Tina Cardone typically blogs at Drawing on Math. She attended the Northwestern Math Conference earlier this month and has recapped her first day and second two days there, including sessions on algebra, algebraic thinking, modeling, problem solving, and engagement.

# Author Archives: Dan Meyer

# [Help Wanted] NCTM 2014

#NCTMNOLA is almost on us and it remains to be seen how much recapping will happen. I’ve posted an excerpt of my own schedule on my personal blog and I intend to recap a lot of it. I have conflicts with *great* sessions in every slot, though, so it’d be great to get some of you to volunteer to recap even *one* session for this site.

Leave your info in the volunteer page if you’d like to help out and I’ll send along some brief instructions. Thanks, everybody, and safe travels to NOLA.

# [Avery Pickford] Proof Doesn’t Begin With Geometry

Avery wins the prize for Best Session Description by sneaking in the totally droll line, “All hail CCSSM MP3.”

He briefly dinged two-column proofs, the Disaster Island where we usually sequester conjecture and argument. I’m all for broading and deepening the definition of proof but I think Avery stretched it too far to include skills like estimation and asking us to justify our answers to the Locker Problem. Is “justify your answer” any different than “prove your conjecture”?

Later, he got into territory I found challenging and really well-thought-out. He showed how simple puzzles like Shikaku could be used to introduce the difference between axioms and theorems. He showed a Shikaku puzzle and its answer (below) and asked us, “What are the rules here?”

“The numbers define the area of a rectangle” and “the side lengths of those rectangles are integers” are axioms, without which the game wouldn’t make any sense. Theorems are the consequences of the axioms, like “Prime-numbered areas result in long, skinny rectangles with side-length 1.”

He also used a variation on the old game Mastermind to create a class-wide context for conjecturing, contradicting, and proving. Great stuff. Wish you were here.

# [Michael Serra] Polygon Potpourri

Five interesting investigations with polygons [pdf]. Michael spent ten minutes prefacing the set, then let us investigate them for twenty minutes, and then asked a volunteer to debrief each one at the end.

If nothing else, it was a nice morning moment to talk about math with Internet friends. That was enough. But I’ve been struck also by how hard it is to make a given math concept more challenging for students *and* more interesting at the same time. We use bigger numbers. We mix in fractions and decimals. We lengthen the problem set. We time it.

For instance, once students understand how to find the sum of the interior angles of a polygon, it’s like, what do you do to make this more challenging *and* more interesting?

Michael introduced *donut polygons*:

Finding the interior angle sum of a donut polygon makes the original task more challenging *and* more interesting at the same time. In particular, it has a great stinger at the end when you find out whether or not a triangle inside of a pentagon has the same angle sum as a pentagon inside of a triangle.

Michael had two questions at the end that asked, basically, “Do your conclusions hold if there’s a dent in the polygon?” Then, “What about *two* dents in the polygon?” This messed me up a little bit, because, no, it shouldn’t matter, but then why would Serra include the *two* questions? Basically, Serra had your correspondent feeling briefly but completely off balance.

# [Allan Bellman] Manipulatives vs. Technology: Bring Your Bias and an Open Mind

Great premise for a session:

Pose two lesson objectives. For instance:

- Students will be able to understand why the angles in a triangle always add to 180 degrees.
- Students will be able to understand how to calculate the shortest distance from a point to a line and then back to another point.

Allan then brought any resource you’d want, from low-tech to high-tech, everything from tracing paper and scissors to a class set of TI-Nspire’s. We used what we wanted to explore those objectives and then debated the merits of the analog and digital technologies.

The debate hit a high register pretty quickly. For the 180 degrees question, people tended to favor the diverse ways you could demonstrate it with paper. (Cutting, tracing, drawing, etc.) With the NSpire, you had one. You downloaded an applet from Allan and moved vertices around, watching the angle sum stay constant.

For the shortest path problem, most people preferred the calculator because you were able to set up the constraints and then drag a point around to see where the distance bottomed out. The only analog method we discussed was to draw the scenario and then use string to measure the different possibilities, slowly narrowing in on the answer.

For my part, I was bothered that we never discussed a) any other digital technology, aside from Texas Instruments calculators, or b) the cost of a class set of *any* kind of technology.

