**Why Grad School**

My first year at Stanford was almost my last. A talk I had given right before I arrived at Stanford rolled past one million views. That opened up a lot of opportunities outside of Stanford, very few of which I declined. During what was supposed to be a perfunctory first-year review, my advisers invited me, with as much grace as I could expect of them, to leave Stanford, to return when I had more focus. I stuck around but I think all of us knew then I wasn’t really cut out for R1 university work. Still, I figured I’d work with teachers in a preservice program somewhere and a doctorate wouldn’t *hurt* my employment odds.

Two years later, just before my dissertation proposal was due, I received a job offer that was really too perfect to pass up, from people who didn’t care whether or not I had a graduate degree. They were nice enough to allow me to defer that offer until this summer.

All of this is to say, I had every incentive to walk, to join the ranks of the ABD. Here’s why I stayed, why I’d do it again even though my new employers don’t care about the letters after my name, and why I’d recommend graduate study to anybody who can make the logistics work: **developing, proposing, studying, analyzing, and writing a dissertation works every single mental muscle you have and forces you to develop a dozen new ones**. It’s the academic centathlon. I know how to ask more precise questions and how to better interrogate my prior assumptions about those questions. I know many more techniques for collecting data and statistical techniques for answering questions *about* those data. I know how to automate aspects of that data analysis through scripting. My writing is stronger now. My presentation skills are more polished. My thinking about mathematics education is more developed now, though still a work in progress.

It’s certainly possible to develop all of those muscles separately, without the extra overhead of a dissertation. (Michael Pershan seems to be making a go of it on Twitter, with Ilana Horn as his principle adviser.) But tying them all together in the service of this enormous project was uniquely satisfying.

**Technology**

If you’re thinking about grad school, take advantage of your tools:

Papers to manage references. Dropbox to sync them across machines. iAnnotate PDF to read and mark them up on an iPad. Google Scholar for everything. Scrivener for writing anything with more than five headings. Google Docs for writing anything else. I couldn’t survive grad school without those six tools.

Google Tasks for scheduling to-do’s. Boomerang for scheduling emails. I couldn’t survive professional life without those two tools.

**The Last Five Years**

- Wrote two books.
- Foster parented three kids.
- Buried my dad.
- Traveled around the world with my wife.
- Learned from the best.
- Collaborated with great people on interesting projects.
- Traveled to a bunch of states and several countries, meeting basically all of you at one point or another.
- Keynoted a couple of big-ish conferences.

**Opportunity Cost**

- Never presented at AERA, PME, or ICMI.
- Never attended AERA, PME, or ICMI.
- Never gave a poster talk.
- Never gave an academic presentation of any kind until my dissertation defense.
- Never taught a course.
- Never TA’d a course.
- Never supervised any of the promising new teachers in Stanford’s teacher prep program.
- Never connected with the people in my research group as much or as often as I would have liked.
- Attended only a small fraction of the lunch talks and job talks and colloquia and dissertation defenses from the great thinkers passing through Stanford.
- Heard “Oh – do you still
*go*here?” way too often.

**Gratitude**

**You guys**. I thanked you all in my dissertation’s front matter and I’ll thank you here. The difference between a happy and sad graduate school experience often cuts on whether or not you like to write. In our conversations here, you guys made me, if not a great writer, someone who likes to write. As much as some of you drive me crazy, our back-and-forths made my arguments sharper and easier to defend in the dissertation. There was also that time that I asked on Twitter for help piloting assessment items in your classes and dozens of you helped me out. You have no idea what that kind of support is worth to a grad student around here.**I never got sick of my dissertation**. I didn’t enjoy some of the logistics of its data collection. I didn’t always have the time I wanted to work on it. But I never got tired of it, which is some kind of gift.**Michael Pershan**. My codes needed interrater reliability, the stuff that says, “Someone else can reliably see the world how I see the world, whether or not that’s the right way to see the world.” I hired Pershan onto my research team (doubling the size of my research team) when my time was crunched. He coded a bunch of data as fast as I needed and also changed “how I see the world” in some important ways.**Desmos**. I had some of the area’s best computer engineers and designers building my dissertation intervention. I got very lucky there.**Jo Boaler**. Jo was my principal adviser for all but my first year of grad school. There were a lot of great reasons to ask for her mentorship, but one of the best is that I never had to hide from her my lack of ambition for a tenure-track research job. As those ambitions faded, a lot of advisers in her position would have waitlisted me, focusing their efforts (rationally) on students who stood a chance to carry their research agenda forward. She invested more in my work than I had any reason to expect and I’ll always be grateful for that.

