Internet. This stuff is great, and it usually works the way it’s supposed to. However, sometimes a site will crash or my router will restart for no particular reason. So I’ve made a habit of drafting conference proposals elsewhere, then copy/pasting once it’s all ready to go.

(This also makes it easier to share if you’re looking for feedback from a colleague before you hit that “finalize” button.)

Here’s my workflow for putting that draft together and ultimately sending it off into NCTM’s servers:

  1. Go to, login, and click “submit a proposal.”
  2. Pull categories (e.g., Title, Description) and requirements (e.g, max character count) into a document via copy/paste.
  3. Write the title.
  4. Write the description.
  5. Write everything else.
  6. Go back to
  7. Paste my responses in the appropriate boxes.
  8. Preview (here I take advantage of the print feature to save a PDF copy for my records).
  9. Press “finalize” and hope for the best.


Here’s my proposal, minus some minutiae (e.g., whether I need a document camera) plus some other details (e.g., connections to NCTM’s Principles to Actions).

Categories are in bold. Details are in italics. My responses/choices are indented.


Type title as it should appear in the program book. Your title should not be all capitals or all lower-case. Limited to 100 characters.

Applying the Five Practices to Visual Patterns

Description of Presentation

Write a concise, specific description of the essential content of your presentation. On acceptance of your proposal, the description will be printed in the program book, subject to editing by NCTM. Use appropriate capitalization. Limited to 350 characters.

In this session we’ll explore a rich context for making connections between multiple representations: visual patterns. Using Smith and Stein’s Five Practices as a guide, we’ll discuss best practices for facilitating classroom discussions around visual patterns, with special attention given to selecting, sequencing, and connecting student work.

Participant Learning

Write the participant learning outcomes of your presentation, including an explicit description of what participants will learn. Please also provide an overview describing how time will be allocated during this presentation. Limited to 1000 characters.

Participants will learn: (1) how to use visual patterns to build arithmetic and algebraic thinking while promoting reasoning and problem solving, (2) how to use the Smith and Stein’s five practices of anticipating, monitoring, selecting, sequencing, and connecting to facilitate productive mathematical discourse, and (3) how to use visual patterns to connect multiple representations including visual, verbal, numerical, algebraic, and graphical.

Participants will spend their time: (1) wearing their “student hats” as they work through a pair of visual patterns tasks, (2) wearing their “teacher hats” as they discuss the student thinking and teacher moves on display during these initial explorations and discussions, (3) engaging in the “anticipating” stage of the five practices as they imagine the various approaches and solutions students will generate, and (4) wearing their “authoring” hats as they create their own visual pattern task.

Focus on Math

What is the key mathematics content that is a focus of this presentation? Be specific. Limited to 500 characters.

Among other things, visual patterns provide a bridge from numerical thinking to algebraic thinking. In this workshop we’ll focus on establishing and strengthening that bridge. Participants will make connections between numerical expressions like 3 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 and 3 + 2 • 10, and algebraic expressions like 3 + 2 • n. Participants will explore recursive relationships and functional relationships in both linear and quadratic settings.

Interactive Workshop

How will you use the provided tables to create an interactive workshop? Be explicit about what participants will do together at tables. Limited to 750 characters.

The success of the session depends on engaged participants, smooth collaboration, and vibrant discussion. The table setting of an interactive workshop offers the most effective setup to support these elements. At their tables, participants will: (1) work through visual patterns tasks with paper, pencil, and manipulatives, (2) reflect on their experience in small groups before sharing out in the larger discussion, and (3) work together to create their own task.

Workshop Audience

6 to 8

(Quick note: I wish we could select more than one grade band. The content in the session is really geared to 6-12. But alas, there’s a one choice limit.)


Teaching, Learning, and Curriculum: Best Practices for Engaging Students

Strand choices (and descriptions) are available here.

Equity and Access

How does your presentation align with NCTM’s dedication to equity and access? Limited to 500 characters.

