Zak Champagne, Mike Flynn, and I are all NCTM conference presenters and we were all concerned about the possibility that a) none of our participants did much with our sessions once they ended, b) lots of people who might benefit from our sessions (and whose questions and ideas might benefit *us*) weren’t in the room.

The solution to (b) is easy. Put video of the sessions on the Internet. Our solution to (a) was complicated and only partial:

**Build a conference session so that it prefaces and provokes work that will be ongoing and online.**

To test out these solutions, we set up Shadow Con after hours at NCTM. We invited six presenters each to give a ten-minute talk. Their talk *had* to include a “call to action,” some kind of closing homework assignment that participants could accomplish when they went home. The speakers each committed to help participants with that homework on the session website we set up for that purpose.

Then we watched and collected data. There were two major surprises, which we shared along with other findings with the NCTM president, president-elect, and executive director.

Here is the five-page brief we shared with them. We’d all benefit from your feedback, I’m sure.

**Featured Comments**

Marilyn Burns on her reasons for attending conferences like NCTM:

I don’t expect an NCTM conference to provide in-depth professional development, but act more like a booster shot for my own learning.

Elham Kazemi, one of our Shadow Con speakers, tempers expectations for online professional development:

]]>I have a different set of expectations about conferences and whether going to them with a team allows you to go back to your own contexts and continue to build connections there. Can we expect conferences and the internet to do that — to feed our local collaborations? I get a lot of ideas from #mtbos and from my various conversations and conferences. But really making sense of those ideas takes another level of experience.

Determining if a relationship is a function or not.

A relationship that maps one set to another can be confusing. Questions like, “What single element does 2 map to in the output set below?” are impossible to answer because 2 maps to *more than one* element.

By contrast, a function is a relationship with *certainty*. Take any element of the input and ask yourself, “Where does this function say that element maps?” You aren’t confused about any of them. Every input element maps to exactly one element in the output.

Pearson and McGraw-Hill’s Algebra 1 textbooks simply provide a definition of a function. Pearson’s definition refers to a previous worked example. McGraw-Hill has students apply the definition to a worked example immediately afterwards. Khan Academy dives straight into an abstract explanation of the concept. In none of these cases is the *need* for functions apparent. Students are *given* functions without ever feeling the pain of not having them.

**What a Theory of Need Recommends**

If we’d like students to experience the *need* for the certainty functions offer us, it’s helpful to put students in a place to experience the *uncertainty* of non-functional relationships first. Here is what I’m talking about.

Put the letters A, B, C, and D on your back wall, spaced evenly apart.

Ask every student to stand up. Then give them a series of instructions.

*Transportation*

If you walked to school today, stand under A.

If you rode your bike to school today, stand under B.

If you drove or rode in a vehicle today, stand under C.

If you got to school any other way, stand under D.

*Duration*

If it took you fewer than 10 minutes to get to school today, stand under B.

If it took you 10 or more minutes to get to school today, stand under D.

*Class*

If you’re in seventh grade, stand under A.

If you’re in eighth grade, stand under B.

If you’re in ninth grade, stand under C.

If you’re in any other grade, stand under D.

These instructions are all clear and easy to follow. Students are *certain* where they should go. Then give two other sets of instructions.

*Clothes*

If you’re wearing blue, stand under A.

If you’re wearing red, stand under B.

If you’re wearing black, stand under C.

If you’re wearing white, stand under D.

*Birthday*

If you were born in January, stand under A.

If you were born in February, stand under B.

If you were born in March, stand under C.

If you were born in April, stand under D.

Perhaps you see how these last two examples generate a *lack* of certainty. Students were lulled by the first examples and may now feel a headache.

“I’m wearing white *and* red. Where do *I* go?”

“I was born in August. There’s no place for me to stand.”

*Now* we gather back together and apply formal language to the concepts we’ve just *felt*. “Mathematicians call these three relationships ‘functions.’ Here’s why. Why do you think these relationships *aren’t* functions?” Invite students to interrogate the concept of a function in different contexts. Try to keep the focus on *certainty* – can you predict the output for any input with certainty? – rather than on the vertical line test or other rules that expire.

**Next Week’s Skill**

Graphing linear inequalities. It’s extraordinarily easy to turn questions like “Graph y < -2x + 5" into the following series of steps:

- Graph the line.
- If the inequality includes the boundary, make the line solid. Otherwise, make the line dashed.
- Test a point on either side of the line. Use (0,0) if possible.
- If that point is a solution to the inequality, shade that side of the line.
- If that point
*isn’t*a solution to the inequality, then shade the*other*side of the line.