Granted, I probably make sport of Texas Instruments too much (and I’m hardly unbiased here) but truly I just find the user experience miserable. From the untouchable, low-DPI screens, to the time it takes me to find the right button out of the millions, to the mindless, button-pushing worksheets teachers have to pass out just to make the devices comprehensible, to the lurching way the cursor moves across the screen in Cabri, I find the whole experience pretty painful.

It would have been great to see the same problems approached with Geogebra or Desmos, for instance. Or even an Excel spreadsheet.

Then there’s the cost. All other things held equal, the high-tech solution will still cost thousands of dollars more per classroom. So we shouldn’t be talking about which solution comes out barely ahead of the other. Technology should shoulder the greater burden of proof here.

# [Breedeen Murray] Telling Stories, Teaching Math

Bree claimed that stories are a useful medium for student learning. She backed this up with lots of citations that I hope she’ll share somewhere. [**Update**: She has.] She posed ideas for filtering our own lessons through the logic of stories – setting context, adding conflicts, etc. She closed by asking us try to create story-based lessons for exponentials and fraction division.

My thoughts went to St. Matthew Island, which I’ll link without elaboration.

# [Robert Kaplinsky] Real-World Problem-Based Learning Using Perplexing Tasks

Robert showed us this image and asked us to figure out how much it cost.

I’ve seen his lesson plan before but it didn’t prepare me for how interesting the math became.

We used the In-N-Out prices for a hamburger, a cheeseburger (a hamburger + cheese), and a Double-Double (a cheeseburger twice over) to figure out the cost of the 100×100 burger.

Our calculation was exactly right but we arrived at it differently. I used a system of three linear equations. My seatmates used a bit more conceptual creativity and got the same answer with a lot less computation. Robert highlighted all of these methods.

My takeaway: it’s really, really hard to describe in a text-based lesson plan all the awesome, heady moments it may provoke. I knew the lesson would be fun and productive. I didn’t know it would be *this* fun and productive. How do we create lesson plans that convey all those highs, rather than just the nuts and bolts of their implementation?

# [Uri Treisman] Keeping Our Eyes on the Prize

On April 19, 2013, the third day of NCTM’s annual meeting in Denver, Uri Treisman gave a forty-minute address on equity that Zal Usiskin, director of the University of Chicago’s School Mathematics Project, called the greatest talk he’d ever heard at the conference in any year. Stanford math professor Keith Devlin would later call it our “I have a dream” speech. At least one participant left in tears.

I’ve personally seen it three times. I got the video feed from NCTM and the slides from Treisman. I then spent some time stitching the two together, resulting in this video. His message is important enough that I’d like to use whatever technical skills I have, whatever time I have, whatever soapbox I can stand on, to help spread it.

If you’re interested in equity, you should watch it.

If you’re interested in teacher evaluation, you should watch it.

If you’re interested in school reform, you should watch it.

If you’re interested in charter schools, you should watch it.

If you’re interested in understanding which student outcomes teachers can control and which they can’t, you should watch it.

If you’re interested in the trajectory of math education in the era of the Common Core State Standards, you should watch it.

If none of those conditions apply to you, well, I can’t imagine the series of misclicks that brought you to my blog. Watch it.

Here’s a fair enough summary from Treisman himself:

There are two factors that shape inequality in this country and educational achievement inequality. The big one is poverty. But a really big one is opportunity to learn. As citizens, we need to work on poverty and income inequality or our democracy is threatened. As mathematics educators … we need to work on opportunity to learn. It cannot be that the accident of where a child lives or the particulars of their birth determine their mathematics education.

That was his destination and the talk took only three stops along the way:

- What did education reform groups like Achieve, the Gates Foundation, et al, recommend in their “Benchmarking for Success” document in 2008?
- How does TIMSS and NAEP data contradict or clarify those recommendations?
- What should we
*actually*do about equity, as teachers and citizens, if those recommendations prove unfounded?