This session will equip participants will skills and strategies to support their work in ensuring that all students have access to a challenging mathematics curriculum, taught by skilled and effective teachers. We’ll also draw out principles for designing and facilitating effective learning experiences that can be applied to a wide range of topics in K-12 mathematics. Participants will also gain access to a large collection of free, ready-to-use visual patterns tasks.

Connection to NCTM’s Principles to Actions:

  • [2] Implement tasks that promote reasoning and problem solving.
  • [3] Use and connect mathematical representations.
  • [4] Facilitate meaningful mathematical discourse.

Your Turn?

I hope that proves helpful to a few folks. Maybe you’ll carve out a couple of hours this week to submit your own proposal? Either way, I hope to see you in Washington, D.C. in April 2018.


How Many Squares?

I had dinner with the kiddos at Round Table tonight. (Side note: yum.)

Must’ve had squares on the brain (thanks Dan and Anna!) because as soon as I sat down with the receipt I folded (and unfolded) it like this:

And then I asked Ainsley (4): “How many squares do you see?”

She counted the top row: “1… 2… 3… 4…” Then a long pause, followed by (pointing along the bottom row): “5… 6… 7… 8…”

We both stared at the receipt for a little while longer. Neither of us spoke for a bit. Then—for better or for worse—I broke the silence, tracing the perimeter of a larger square consisting of a 2-by-2 array of smaller squares. She helped me count this 9th square plus two more just like it, and we landed on a total of 11.

It’s hard to tell if those last three squares were lost on her, even after we traced them together. In all likelihood, they probably were. Anyway, that’s not the point of the post. (Nor is her original counting of eight, two rows of four at a time, though I’m confident there’s more than enough material for a blog post in the “how do you see it” conversation.)

A Familiar Problem

Here’s where I am headed with this.

I found myself recalling the classic problem “how many squares on a chessboard?” And then I wondered, “how many squares on an n-by-n board?” I’ve explored both of these questions before, and while I cannot recall the generalization off the top of my head, I’m confident I could find an explicit formula if I tried.

A New (to me) Problem

That sequence of thought led to a new question, or rather, a question that’s likely been posed many times by others but is brand new to me:

How many squares on an m-by-n board?

I’m sharing it here sans answer because I don’t yet have an answer (and it’d be no fun to spoil your fun in tracking one down). But I’m excited to start exploring. It may not be the sexiest or most challenging problem in the world, but it’s grabbed my attention nonetheless.

And if I’ve piqued your interest, I’d love if you gave it a try as well.

In which case, drop a line in the comments describing your approach, your answer, and (bonus!) another question about rectangles/squares/boards that interests you.

Beyond Explaining, Beyond Engaging

In my first few years in the classroom, I held the notion that the best way to improve as a teacher was to hone my explaining skills. I figured that if I could explain things more clearly, then my students would learn more.

It only took me a few years to realize that this philosophy of personal development was woefully incomplete. (Quick learner, right?!) So I turned my attention to a more noble pursuit: engaging my students.

In version 1 of this approach, I tried to find ways to get my students to pay more attention to my better-than-they-used-to-be explanations. The net result? Nothing much changed.

In version 2, I put my energy into engaging students not with explanations, but with mathematics. I tried—and still try—to create opportunities for students to engage productively with a problem or a concept. As with everything else in my life, it’s a work in progress. But I’ve seen some promising results.

Last week at NCTM, these percolating thoughts combined with several tweets, sessions, and conversations and led to this thought:

The best way to grow as a teacher is to develop my capacity to listen, to hear, to understand.

(Quick aside: I suppose I might replace the word teacher with husband, father, neighbor, colleague, or stranger, and the statement would still hold.)

This doesn’t mean that I’ll stop working on those other skills. But it does mean I have a new passion for learning about listening—really listening—to students and their thinking.

If you know of any books, articles, or blog posts that might help me along, please share them in the comments. Or maybe you disagree with my thoughts above as some combination of wrong or incomplete? I’d love to hear your pushback in the comments as well. Thanks in advance!


Should Teachers Post Objectives?

Last week I shared a little resolution: to post more questions on the blog.