Students can become quite capable at executing that algorithm without understanding its necessity or how it figures into algebra’s larger themes.

What can you do with this?

**BTW**

Kate Nowak encouraged me to look at other textbooks beyond McGraw-Hill and Pearson’s. She recommended CME, which, it turns out, does some great work highlighting this need for functions. It asks students to play a “guess my rule” game, one which has a great deal of certainty. Each input corresponds to exactly one output. Then the CME authors offer a vignette where a partner reports multiple outputs for the same input, making the game impossible to play. Strong work, CME buds.

]]>8 names are read per min at the CHS graduation, I will be here for another 36min.. slooooow.

— andreabonilla (@andreabonilla) June 9, 2011

My friend has taken a problem from the world that was personal to her, identified the variables that are essential to the problem, selected a model that describes those variables, performed operations on that model, and re-interpreted the result back into the world. And tweeted about it.

That is *modeling* – the process of turning the world into math and then turning math back into the world. My friend probably wouldn’t wouldn’t label her experience like that but that’s what she’s doing. That’s what people who do math in the world do.

We know how this looks in many textbooks, though.

The amount of time (t) it takes a number of graduates (n) to cross the graduation stage can be modeled by the function t(n) = n/8. How long will it take all 288 graduates to cross the stage?

Here students would simply perform operations on real-world-flavored math while the important and interesting work is in turning the world *into* that math and turning that math *back* into the world.

Here is an alternate treatment, one that has students modeling as the practice is described in the Common Core.

**Show this video**.

Ask: “If I want to set an alarm that’ll let me take a *long* nap until just before my cousin Adarsh crosses the stage, how should I set the alarm?”

By design, it’s a short video. I’d like it to be boring enough to provoke my friend’s modeling but not *terminally* boring.

By design, it lacks mathematical structures because we’d like students to participate in the process of developing those structures. They won’t do that unassisted.

Before we get to the algebraic model, we can ask some important and interesting questions.

**How long do you think it will take my cousin to graduate? Just estimate.**

I asked that on Twitter and received the following estimates:

These guesses interest us in a *calculation* and also prepare us to evaluate whether or not that calculation is correct.

**Sketch the relationship between the number of graduates and time.**

Asking students to sketch the relationship, rather than plot it precisely, asks them to think relationally (“how do these two quantities change together?”) rather than instrumentally (“how do I plot these points?”).

Many students will *assume* the data is linear. But this prompt may invite some students to consider the possibility that the data is non-linear.

**Collect data. Model the data. Get an answer.**

Ask students to create a table of values. Ask students to plot the data in Desmos. Regress the data. Give them the graduation program. Calculate an answer.

I plotted the first ten names and modeled their times with a linear equation. (“Time v. names read” was my model, though commenter Josh thinks “time v. number of syllables read” would be more accurate.) The calculation for cousin Adarsh’s 157th name is 19 minutes. I would be foolish to rely on that calculation, however.

**Ask your students to “Assume your answer is wrong, that something surprising actually happens. Anticipate that something and fix your mathematical answer.”**

George Box: “All models are wrong, but some are useful.”

This is where we turn the math back into the world. This is where we make some math teachers uncomfortable, admitting that the world and the math don’t correspond exactly and that the *math* needs modification.

Watch all of these math teachers make *exactly* those modifications in the comments of the preview post. They perform mathematical operations and then proceed to describe why the results of those operations are *wrong*.

- Scott: “Add the bit of time prior to starting and a few seconds for a switch in readers as tends to be customary in larger groups like this … “
- Sadler: “14 minutes and 10 seconds but given that it is better to wake 10 seconds early than miss it, I would submit 14 minutes.”
- Scott #2: “You would probably want to set your timer a little earlier so you are fully awake when your cousin’s name is called.”
- Julie Wright: “As an embittered W, I am aware that there is lots of ponderous gravity for A’s and B’s, then everybody gets bored and speeds things up.”

**Validate (or invalidate) the answer.**

Commenter Mark Chubb, at the end of his modeling cycle: “Can’t wait to see Act 3.” Act 3 is the *reveal* in this task framework I call three-act math. It isn’t enough for Mark to simply read the answer in the back of the book or hear it from me. He wants to see it. So:

If you built a linear model from the first ten names, your answer winds up too large. Instead of 19 minutes, my cousin graduates at 17:12, *sooner* than the math predicted.

Why?