So that’s that. On to the next thing.

]]>**Title**

Functionary: Learning to Communicate Mathematically in Online Environments

**Bloggy Abstract**

I took a collection of recommendations from researchers in the fields of online education and mathematics education and asked our friends at Desmos to tie them all together in a digital middle-school math lesson. These recommendations had never been synthesized before. We piloted and iterated that lesson for a year. I then tested that Desmos lesson against a typical online math lesson (lecture-based instruction followed by recall exercises) in a pretest-posttest design. Both conditions learned. The Desmos lesson learned more. (Read the technical abstract.)

**Mixed Media**

You’re welcome to watch this 90-second summary, watch my defense, read it if you have a few minutes, or eventually use it with your students.

**Process Notes**

True story: I wrote it with you, the reader of math blogs, in mind.

That is to say, it’s *awfully* tempting in grad school to lard up your writing with jargon as some kind of shield against criticism. (If your critics can’t *understand* your writing, they probably can’t criticize it and if you’re lucky they’ll think that’s their fault.) Instead I tried to write as conversationally as possible with as much precision and clarity as I could manage. This didn’t always work. Occasionally, my advisers would chide me for being “too chatty.” That was helpful. Then I stocked my committee with four of my favorite writers from Stanford’s Graduate School of Education and let the chips fall.

Everything from my methods section and beyond gets fairly technical, but if you’re looking for a review of online education and the language of mathematics, I think the early chapters offer a readable summary of important research.

]]>**The lesson asks for an imprecise sketch rather than a precise graph.**

This is so rare. More often than not, our curricula rushes past lower, imprecise, informal, concrete rungs on the ladder of abstraction straight for the highest, most precise, most formal, most abstract ones. That’s a disservice to our learners and the process of learning.

You can always ask a student to move higher but it’s difficult to ask a student to move lower, forgetting what they’ve already seen. You can always ask for precisely plotted points of a model on a coordinate plane. But once you ask for them you can’t *unask* for them. You can’t then ask the question, “What *might* the model look like?” Because they’re *looking* at what the model looks like. So the Times asks you to sketch the relationship before showing you the precise graph.

Their reason is exactly right:

We asked you to take the trouble to draw a line because we think doing so makes you think carefully about the relationship, which, in turn, makes the realization that it’s a line all the more astonishing.

That isn’t just their intuition about learning. It’s Lisa Kasmer’s research. And it won’t happen in a print textbook. We eventually *need* students to see the answer graph and whereas the Times webpage can *progressively disclose* the answer graph, putting up a wall until you commit to a sketch, a paper textbook lacks a mechanism for preventing you from moving ahead and seeing the answer.

This isn’t just great *digital* pedagogy, it’s great *pedagogy*. You can and should ask students to sketch relationships without any technology at all. But the *digital* sketch offers some incredible advantages over the same sketch in pencil.

For instance:

**The lesson builds your thinking into its instruction.**

Once it has your guess – a sketch representing your best thinking about the relationship between income and college participation – it tailors its instruction to that sketch. (See the highlighted sentences.)

The lesson is the same but it is *presented* differently and responsively from student to student. All the highlighted material is tailored to my graph. I watched an adult experience this lesson yesterday, and while she read the personalized paragraph with interest, she only *skimmed* the later prefabricated paragraphs. It should go without saying that print textbooks are entirely prefabricated.

**It makes your classmates’ thinking visible.**

The lesson makes my classmates’ thinking visible in ways that print textbooks and flesh-and-blood teachers cannot. At the time of this posting, 70,000 people have sketched a graph. It’s interesting for me to know how much more accurate my sketch is than my classmates. It’s interesting to see the heatmap of their sketches. And it’s interesting to see the heatmap converge around the point that the lesson gave us for free, a point where there is much less doubt.

In a version of this article designed for the classroom, students would sketch their graphs and the textbook would adaptively pair one group of students up with another when their graph indicated disagreement. Debate it.

I’m not saying any of this is easy. (“Sure! Do that for factoring trinomials!”) But we aren’t exactly drowning in great examples of instruction enhanced by technology. Take a second and appreciate this one. Then let me know where else you think this kind of technology would be helpful to you in your teaching.

**Featured Comment**

]]>And as far as I know, even with Apple proclaiming “Textbooks that go beyond the printed page” since 2012?, there isn’t a single digital math textbook doing this yet.

They gave that talk to a room of maybe 50 teachers. It should have been 500, or however many math teachers there are in the world. (I assume around 500.) I hassled the Global Math Department to set them up with a forum, so here they are.