I also put out a call for questions, and within hours Chase Orton delivered, sharing a question he had been discussing on Twitter:

Is it better to clearly post/state a lesson objective at the start, or to allow students to discover it during the course of the lesson?

Feel free to treat this as an always/sometimes/never, or to suggest a third approach not described above.

More Questions

I shared a pedagogy vs content knowledge question last week that made me realize a couple of things.

  1. I learn quite a lot from wrestling my own thoughts on a topic into blog post form. I learn even more when I pose a question and have the privilege of reading others’ thoughtful responses.
  2. I miss #slowmathchat.

And so, a tiny resolution: To post more questions on my blog. Stay tuned. I hope you’ll join in the fun.

P.S. I have a few questions of my own queued up, but I’d love to know what questions you’d love to see in the mix. Drop me a line (mjfenton at gmail dot com) with your contributions. Thanks in advance!

Pedagogy and Content Knowledge, Part 2

Yesterday I asked which is more important for a math teacher: teaching ability or content knowledge.

Several commenters reframed the question as pedagogy vs content knowledge. I found that shift helpful, so let’s run with that.

My original plan was to highlight a few comments here, and then offer a twist (or two) on the original question. But the post received way more than a few comments, and all of them are super thoughtful. Instead of including a few excerpts here, I’m going to push against conventional Internet wisdom and encourage you to go read all the comments. (Seriously. It’ll only take a few minutes, and it’ll be worth every second.)

New Questions

Alright, welcome back!

This discussion has raised a host of new questions for me. I’ll rattle off three of them here (and possibly more in a future post). Feel free to chime in on one or all. Or just lurk. That’s cool too. 🙂

  1. Is it possible to separate pedagogy from content knowledge?
  2. What are the best ways to develop one’s pedagogy?
  3. What are the best ways to develop one’s content knowledge?

Bonus Question

At the risk of damaging the discussion by asking too many questions at once, here’s one more:

  1. Suppose you’re involved in hiring for your department. What are the top three qualities/skills you look for in a math teacher? (Double bonus: Rank them.)


Teaching Ability and Content Knowledge

Last weekend my wife and I had the privilege of sharing a meal with Scott Miller, David Sladkey, and each of their wives. Awesome food, excellent conversation. It was the highlight of a great weekend at the DuPage Valley Math Conference.

At one point, David asked an intriguing question that sparked a lengthy discussion. I’ll share the question here in the selfish hope that I’ll be able to hear a few more opinions, and continue my own pondering (and learning) in the process.

Here’s the question:

Suppose a teacher gets to divide 100% between two categories: teaching ability and content knowledge. What’s the ideal breakdown?

Update: Several commenters reframed the question as pedagogy vs content knowledge. I find that shift in language helpful. If you prefer that formulation—or if you’d like to push back and offer your own related question—go for it.

Update: I want to learn more about this, shared by elsdunbar: “Deborah Ball describes a horizontal content knowledge as ‘[A]n awareness of how mathematical topics are related over the span of mathematics included in the curriculum.'”

Related post: These thought-provoking comments have sparked a new set of questions for me.

Update: Mark Chubb addresses this question—and a great deal more—in his recent post Professional Development: What Should It Look Like? It’s fantastic. Go check it out.

Some Thoughts on Modeling

Here’s a definition of modeling I’ve offered in a couple of conference sessions:

Describing the world with mathematics, in order to make reasoned predictions and decisions.

That I borrow heavily from Dan Meyer is readily apparent, especially when considering some of the activities I’ve created over the last couple of years (e.g., Charge!, LEGO Prices, Predicting Movie Ticket PricesMocha Modeling).

In each of those activities, students build a model in order to make a prediction, ideally a more precise one than the wild estimate I typically call for at the beginning.

A couple weeks ago, a colleague of mine (Jason Merrill) invited me to expand my definition a bit by considering how modeling often plays out in physics. Rather than a method for making precise predictions, modeling in the physics classroom (or laboratory) may sometimes offer a process for inferring material properties and physical constants.

Physics Teachers, Help Me Out

Do Jason’s comments resonate with your experience? If so, can you share any exemplar activities in that inferring properties and constants vein?