In the video, you can hear the validation of Julie Wright’s hypothesis above. The A’s and B’s get a lot of pomp, and then the commencement reader races through the rest.

Many congratulations to Megan Schmidt for her guess and to Scott and Kyle Pearce for their calculation. They all put down for 18 minutes. Special mention also to aga bey for 16.3 minutes. That commenter’s method? “I took the average of all submissions upthread.” Strong!

Again, if mathematical modeling requires the cycle of actions we find here, our textbooks typically only require one of them: performing operations. The purest mathematical action. The one that is often least interesting to students and the least useful in the world of work. So let’s offer students opportunities to experience the complete modeling cycle. Not just because those are the skills that most of the fun jobs require. But because modeling with math is fun for students *now*.

**Featured Comment**

]]>I ran into this when working up an exponential growth problem for my son’s precalculus class. The CDC had data on the number of Ebola cases which could be modeled with an exponential growth curve at the time. However, the math needed correction because of a sudden increase in cases. The CDC readily admitted they believe the cases were unreported by a factor of 1.5 to 2.5. Thus, a human eye on the data to recognize that and make an adjustment was necessary.

Later, when the curve could be modeled nicely by a logistics curve, the equation was still incorrect in predicting the end of the epidemic. As teachers we would like to be able to button everything up and wrap it in a bow, but the real world seldom works that way.

Rules like these are too quickly abstracted, memorized, confused, and forgotten.

We can attach to them meaning and purpose by asking ourselves, why did we come up with these shortcuts? If these shortcuts are aspirin, then how do we create the headache?

**What a Theory of Need Recommends**

Again, with Harel’s “need for computation,” students need to experience the “longcut” before they learn the shortcut. Otherwise it’s just another trick in the endless series of tricks students call “math class.”

Several people suggested the same in last week’s thread, of course. Ask students to calculate expressions like these the long way before discussing shortcuts the students may have noticed (or that *you* may have noticed as a member of the class also).

Chris Hulitt was one of my workshop participants in Norristown, PA, and his group suggested an important addition to this idea. Ask students to calculate *this* expression instead

Looks the same as the last, right? But whereas the last expression resolves to 16, this expression resolves to 1. That headache is a *little* bit sharper. How did this gangly mess of numbers result in such a simple answer? Could I have realized that in advance?

Again, this isn’t real world, or relevant, per our usual definitions of the term. And yet this approach may *still* endow exponent rules with a purpose they often lack.

**Next Week’s Skill**

Determining if a relationship is a function or not.

This is another skill that can become quickly instrumental (run a vertical line over the graph, etc.) and obscure why it is aspirin for a particular kind of headache.

Let us know your ideas for motivating the definition of a function in the comments.

**What You Recommended**

]]>I think of my 6th grade students writing down the prime factorizations of whole numbers (Why learn that? Oh yeah, to improve number sense and seeing structure behind numbers, among other reasons). When you work with something simple like 24, writing out 2*2*2*3 is not so bad. When you up the stakes to something like 256, writing out 2*2*2*2*2*2*2*2 becomes annoying. It’s not so much a headache as a tedious process that any normal person would like to make quicker and easier. At this point, I discuss exponents as a means of communicating all of that multiplication without having to write it all.

Here are three minutes of a Harvard graduation ceremony and the relevant program. My cousin Adarsh is graduating and his name is *quite a ways down*.

I’d like to take a nap but I’ll set a timer first so I won’t sleep through my cousin’s walk across the stage.

What time should I set the timer for?

Tell us the time and your method in the comments. The winner is whoever comes closest to the time my cousin walks across the stage, without going over.

]]>Factoring quadratic trinomials.

eg. We can express quadratic trinomials like x^{2} – 7x – 18 as the product of the two binomials: (x – 9)(x + 2).

If you find that language disorienting, if it makes you wonder why anyone would even bother on a sunny day like today, you’re in good company with lots and lots of math students. At the secondary level, there are few skills that seem less necessary to students and few skills that seem harder to motivate for math teachers than factoring quadratic trinomials. (Sample their stress.)

**What You Recommended**

Mercifully, very few of the 70-ish comments on my last post suggested an instructional strategy for this skill without *also* describing the theory that gave rise to the strategy. We need more of that kind of conversation, not less.

Here are three theories I found particularly interesting:

- Tie the skill to its historical roots. “The thing I find interesting about factoring quadratics is the idea that the Babylonians started using it with taxes.”
- Tie the skill to its later uses in math. “We want to know factors because they explain why polynomials and rational functions work the way they do.”
- Turn the skill into a puzzle. “… they knew that finding integer roots can be a fun and satisfying puzzle to solve.”