The live show is Tuesday, May 5, 6:00 PM Pacific Time. The recording will live on forever. Do yourself and your students a favor and watch it.

**2015 May 4**. I’m informed there’s a 100-person cap on live attendance, which means 400 of the world’s 500 math teachers will have to watch the recording. I’ll be sure to add that here once it’s available.

So here’s that directory. Or you can download all of the handouts in one big swipe [360MB] if you’d rather.

If you find anything interesting, do let us all know in the comments. It’s almost like you were there!

**BTW**: My slides don’t make a bit of sense without my voice attached so I don’t tend to upload them or send them around. I *will* be editing together video from my talk and posting it later this summer, though.

**2015 Apr 26**. In the comments, Eric Henry asked if my script could be easily tweaked to download all of the NCTM *Research Conference’s* handouts. It *was* easy! Man, code is cool. Directory + giant ball of handouts.

**Tuesday**. 9:00A. Beyond Real World & Relevance: Stronger Strategies for Student Engagement. Hanover, MA.**Wednesday**. 7:30A. The Future of Math Textbooks (if Math Textbooks Have a Future). NCSM.**Wednesday**. 10:30A. Plenary. NCTM Research Conference.**Wednesday**. 1:30P. Mathematical Modeling with Multimedia NCTM.**Thursday**. 3:30P. The Express Lane Is for Suckers, or Is It? (Yes, it Is.) [Ignite]. NCTM.**Thursday**. 5:00P. ShadowCon. NCTM.**Thursday**. 7:00P. Desmos / Mathalicious Happy Hour. NCTM.**Friday**. 9:30A. Fake-World Math: When Mathematical Modeling Goes Wrong . NCTM.

I really think we should spend Thursday evening together.

From 3:30 on you have a) the Ignite sessions, which are always fun, then you have b) ShadowCon, which is basically the future of NCTM conferences, after which c) your two favorite math education companies want to buy you a drink.

If you’re looking for help with the *rest* of your schedule, have a look at this list of Internet-enabled presenters as well as any of the names from my list last year.

For my part, if I can only make it to one session, it’s going to be this one.

]]>Blog monster hungry. Feed me.

I’m not big on the retroactively “sorry I haven’t been blogging” posts. I’d rather proactively explain why I’m not going to be around here for the next several months.

It *isn’t* for lack of interest in math education or for lack of interesting things *happening* in math education. For instance:

- My research group had a conversation about successful teacher professional development with Tom Carpenter, lead author of one of the most successful teacher professional development programs ever. Lots to share there.
- There were two especially useful sessions at CMCN14 I’ve wanted to share for months.
- RER published a meta-analysis of computerized feedback (useful summary) that I’m sure would find us all in a productive fight.
- Lots of
~~snark~~trenchant analysis of a current Khan Academy initiative. - Two curriculum confabs.
- Lots of great classroom action to talk about.

But my dissertation hearing is scheduled for mid May. I’m in the middle of data collection with lots of writing and analysis ahead and I’m sure I need to become a bit more ruthless in managing my time and writing.

So I’ll see you on Twitter (can’t quit *that* obviously), at NCTM and other conferences. But I won’t see you around here for a few months.

Please use this as an open thread to talk about whatever while I’m off dissertating. Also here’s all the great classroom action I haven’t written about over the last twelve months. Plenty of food for the blog monster there.

]]>Here is the temperature in the United States today (Fahrenheit):

So basically business is bad. No one wants your frozen treat.

So what do you do? You lower prices. An across the board cash discount? Maybe. But if you’re Gelato Fiasco, you institute The Frozen Code:

On each day that the temperature falls below freezing, we automatically use The Frozen Code to calculate a discount on your order of gelato dishes. [..] You save one percent for each degree below freezing outside at the time of purchase.

But how do you write this code using the language of variables that your pricing system understands? (Click through for Gelato Fiasco’s answer.)

How would you set this up as a mathematical learning experience for your students?

[h/t reader Nate Garnett]

*This is a series about “developing the question” in math class.*

**2014 Jan 8**. Updated to add this important exchange with Gelato Fiasco on Twitter.

@Veganmathbeagle @ddmeyer At -68 we would take 100% off, but we would likely not be able to open. Wind chill is not a variable.

— Gelato Fiasco (@gelatofiasco) January 9, 2015

**Featured Comment**

John:

]]>With the clear correlation between temperature and number of chain gelato shops, CLEARLY the temperature causes more gelato shops to be built…or does the number of gelato shops cause the temperature to decrease…am i right?

