Here are three items that crawled into my head some time during the last few months and didn’t find their way out yet. This post is brain surgery.

Lesson Exploder

I have thought about this tweet from David Coffey at least once per week for the last five months.

The Song Exploder podcast interviews artists about the craft of songwriting. The artists describe their motivations for creating their songs, what they were trying to accomplish, and how they tried to accomplish it, all while the Song Exploder team teases out key elements of the song for illustration. I feel smarter about the craft of songwriting whenever I listen to it. Maybe not as smart as if I had spent a year at the Oberlin Conservatory of Music, but for how much smarter I’m feeling, it’s hard to argue with Song Exploder’s cost (free) and scale (internet-sized).

Now swap “teacher” for “artist” and “lesson” for “song.” I know what we can swap in for “Oberlin Conservatory of Music.” Classroom visits. Lesson studies. Problem solving cycles. Professional learning communities. Those are all very effective and also very expensive. I don’t know what to swap in for “Song Exploder,” though — an option that is less effective but basically free and scales with the internet.

What kind of digital media could we use to a) highlight something significant and useful about the craft of teaching b) as quickly as possible c) distributed as widely as possible d) in a form that’s replicable and episodic? (Song Exploder is up to 133 episodes right now.)

What current examples can we find? Teaching Channel videos? Blog posts? Lesson plans? Unedited classroom video? Marilyn Burns distills classroom anecdotes into really popular tweets.

What inspiration can we take from other fields? Delish videos? NFL Red Zone / Mic’d Up? Mystery Science Theater 3000? Twitch streaming?

New Jersey Turnpike

I can’t figure out the tolls on the New Jersey Turnpike.

If you don’t come from turnpike territory, how it works is you enter the turnpike somewhere and you exit the turnpike somewhere else. You pay depending on where you entered and exited.

My assumption is that the pricing would look pretty linear as a function of the miles traveled. Like this:

But it doesn’t. It looks like two linear functions with the second piece starting maybe at the Garden State Parkway. (Why?) And the Pennsylvania Turnpike exit is also way more expensive than a linear function would predict. (Why again?)

Here is the website that tells you the cost of different trips on the turnpike. Eric Berger, our CTO at Desmos, helped me type code into my browser’s Javascript console that returned all the data. Feel free to dig in. I’m looking for answers to my questions about pricing and I’m also interested in possible classroom applications of the data.

Cape Town’s Zero Day

Cape Town has a water crisis and a website that until recently calculated a “Zero Day” for their water reserves, a day when faucets will run dry and people will collect a daily allotment of water from central locations throughout the city.

That’s either terrifying or mathematically interesting, depending on which part of my brain I subdue while I’m thinking about it. How do they calculate that zero day? How can we put students in a position to appreciate, replicate, and even adapt those calculations for their own contexts?

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Re: lesson exploder… How do we do number talks? Would something like this work for planned lessons that turned into a vlog?

  2. Having done the Jersey Turnpike to PA turnpike thing many times (it’s how I get to Atlantic City – don’t judge), here’s a partial explanation. When you leave that exit, you travel a few more miles then cross a bridge to get into PA. Thus your toll not only gets you off the exit, but also includes the troll toll. Also, you pay extra to escape the smell.

  3. Bob Lochel again

    April 20, 2018 - 5:46 pm -

    But here is a less snarky answer to the turnpike question. There do seem to be 2 semi-clearly defined functions: 1 pricing scale for “south” jersey towards central (from mile marker 1 at Delaware to around mile mark 90), then we transition to the NYC metro area, where the price structure shifts to the steeper slope. Ask Jason…he knows this.

  4. Hi Dan,

    This is very nice!

    Lesson Exploder:
    I always desire and dream to grow as a math teacher with reasonable cost. I am very thankful that I live the in the 21st century where #MTBOS, ShadowCon, Mount Holyoke math, Robert’s online workshop, Kyle’s online workshop and online workshops are at hand.

    I echoed what you’re saying. If we can have something like the Song Exploder, more teachers can benefit from it, and maybe we can eventually fight poverty for some countries through education.

    New Jersey Turnpike:
    Honestly, I had the same experience a while ago when I first immigrated to the country. I sensed the pricing on the Pennsylvania Turnpike and told my husband that we should go down and avoid several session of Turnpike to save some money. He didn’t want to, but he agreed to do it. Eventually, lady TomTom guided us to some rough area around inner-city Philadelphia where my husband found driving being very stressful.

    So, after that, we stop saving money on the Turnpike. If the Turnpike asks me even 50 USD, I will pay just so if it’s cheaper than flying. Tell you what, I will pay if traveling is necessary for me.

    Cape Town’s Zero Day:
    I have always been curious about issues like this: 1 child dies every four minutes because of poverty or the speed of the virus spread, but honestly, I have never thought of using the math skills I have to solve them. It seems like it’s too far away and it’s something only the Statisticians can do.

  5. Chester Draws

    April 21, 2018 - 4:31 am -

    Not lesson plans.

    It would be a fascinating experiment to give ten teachers exactly the same lesson plan, tell them to stick to it pretty closely, then view the different results.