The first two solutions seem to me very clearly defined and very easy to implement but also very far-fetched.

Why am I interested in the *history* of a topic that has terrorized me for years? I’d like to know *less* about its history than more. (Even Andy, who suggested the idea, admits its vulnerability.)

And if learning this tough skill *now* helps me learn some tougher skill *later* (what Josh G. described as “passing the buck upwards“) I can save myself the trouble twice over and learn neither.

The last design theory has a lot of promise. People love puzzles after all, even the kind that are far from the “real world.” But it seems very difficult to implement.

I consulted two textbooks – one Algebra 1 text from McGraw-Hill and Pearson each. One attempted to tie the skill to the real world. The other said, “Look we’re just going to teach you this stuff okay.” The latter approach is honest, if uninspired. The former approach seems overly self-satisfied. After checking off the “real world” box, they proceed to teach the skill abstractly through a series of worked examples.

**What a Theory of Need Recommends**

First, ask yourself, if the skill of factoring quadratics is aspirin, what is the headache? How does the skill relieve a mathematician’s pain? The strongest answer, I think, is that *the skill helps us locate zeros*. The number that evaluates the expression x^{2} – 7x – 18 to zero isn’t obvious. In its factored form however – (x – 9)(x + 2) – the zero product property tells us the answer quickly: x = 9 and x = -2.

This is Guershon Harel’s “need for computation,” particularly the need for efficiency in computation.

Once that need is clear, the activity becomes much easier to design. Students should experience *inefficient* computation before we help them develop *efficient* computation.

So with nothing on the board, ask your students to: “Pick a number between 1 and 10. Write it down.”

Not a problem. Now put the expression x^{2} – 7x – 18 on the board and ask students to evaluate it for their number.

This is an unreal and irrelevant task, admittedly, but no one asks “When will I ever use this?” because students tend to ask that question when they feel disoriented and stupid. This prompt is relatively clear and accessible.

Now you ask: “Who got zero? Anybody? Anybody? Raise a hand. Nobody? Okay. Not a problem. Try a different number. Try a different number. Try a *different* number. Don’t stop until you get a zero. Call me over when you do.”

If someone *did* get a zero, ask them to get another one. (Later question: how do they know there are only two solutions to this equation.) Record the solution next to the quadratic on the board. Put up three more. Ask them to find more zeros.

Tease the possibility that a more efficient method than guess-and-check exists.

After 5-10 minutes of guess-and-check, help them learn that method.

**What This Is and Isn’t**

I’m not saying this activity will be your students’ favorite day in your class all year. Factoring quadratics was never going to be that.

But I’ll make a mild claim that this activity will be motivating for students. We’ve created a task with a clear goal state and a low entry and a high exit, a task that is iterative with timely feedback. These features are all common to the most intriguing puzzles. Of *course* a student could ask, “Why do I care about finding zero?” But they could ask similar questions about Sudoko, Tetris, and other puzzle games. They don’t because puzzles are, by definition, puzzling.

I’ll make a strong claim that this activity will endow factoring quadratics with a sense of purpose that it often lacks. Not purpose in the world of work or surfboards or trains leaving Philadelphia traveling west, but a purpose in the world of math. By tying the skill of factoring quadratics into a network of older skills (especially “guess and check”) we strengthen all of them.

I’ll make a strong claim that this is an example of taking a theory of instruction and enacting it. Finding a workable theory of instructional design is hard enough. Enacting it is even tougher. I love that work.

Doug Mackenzie asks an important question which I’m about to ignore:

Is it bad/good theory to expect that they will “construct” their own aspirin? (Do we leave them in disequilibrium until they get themselves out?) Is it good/bad theory for teachers to deliver the aspirin, or should students only get aspirin from other students

I’m not making any recommendations here about *how* students should learn that more efficient method for finding zeros. Tell them that method directly. Let them discover it. I know what I would do. We can draw from research. But that isn’t what this series about. This series is about creating the *need* for new learning, not *satisfying* it.

**Next Week’s Skill**

It’s like foldables were *invented* for exponent rules. Students can memorize a bunch of rules and write them down in something organized and pretty.

But why do we *need* them? If exponent rules are aspirin, then how do we create the headache?

**Other Great Comments About Factoring Quadratics**

It turns out I was on a similar frequency as Eric Fleming, Joshua Greene, Chuck Collins, and others.

