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.

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.

Charge! v2 – Activity Makeover

Background. I’ll share an activity. Offer some ideas on what’s wrong. Invite you to share your own diagnosis/treatment. Then (end of week) share an upgraded version of the activity. (More details about this series are available here.)

Several Mondays ago, I shared the Desmos Activity Builder version of Charge!, a linear modeling task where students predict how long it will take for a phone to become fully charged. Here’s the diagnosis.

In that post I suggested that the Activity Builder version of this activity was inferior to its original slide-deck-driven version, in part because it struggles with these principles from the Desmos activity building code:

  • #5 – Give students opportunities to be right and wrong in different, interesting ways.
  • #8 – Create objects that promote mathematical conversations between teachers and students.

Featured Comments

Dave Johnston: “Can we give students more chucks of the data and give them an opportunity to revise their model? What if different groups of students had different data points along the way & they discussed the models they came up with?”

Nathan Kraft: “I’m trying to figure out what Desmos adds to the activity… This activity pigeonholes students into one way of doing it, and even for students in higher grades, I’d like to give that option to explore it differently… In the end, maybe this doesn’t work in activity builder.”

Elizabeth Raskin: “One of the beautiful things about 3 acts is allowing students to determine what information is important (to an extent) and what to do with it.”

Mark Kreie: “Utilizing the Classroom Conversation Toolkit w/ teacher pacing and pausing might be a benefit to this activity.”

My Upgrade

I find myself wondering what several commenters suggested: maybe the best version of this task doesn’t live in Activity Builder. I’m definitely open to that possibility.

That being said, I still wanted to see how far forward I could push the AB version.

I recommend opening up Charge! v2 while you read through the rest of the post. Here is what’s new:

  1. Fresh artwork, all throughout, including a video reveal. This doesn’t shift the pedagogy, or address the weaknesses above, but it adds a little polish to the existing structure. And when it comes to the video, that adds a little anticipation, which is often a good thing in a math classroom.
  2. Condensed opening. I love asking students “what do you notice/wonder?” But I haven’t figured out how to open an Activity Builder activity with that approach. So I’ve trimmed the opening to make room for more discussion later on, a la Principle #8 (“Create objects that promote conversations…”).
  3. “Informalized” the sketch on Screen 2. Goodbye axis scale. Goodbye grid. Goodbye sample data points. Hello increased variety of student responses. That’s the plan, anyway. I’m anticipating an uptick in variety here, ranging from type of graph (linear vs nonlinear) to features of specific sketches (e.g., steepness, starting point). And that could lead to some nice interplay between Principle #5 and Principle #8.
  4. Condensed middle. The core here is much tighter. It’s now: Build a model. Use the model. Interpret your model. This condensing improves our conversation-to-screen ratio, which has been a theme around Desmos HQ for the past few months. For so many activities—this one included—carving out time to explore what they’ve done on each screen is crucial. There just isn’t enough time to do a 20-screen activity and discuss it. By cutting the screen count in half (from 18 to 9), I’m hopeful that there’s sufficient room for those important discussions to unfold.
  5. Different ending. I pulled the “why do you think it charged that way” question forward to the new Screen 8, which makes room for a new extension on Screen 9. Note two differences: (1) the phone is different, and (2) the direction of the question is different. Instead of “given the charge, what is the time,” it’s reversed: “given the time, what is the charge?”

(Pssst. Did you see that video reveal?)

Lingering Concerns

Is v2 better than v1? In my opinion, absolutely. Is v2 perfect? Not even close.

I think this new version better addresses Principle #8 (“Create objects that promote conversations…”). Fewer screens. More room for conversation.

But I don’t think I’ve addressed the more difficult Principle #5: “Give students opportunities to be right and wrong in different, interesting ways.”

There’s still just one way to move through this activity. It still feels too scripted. Maybe it does live best outside of Activity Builder. Or maybe not. I’ll continue thinking and tinkering. Hopefully most of my steps are forward.


What do you think of the upgrade? Which of the featured comments resonated with you? What could be done to better address Principle #5?

Let me know in the comments, or drop me a line on Twitter.


LEGO Prices vs Marriage Age – Contrasting Activities

I spend a fair amount of my time at Desmos creating activities with the Activity Builder. In my mind, the job becomes more and more interesting over time for at least two key reasons:

  • Over time, the toolset expands.
  • Over time, our sense of how to use the toolset expands. That is, our pedagogy improves.

