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.
This Week’s Skill
Factoring quadratic trinomials.
eg. We can express quadratic trinomials like x2 – 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:
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 x2 – 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 x2 – 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=(x2 + 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.
Several months ago, I asked you, “You’re about to plan a lesson on concept [x] and you’d like students to find it interesting. What questions do you ask yourself as you plan?”
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.
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.
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 x2 + 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?”
2019 Jan 18. The Directory of Mathematical Headaches.
I’m signing on with Desmos as their Chief Academic Officer. Job one is producing the best digital math curriculum in the world. We’ve started that project already.
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.
This is the post I’ll re-read when I want to remember my five years at Stanford’s Graduate School of Education.
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
Opportunity Cost
Gratitude
So that’s that. On to the next thing.