Tim Hatman does some really impressive work exploiting Harel’s “need for certainty”:

So here’s my headache. Graph y=(x

^{2}+ 7x + 10)/(x + 5)Without factoring, the only way to graph this is to just start plugging in x’s and making a table – that’s a headache! But when you start plotting the points…Whaaaaaaaat?!? It’s a straight line! How did that happen? What’s the equation of that line (why is one point missing) and how can I get there through a shortcut?

Malcolm Roberts names a central dilemma to *all* theories of instruction, not just this one:

Given that all learners are different, and that the context of learning varies every time we teach, it seems to me to be a near impossible task to create a situation that will be headache inducing for all (maybe even the majority of) students all (maybe most of) the time.

Simon names one key misunderstanding of factoring quadatics:

I think one of things that’s important is that our students understand that 1) factorising doesn’t change the value of the expression and 2) why it is more useful. Too often I find students thinking (x-3)(x+2) only ‘works’ for x = {-2,3}.

Chuck Collins names a second:

you’d be surprised how many college students don’t realize that the quadratic formula gives the same solutions that you get from factoring

Scott Hills recommends “diamond puzzles” in the weeks running up to this instruction:

]]>I start out, about 2 weeks before factoring becomes a part of the math lexicon, with “diamond puzzles” in which students must first identify what 2 numbers add to a particular sum while multiplying to a particular product. The puzzle being the point, no mention is made of factoring.

There were nearly 100 responses and they said a great deal about the theories of learning and motivation that hum beneath everything we do, whether or not we’d call them “theories,” or call them anything at all.

- “How can [x] help them to see math in the world around them?”
- “How can I connect [x] to something they already know?”
- “How can I explain [x] clearly?”
- “What has led up to [x] and where does [x] lead?”

You can throw a rock in the math edublogosphere and hit ten lessons teaching [x]. They might all be great but I’d bet against even one of them describing some larger theory about learning or mathematics or describing how the lesson enacts that theory.

Without that theory, you’re left with one (maybe) great lesson you found online. Add theory, though, and you start to notice other lessons that fit and don’t fit that theory. When great lessons don’t fit your theory about what makes lessons great, you modify your theory or construct another one. The wide world of lesson plans starts to shrink. It becomes easier to find great lessons and avoid not great ones. It becomes easier to *create* great ones. Your flywheel starts spinning and you miss your highway exit because you’re mentally constructing a great lesson.

Here is the most satisfying question I’ve asked about great lessons in the last year. It has led to some bonkers experiences with students and I want more.

- “If [x] is aspirin, then how do I create the headache?”

I’d like you to think of yourself for a moment not as a teacher or as an explainer or a caregiver though you are doubtlessly all of those things. Think of yourself as someone who sells aspirin. And realize that the best customer for your aspirin is someone who is in pain. Not a lot of pain. Not a migraine. Just a little.

Piaget called that pain “disequilibrium.” Neo-Piagetians call it “cognitive conflict.” Guershon Harel calls it “intellectual need.” I’m calling it a headache. I’m obviously not originating this idea but I’d like to advance it some more.

One of the worst things you can do is force people who don’t feel pain to take your aspirin. They may oblige you if you have some particular kind of authority in their lives but that aspirin will feel pointless. It’ll undermine their respect for medicine in general.

Math shouldn’t feel pointless. Math isn’t pointless. It may not have a point in job [y] or [z] but math has a point in math. We invented new math to resolve the limitations of old math. My challenge to all of us here is, before you offer students the new, more powerful math, put them in a place to experience the limitations of the older, less powerful math.

I’m going to take the summer and work out this theory, once per week, with ten skills in math that are a poor fit for other theories of interest and motivation. As with everything I have ever done in math education, your comments, questions, and criticism will push this project farther than I could push it on my own.

The first skill I’ll look at it is **factoring trinomials with integer roots**, ie. turning x^{2} + 7x + 10 into (x + 5)(x + 2). All real world applications of this skill are a lie. So if your theory is “math is interesting iff it’s real world,” your theory will struggle for relevance here.

Instead, ask yourself, “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”

]]>This is an easy call. I need a question to carry me through my thirties and I can’t think of a better one than, “What does the math textbook of the future look like?”

I’ve known for awhile I need a certain set of collaborators for that project. I have worked with Eli, Eric, and Jenny for the last three years. We need each other. They need what I do (the math teaching stuff) and I need what they do (the computery stuff). They’re great at what they do and we get along great. Stay tuned.

]]>**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.

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