**Highly Quotable**

- [A]s math people we know that if we’re going to work on a problem, we have to formulate it clearly. And as math people are wont, we need to swaddle ourselves in the numbers and the data because that’s what gives math people direction, strength, and courage.
- Let’s look at “Benchmarking for Success” and see its analysis of the problem. Then let’s look at the data and see how it actually lines up with what we know today. And then let’s see where we need to go to really enact the vision of NCTM for equity.
- So the notion was: “Let’s focus on teachers as the central driver of reform and rethink how we evaluate teachers.” They had the view that teachers were the single most important in-school factor in student achievement. And math people know that was just an artifact of the way they modeled the problem.
- I’m now going to show you two graphs that I don’t believe anyone in the math community has seen. It’s the PISA data disaggregated by child poverty rates.
- About one half of students who go from high school to college are referred to remediation and mostly developmental math. Fewer than a quarter of those students will ever get a credential. Those students are more likely to end up with debt than a credential. [..] Those remedial programs are burial grounds for the aspirations of students. And it’s mostly math that’s the key trigger. 35,000 students in California two years ago repeated a developmental course for the fifth or greater number of times. So no one can say those students don’t have persistence.
- So states – where you go to school – are a profound influence on what you actually get to know.
- Low income student scores in Texas were the top in the country in 2011. It’s really good for Texas to be the top of the country. Because whenever Texas does something well, everyone else is
*positive*that they can do better. When Massachusetts is at the top, people go, “Ah, it’s just Massachusetts.” - Again, two and a half years difference in opportunity depending on where you happen to go to school. This is something that, as a math teaching profession, we can influence. Poverty is something we need to work on as citizens. Opportunity to learn is something we need to work on as math educators. That’s a core message for this talk.
- So you would think that charters would fix this. Almost all the charters in Texas produced 0% of students who are college-ready. There are a few of them – one KIPP, one YES Prep, one IDEA, one Harmony – that are pretty good. Most of them are well below the public schools. So this theory of Achieve, NGA, CCSSO, Race to the Top, that charters were the answer? Not so clear when you actually climb into the numbers. The reverse looks true.
- When you visit most math classrooms it’s like you’re in a Kafkaesque universe of these degraded social worlds where children are filling in bubbles rather than connecting the dots. It’s driven by a compliance mentality on tests that are neither worthy of our children nor worthy of the discipline they purport to reflect. That is the reality. That’s something that we as math educators can control.
- What this shows is that the current theory about school improvement – that charters, Common Core, value-added measures of teaching are going to solve the problem – is profoundly wrong. That doesn’t mean we can’t use the Common Core powerfully to reboot our systems but it’s not the solution to the basic problems of schooling.
- Guess what? Poverty really sucks. It’s incredibly hard. All the lifespan studies going back to the 1920s show that poverty and youth is a very hard force. We need to build fault-tolerant schools and systems if we’re actually going to address equity.
- Just think about it. The great majority of our children finish our schools
*positive*that there’s a whole list of things they’re not. They come out of schooling believing they’re not mathematical, they’re not artistic, they’re not philosophical, they’re not athletic. And these self-imposed beliefs undermine your sense of personal freedom, the font from which all freedoms come. - You have to remember that when the Common Core was created, they didn’t come to NCTM. They got David Coleman to write it and he brought his friend Jason Zimba to do the math. They did not come to NCTM. It’s time for us now – the professional societies – to talk about what standards should be and how to reshape the Common Core so that it reflects our best practice knowledge of schooling. Hard message, but a necessary message.
- What is the determinant of whether you have a high-skill job in the US? Overwhelmingly it’s mathematics. It’s the single biggest factor in upward social and economic mobility. It’s our beloved subject. It would be wonderful if it were music instead of math. Think how great the country would be if everyone were striving to learn to play an instrument instead of factor quadratic equations but the fact is it is our discipline that is the primary determinant.

*Dan Meyer is the editor of MathRecap.com. He blogs at dy/dan and tweets @ddmeyer.*

# [Help Wanted] Recappers Needed For NCTM13

NCTM 2013 is on us in two weeks.

**It’s a banner year for speakers.**I’ll post a few recommendations shortly but you have all the usual institutions plus a few new upstarts from the blogosphere. I’m looking forward to it.**It’s a very expensive ticket.**No two ways about that.

So if you’ll be attending NCTM, consider recapping a session or two here at MathRecap. A photo and a few paragraphs is all it takes to open the conference up to the 99% of math teachers worldwide who can’t attend. Leave your details at the volunteer page if you’d like to help them out.

# Recaps Around the Web

- Henri Piccioto recaps Avery Pickford’s session on student-posed problems at CMC-North 2012. (Part one & part two.)
- Tom Ward recaps the best of NCTM 2012 in Chicago, with posts on sessions by Sarah Greenwald, Adam Poetzel, Ron Lancaster, and Jen Szydlik.