These two developments play off one another, and we find ourselves with more and more opportunities to build increasingly interesting things.

One way we push our pedagogy forward is by holding up two activities and asking questions like:

  • How are they similar? How are they different?
  • What are their strengths? What are their weaknesses?
  • In what ways could the Activity Building Code inform these activities? In what ways could these activities inform the building code?

Recent Comparison

Earlier this week, we took a close look at two linear modeling activities: LEGO Prices and Are People Waiting to Get Married?

If you’re up for it, I’d love to hear your analysis (similarities/differences, strengths/weakness, relationship to the building code) in the comments.

My Thoughts

Each activity has its own strengths, and each activity has the potential to generate interesting class discussion. However, I think LEGO Prices does a much better job with creating problematic activities (Principle #4 if you’re counting).

Are People Waiting to Get Married? nibbles at the edges of a context in a largely disconnected way.

  • Students make a prediction, but it never resurfaces.
  • Students sketch on a graph, but we’re not told why.
  • Students make another prediction, but that too never resurfaces.
  • Students write equations. Why? Because we tell them to.

LEGO Prices hits you right out of the gate with a single, overarching question that will drive the rest of the activity: How much did that LEGO set cost?!

  • Students make a prediction, and it resurfaces later in order to compare the power of wild guesses with the power of mathematics.
  • Students sketch in order to better understand the relationship.
  • Students build and then use a model in order to refine their prediction.

The key phrase for me here is in order to. That’s the difference. As much as Are People Waiting… has going for it (and, by the way, I rather love Screen 8), there’s no in order to attached to the tasks we’ve given students. Instead, it’s piecemeal. Screen-by-screen. Asking because we can, not because we must. Questions that serve themselves, rather than a single, coherent pursuit.

LEGO Prices suffers in other ways. For example, I think it does a fairly awful job with Principle #5: “Give students opportunities to be right and wrong in different, interesting ways.” It’s narrow-minded in that sense. There’s really just one way to move through the activity. I’m discouraged by that, and hope to discover creative solutions around that weakness in future activities.

And yet while it struggles with that, it doesn’t struggle with this: clarity of purpose. With the exception of Screen 5 (interpreting parameters), everything students do is done in the service of making the most insightful and accurate price prediction they can.

And that’s something I’ll be trying to infuse in similar tasks in the future.


What did you see in the activities? What did you think of my analysis? Where did it resonate? Where do you think it’s off base? I’d love to hear your thoughts in the comments.


CMC North 2016

Math teachers are amazing.

Exhibit A: Last weekend I spent an hour with a group of math teachers who woke up early on a Saturday morning to hone their craft. We were just steps away from the beach in Monterey, and yet they’re willing—hungry, even—to gather in a room to reflect on their own learning and teaching, even at 8 am. That’s just awesome.


If you weren’t able to join us, here’s a quick snapshot of the session:

  1. Some of the ways we use technology in the math classroom waste the human potential in the room. Picture computer cubicles. Kiddos wearing headphones. Teachers grading papers while students work in isolation.
  2. Let’s build something better. Let’s build activities—digital or otherwise—that spark more conversation, more discussion, more human engagement. And let’s build tools to help teachers facilitate those activities more effectively.
  3. With that in mind, the team at Desmos built a Classroom Conversation Toolset. You can read about it here.
  4. We’ve also spent considerable time thinking about the strengths and weaknesses of the activities we’ve built over the last year or so, and codified those thoughts into our Activity Building Code. You can read about it here.

Activities + Principles

Here’s a list of the activities we looked at during the session, along with the principles I had in mind for each one:

My hope is that these activities serve as exemplars for the principles, and that folks in the session walked away with a sense of how the principles might apply in other situations (whether Desmos or non-Desmos, digital or non-digital). We ended with a brief quiz to encourage folks to wrestle with these principles in new contexts.


Here’s the quiz. Take a look at each task, and let me know in the comments which principle(s) you think each activity exemplifies.

  1. Lift the Rainbow
  2. Marcellus the Giant
  3. Split 25

(To check your answers, turn the cereal box upside down. Wait, no. That’s something else. Never mind.)


The original presentation included quite a lot of video (to quickly show off the activities linked above), as well as a presenter (hi there!). So this version of the slide deck (which includes neither) may not be very helpful. But, in case it is, you can download it here.