    I am positive that there would be ten different ways of following it “closely”.

    Short unedited videos with subtitled comments I would watch. And probably rewatch if they were good. I like to see other teachers in action.

    But it would have to be normal lessons, not some “show” lesson of how the teacher would like to teach if they had infinite prep time and beautifully crafted resources that they’d practiced with and with students bribed to perform.

    I like to see how teachers deal with all the little day to day issues — students arriving late, phones out, slacking — while still delivering a lesson.

    • Helpful perspective here. Seeing the same dilemma or technique applied across ten different lesson plans or seeing the same moment from the same lesson plan taught ten different ways — those both seem really appealing (if also 10x more expensive to produce).

      You also highlight a significant limitation of the Teaching Channel model — that it tends towards the extraordinary and the hyper-prepared. In my ideal world, I think the teacher is just wearing a single mike and there is a single class mike somewhere in the center and everyone forgets about it pretty quickly.

  6. Nice distinction.

    Ah, I love Song Exploder! The challenge I see for “Lesson Exploder” is (to put it daftly) that lessons are different from songs. A musical recording is a polished, hyper-compressed, mass-distributable product that takes 4 minutes to consume. A lesson is a ragged, half-improvised, interactive, in-person event that lasts 45 minutes. Before you could “explode” a lesson, you’d first have to compress it, so that the audience knows what exactly is being explained.

    I imagine it’d be easier to explode an *element* of a lesson, rather than a whole lesson itself. An introductory hook? A tight sequence of questions? An assessment?

    One fun thing about Song Exploder is seeing so much ingenuity and effort devoted to each tiny aspect of the final song. So even though teaching is largely made up of unplanned moments, I think the plannable elements would lend themselves better to that kind of treatment.

    • That’s a helpful distinction in the first paragraph. I’m curious about two things:

      (1) What the experience of Song Exploder is like if you’ve never heard the song.
      (2) How the podcast gives the audience the minimum context necessary to appreciate the analysis that follows.

      I’m wondering how hard it’d really be to compress the lesson plan down initially to a capsule summary and then decompress it through a few minutes of analysis and live audio.

  7. I’ve had similar thoughts about capturing the breaking down of lesson planning. But…more like…how can we capture the process I see taking place during organic tweet sessions that are tackling problems of practice? Virtually any day on Twitter, among the MTBOS community, you can find a group of people tackling a problem. These are so very, very rich. It’s a live PLC that occurs across the world. I have heard of things like Storify, but haven’t used it. But, there needs to be a one-stop shop that is searchable where educators can go and look for similar problems and “listen in” on educators tackling similar issues. Someone needs to make that. Not me…I don’t know code or programming. But there are many of you out there that do!!!!

    And..can I just add…this post is one more reason I love the #MTBOS #ITEACHMATH community. I really am not weird, if one chooses to use Dan as a measuring rod. Things get stuck in my head and I carry out these mini thought experiments all the time, several at a time, and I need a way to put those out into the world, yell them out into the world, and see what reverberates.

    • That’s sounding sensible to me – a quick (maybe 1-minute) encapsulation of the lesson plan, followed by 10-20 minutes of audio (or video?) showing how the lesson played out in practice, cut together with the teacher commenting and reflecting on particular twists/turns/decisions.

      In a sense, that “Exploding” process is the mirror image of Song Exploder. Where they start with all the little sonic elements and creative choices, and then reveal at the end how they form a unified song, you’d start with the unified plan, and then reveal how it entails all the little pedagogic elements and creative choices.

    • (Sorry, Rene – meant to reply to my own comment above! Maybe Dan can move the comment?)

      (And Storify is pretty cool – I haven’t used it much myself, but it seems like a nice tool for crystallizing and saving that Twitter give-and-take we all enjoy!)

  8. Don’t you think that it’s just a higher demand area at one side (probably the east side of the PA turnpike, for example)? So higher demand allows you to charge more.

  9. Exactly what I need!

    Dan, Regarding the Cape Town crisis, I found this NBC News article more accessible to 6th graders, plus it includes a link to a water usage calculator which I used in a recent lesson. I wrote about the lesson here.
  10. We did a water project after looking at some facts we gathered related to the Cape Town water crisis – students had to use 2 percentages, 2 conversions, and 2 rates to discuss some aspect of water usage. Students were surprised that Americans use 80-100 gallons a day. Students compared water usage in various cities, water usage for various tasks, and the water cycle and the amount of water on the earth in general. Projects culminated in a information poster. Next year we plan to incorporate estimation and make it more 3-act worthy.

  11. Tofer Carlson

    May 8, 2018 - 8:12 am -

    A similar relationship shows up on the Massachusetts Turnpike with flat areas due to passing through cities, and a spike at the end due to going through a tunnel with a high fee.

    I would love to see more data like this and use it to build out an Algebra II start of the year unit focused on understanding linear-ish relationships.

  12. If you have ever taken the “off ramp” to go from NJ to PA is long. It’s been a dozen or so years since I have been on it but it’s hella long – probably more than 5 miles. I think that the fact that they had to create a new freeway to create the connection to the bridge explains the cost.