Math Teachers, What Say You?

How often does your modeling work serve as a means for making predictions? How often does it serve as a means for something else? How would you expand (or revise) the definition of modeling I’ve offered above?

Heartbreak, Relief, Shame

I started this blog to explore how I might become a better math teacher. Today I’m writing to explore how I might become a better person. A better husband. A better father. A better neighbor. A better stranger.

I have more questions than answers.

I’m not quite sure what to say, or how to say it.

Anyway, if you’re interested in following along as I stumble along, here goes.

Alton Sterling on July 5.

Philando Castile on July 6.

Lorne Ahrens, Michael Krol, Michael Smith, Brent Thompson, and Patricio Zamarripa on July 7.

Micah Johnson on July 7.

I’m halfway across the country. White and insulated. Unaffected. Yet entirely affected.

Trying not to return to “life as usual” so quickly this time. (Confession: That’s what I’ve often done.)

Trying to make sense of it all. (Spoiler: I cannot.)

Trying to learn mourning and lament. (Baby steps, thanks to my wife.)

Several days later, another news headline. A new tragedy. More dead.


I read further. It’s somewhere else. The Sudan, I think. Anyway, it’s not here. Not us.


And then…


Shame because I gave myself permission to view another human being as other and unworthy.

Other than myself, my family. Unworthy of my tears, my concern.

The better version of myself doesn’t look away. It mourns. It listens. It seeks to understand.

I’m not there yet. But I want to move in that direction.

Questions and Convictions

I’m not sure how to balance that desire to be compassionate against the reality that my conscience can’t bear the collective weight of human suffering, not even a fraction of it.

Nor am I sure how to balance the tension between compassion for others and caring for my family. (More on this in a future post, I think.)

But I am convinced of this: Christ is not honored when I look away from the plight of the orphan and the widow. He is not honored when I ignore the needs of others because it might cost me something.

So what does that mean for me? In the words of Francis Schaeffer, echoed often by one of my former colleagues, “How then shall we live?”

I don’t know. I’m in the process of figuring that out. But I do know that if my answer doesn’t honor the heart of James 1:27, if my answer doesn’t take into account the voices of the Old Testament prophets, who cry out loudly on behalf of the weak and the poor, then my answer doesn’t match God’s answer for how I should live in these challenging days.

Silent? No more.

I began tweeting at the end of 2012. I began blogging a few months later. Since then, nearly everything I’ve shared in this medium has been strictly related to mathematics and education. The balance? A few quips about my kids and running.

Up to this point, I’ve not written anything overtly political. In fact, for most of my adult life I’ve been so disillusioned by the American political landscape that I considered myself apolitical.

Also, beyond the first line of my Twitter bio—which reads Follower of Christ—I’ve shared almost nothing that springs from my faith, the very foundation of my life.

In a sense, I’ve been loud on mathematics/education/technology, and silent on everything else.

Over the past year, I’ve become increasingly uncomfortable with that silence.

Our professional, personal, and political thoughts are intertwined. For some time, I’ve been pretending that I can keep them separate in my own life. I cannot. Nor do I want to.

So as Mr. Trump imposes his own set of bans, I’ll lift one of my own. As I wrestle with how to live as a faithful follower of Christ in this strange new world, I will no longer wrestle silently. Where I see bigotry and hatred, I will stand and speak against it, especially if that bigotry and hatred spews forth from a position of power.

There’s a pretty good chance everyone reading this post will disagree with something I share over the coming months. That’s fine. I invite your pushback, your perspective. I’m open to dialogue. If you’d prefer I keep my thoughts to myself, that I just “stick to education” as some of my friends and colleagues have been advised, you’ll likely be disappointed and may want to give the unsubscribe/unfollow button a try.

I’ll leave you with some words that have been troubling me in a most helpful way:

“They have healed the wound of my people lightly, saying, ‘Peace, peace,’ when there is no peace.”

– Jeremiah 6:14

“In the end, we will not remember the words of our enemies. But the silence of our friends.”

– Martin Luther King, Jr.