Here is one of the five, taken from Scott Farrand’s presentation at CMC North.

Here are some points in the plane:

(4, 1), (17, 27), (1, -5), (8, 9), (13, 19), (-2, -11)

(20, 33), (7,7), (-5, -17), (10, 13)Choose any two of these points. Check with your neighbor to be sure that you didn’t both choose the same pair of points. Now find the rate of change between the first and the second point. Write it on the board. What do you notice?

From Henri Picciotto’s review of Farrand’s session:

Students are stunned to learn that everyone in the class gets the same slope. This sets the stage for proving that the slope between any two points on a given line is always the same, no matter what points you pick.

In an email conversation with Farrand, he proposed the term “WTF Problems” because they all, ideally, involve a moment where the student exclaims “WTF”:

Set up a surprise, such that resolution of that becomes the lesson that you intended. Anything that makes students ask the question that you plan to answer in the lesson is good, because answering questions that haven’t been asked is inherently uninteresting.

These seem like essential features:

- These problems are all brief. They slot easily into an opener.
- They look forward
*and*backward. They fit right in the gap between an old concept and the new. They review the old (slope in this case) while setting up the new (collinearity). - Students encounter an unexpected result. The world is either more orderly (the slope example above) or less orderly (see problem #2) than they thought.

And the *weirdest* feature:

- They require the teacher to be cunning, actively concealing the upcoming WTF, assuring students that, yes, this problem is as trivial as you think it is, knowing all the while that it isn’t.

When did they teach you *that* in your teacher training?

It’s striking to me that the history of mathematics is driven by the explanations following these WTF moments:

- We knew how to divide numbers. We didn’t know how to divide by zero. Enter Newton & Leibniz explanation of calculus.
- We knew how to find the square roots of positive numbers, but not negative. Enter Euler’s explanation of imaginary numbers.
- We knew what Eucld’s geometry looked like, but what if parallel lines
*could*meet. Enter the explanation of hyperbolic, spherical, and other non-Euclidean geometries. - There are lots of WTF moments that
*haven’t*yet been explained.

In school mathematics, though, we simply *give* the explanations, without paying even the briefest homage to the WTFs that provoked them.

What Farrand and you and I are trying to do here is restore some of that WTF to our math curriculum, without forcing students to re-create thousands of years of intellectual struggle.

So help me out:

**Have you seen other problems like these?****Who else has written about these problems?**I believe we’re talking about disequilibrium here, which is Piaget’s territory, but I’m looking for writing local to mathematics.

**Featured Comments**

David Wees cautions us that the effect of these problems depends on a student’s background knowledge. If you don’t know how to calculate slope, the problem above won’t surprise, just confound. I agree, but the same is true of textbooks and nearly every other resource.

Michael Pershan worries that the “twist” in these problems will become overused, that students will become bored or expectant. (Clara Maxcy echoes.) I demur.

Dan Anderson offers other examples. As do Mike Lawler, Federico Chialvo, Kyle Pearce, Jeff Morrison, and Michael Serra.

Franklin Mason critiques my math history without (I think) critiquing my main point *about* math history.

Scott Farrand, whose presentation at CMC-North inspired this post, elaborates.

Ben Orlin summarizes the design of these problems in four useful steps.

Terri Gilbert summarizes this post in a t-shirt.

**Featured Tweet**

]]>WTF-based Learning (WTF-bL for short) in the math classroom http://t.co/cptgRZvo7J pic.twitter.com/X6qX4HAHxE by @ddmeyer

— Nacho Santa-María (@nacheteer) January 11, 2015

Here were my new blog subscriptions in December 2014, some of which might interest you.

**Steve Wyborney**posted his animated multiplication table, a very thoughtful tool highlighting interesting patterns for young math students. So I subscribed.**Steve Leinwand**has a blog! I’d subscribe if I knew how. [**BTW**. Commenter Sadler found the feed.]**Thinking Math**is a group blog written by four elementary math educators. Go encourage them to post more.**Kassia Omohundro Wedekind**blogs about elementary math education also (subscribed!) and posts interesting observations and analysis to her Twitter feed (followed!).- Khan Academy’s
**crack data science team**has a blog, which might be Khan Academy’s most valuable contribution to my life so far. Fancy tools, sharp analysis, well-written. - Off
**Anthony Carabache’s**post, Education Should Step Away From Apple Devices, he seems like a cautious, thoughtful technologist, which is my favorite kind.

What did you fill your head with in December?

]]>