There must have been forces at work outside of you that contributed to that mistake, right? It seems like a pretty common mistake. Are those forces still at work? What can we do about them?

“What will I do differently next time is a good starting point,” as I am wondering what to do now. Start a blog.

Our 100-student public high school (call it alternative) has come under the watchful eye of Debra Gist and the state regulators for having low test scores. We now have to have math classes in order to lift scores. Previously all math was through projects. And while we are cramming our 11th graders in an effort to improve NECAP scores we are very much interested in, long term, elevating Mathematics as an art form so students are, as you say, are engulfed in the mysterious, captivating, perplexity of it all. Math doesn’t need to be relevant as long as it is inspiringly perplexing.

I operate under the premise that all humans are attracted to some for of problem. We want to puzzle, but we do not all gravitate toward the same puzzles. My work is also inspired by Paul Lockhart, author of A Mathematician’s Lament. The scathing indictment of education (especially Math) I’ve read.

Lockhart ran a 3-day symposium of sorts in Prospect Park in Brooklyn in mid-August. I brought four students from Rhode Island and our principal time and we had a blast doing math in the park. There were people of all ages drawing lines, rectangles, taking apart locking puzzles, working with shapes, basically immersing themselves in some dimension of inspiring perplexity.

This is the culture we are hoping to build in our small school in Newport, RI. Here are some of our questions:
What is Math? What is Mathematics? What should students study, Mathematically speaking, and be expected to know upon graduation from high school? Should students be required to pass a test in order to graduate?

You get the drift. I have posted some thoughts in an effort to solicit wonderful ideas on what to do next.

Steve Da Catalyst

Four years is a HUGE milestone. I just realized that I’ve been going at it for three years, since I’ve started teaching. And that’s largely thanks to you. I read your blog when I was on the search for a teaching job, and it was from reading you that I started blogging. Because I wanted to join in the conversation too! The conversations you were starting.

So CONGRATULATIONS.

You rock.

(And some good larnin’ can start with a worksheet!)

If you take a worksheet as a placeholder concept for lesson design/instructional design, you were correct. It seems much of your ethos is around generating compelling and engaging problems, and working through them in an engaging way. It’s a way down the road from a good worksheet, but it is the same road.

Congratulations on going strong for four years, and for all the accolades that you’ve deservedly received. I know that all of your readers appreciate your thoughts, and look forward to even more.

I agree with Sam’s parenthetical that worksheets can actually have a valuable role, insofar as they can provide structure to both students and teacher. I’ve often sensed from this blog a certain bias against scaffolding, or at least the explicit variety. Of course, even the most seemingly constructivist lesson is very structured; if it weren’t, things would quickly spiral into chaos, and “let’s see what we come up with” isn’t a viable practical strategy in a standards-based classroom. Still, when you taught linear equations, say, certainly you knew exactly where you were going, and how you planned to get there, which is why you were likely very successful as a teacher. Which is to say: there was a worksheet. You just didn’t print it.

It’s worth remembering, Dan, that not everyone is where you are. Yet. In the next few years, we’ll see more new and younger teachers than at any other time in recent history. Many of these teachers needs the structure of worksheets, of handouts, of the more “traditional” components of classroom instruction that are easy to bemoan retroactively.

Like many of your readers, I’m grateful for the role you’re playing in helping experienced teachers stretch their limits, and see you as a clear leader in instructional design: the “how” of teaching. At the same time, please don’t discount the efforts that others are making on the content side. People need different things, and I’m sure many of your readers–particularly the less experienced ones–need reminders from folks like you that, yes, it’s a process. That, no, it doesn’t happen all at once. And that there’s no singular, identifiable and *right* way to do this. If worksheets help, let’s just make sure that they’re actually good.

We need poets, who need poetry.

One of the finer posts (and follow-up comments by Dan himself) re: the life of a teacher + blogger I’ve read in a long time. Inspired by a 4-year anniversary, but inspiring much more. Thanks, Dan. For the mistake (I suppose), the ability to reflect, and the willingness (no, demand) to *fail* publicly. And to remind us to do the same.

i have to say… i’ve learned so much more from you here Dan – than the math. and i love the math.

but the class and tact and honesty and reflection and wisdom and push back and transparency you display with every post – is golden. that’s pd for everyone… and for life.

thank you for that.
and also for the very cool mathematical thinking..

I was a student of Paul’s at UCSC- he passed along A Mathematician’s Lament to me several years ago. It’s incredible. Thank you for this blog- it’s where I go when I need a kick in the butt. Unfortunately, after all the inspiration, I’m left trying to balance my own personal life on one side with this intense desire to come up with amazing problems like Dan. For now, I’ll plagiarize. :)

Four years, Dan, great work.

Sorry for pissing you off so unexpectedly throughout the years. Sometimes when I speak (write) my mind, it doesn’t go over well in some crowds. I believe in your post here you call that “failing faster, more often, and more productively.”

I’ve been trying to call it “what I will do differently next time.”

As a result, I’m learning that I’ve also become much more reserved online, measured in my words, attempting a constant gut-check on all that I write. Being imperfect, like we all are, I fumble a lot. However, I hope you understand how impressed I’ve been with your writing style, your willingness to share, and your ability to keep technology in check while endlessly weaving it throughout your instruction.

I highly doubt your students really know how good they’ve got it.

I remember that angry period. I had only been blogging about a month longer than you had. Was actually worried about you (not sure why!)…

So glad you’ve continued to write and share and provoke our thinking. Never thought I’d subscribe for so long to a guy who writes about teaching math but here I am years later…

Looking forward to much more from you in the years to come!

Hi Dan:

You’re an enormous influence. Thank you. Used the ‘How I met Your Mother’ bit in my summer school section. It uplifted an otherwise dry five hours. Curiously, they didn’t need clarification that the data was made-up. Of course it is, one girl said, no way you had ten girlfriends.

I am interested in your vitriol towards textbooks. Textbooks are created by experts in the field- a panel comprised of dozens and dozens of committed, educated professionals- many at the forefront of math education. While their design and layout is troubling (see dog, handkerchief) there are often excellent activities that provide structure around important concepts.

In other words, it’s not all 3x + 12 = 9.

Point taken.

Many textbooks present artificial (often comically bizarre) situations for systems . From your writings, I know this drives you mad. It does me, as well. Something like:

‘You sell 400 tickets to the school production of ‘My So Called Life.’ You made $1700. Student tickets cost $2 and adult tickets cost $5. How many adult tickets did you sell? How many student tickets did you sell?’

First, why is this data meaningful? The short answer: it is not. But to assume that the student is not engaged in or cannot learn something algebraically meaningful is not true.

Second, there are dozens more creative ways of figuring this out than setting up a system of linear equations.

Let the students explore them! Provide them with a highly structured problem like this. Have them solve it in three ways. In six ways. Have them solve it in a way that their little sister would understand. Have them solve it in a way that would impress a math teacher. Have them solve it as if they were trying to explain it to their past selves.

And if you much teach systems, and you must use this problem, dig deeper into it. Do not make your students robots so that they recognize a problem-type, rather than a context. Persistently- annoyingly even- make them think about what each component of the problem represents. Engage the process and not the answer.

I do not like this problem. There is not a sillier, more contrived context. But, in fairness to you, how different is that from seconds gained or lost taking the stairs?

Your original statement was correct, but let me translate it:
“The best learning occurs through solving the right problems.”

Problems are the fundamental unit of measurement from which all organization flows for math instruction. If you research top scoring countries, its not their method of instruction that has commonalities, its their problem design.