All posts by Dan Meyer

[Uri Treisman] Keeping Our Eyes on the Prize

By Dan Meyer Published May 16, 2013

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.

You should watch 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:

  1. What did education reform groups like Achieve, the Gates Foundation, et al, recommend in their “Benchmarking for Success” document in 2008?
  2. How does TIMSS and NAEP data contradict or clarify those recommendations?
  3. 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.

Posted in General Interest, NCTM, Treisman, Uri | Leave a comment

[Help Wanted] Recappers Needed For NCTM13

By Dan Meyer Published April 1, 2013

NCTM 2013 is on us in two weeks.

  1. 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.
  2. 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.

Posted in Uncategorized | Leave a comment

Recaps Around the Web

By Dan Meyer Published December 7, 2012
Posted in 6-12, CMC-North, General Interest, K-5 Tagged: , , , , , | Leave a comment

[Michael Shaughnessy] Reasoning and Sense-Making: Keys to Student Engagement

By Dan Meyer Published December 3, 2012

Shaughnessy’s morning keynote covered Reasoning and Sensemaking (RSM). He spent the first ten minutes reviewing the different resources NCTM offers. Then he illustrated RSM with four tasks he had us work on and debrief. You can find all the problems in his slidedeck but the fourth one interests me most.

Problem 4

A “data detective” exploration. What do you notice in these tables? What do you wonder about?

As we noticed sums between rows and columns and speculated what they could be describing, he progressively added column and row headings, which in turn sent our reasoning down new, different paths.

His point was that RSM can and does occur in “naked number” contexts.

Resources

Posted in CMC-North, General Interest, Shaughnessy, Michael Tagged: , | 1 Comment

[Karen Arth] Mathematical Modeling

By Dan Meyer Published December 3, 2012

This is the second session I attended from Karen Arth at this conference and I’ll re-iterate again that I appreciate her communication skills. She also has a lot of good ideas, even though I disagreed with one of her premises in this particular session.

She started by asking for our definitions of “modeling” and I thought the audience’s suggestions were astonishingly perceptive. Two well-turned phrases:

  1. “Start with life, do the math, check back with life.”
  2. “Approximating the world.”

“So what I’m hearing from you guys is ‘application’,” Arth said, and I made a note to ask her a question about that later.

She gave the following Algebra II problem, which she called “traditional”:

Write the equation of a parabola going through the points (0,0), (20,50), and (40,0).

a) Use the form y = a(x-h)^2 + k.
b) Solve for y when x = 25.

Then she gave us the following revised problem, which she implied was the same task only with modeling:

McDougal’s Restaurant has a play area for children under and around their giant arch (in the shape of a parabola with negative orientation). They plan to set up a new activity that allows children to bungee jump from the arch. The manager, upon hearing of your team’s expertise, hires you to calculate the maximum stretch of the rope that will keep the kids safe. The arch is 50 feet high and 40 feet wide at the base. The jumping location will be 5 horizontal feet away from the axis of symmetry of the arch.

a) Write an equation to model the shape of the arch.
b) What’s the maximum length to which the cord could stretch to keep McDougal’s safe from lawsuits?

She had us create a graphic organizer for our work that included sections for “Notes,” “Labeled Picture,” “Table,” “Estimates,” “Assumptions,” “Calculations,” and “Recommendations.”

“If I look at your picture, if I look at your table, if I look at your notes, I want to be able to tell the whole story,” she said.

We worked on the task in groups and presented it. Then I asked my question. “Earlier you equated modeling with applications. Is there ever an applied task that doesn’t involve modeling?”

She told us about several SBAC performance tasks in the modeling strand. Bruce Grip asked if my question had been answered. I said I was still unclear on what features of those performance tasks made them modeling. Is it enough to have “real world” objects in your problem?

Our own definitions of “modeling” are interesting, of course, but the CCSS and other documents have offered their own, very specific definition, which includes, among other skills:

  • Identifying essential variables.
  • Formulating a model that uses those essential variables.

Arth had us talk about our model’s assumptions, which is a core component of modeling, but her task also gave us the essential variables (the height of the arch, the width of the base) and it gave us the model also (“the arch is in the shape of a parabola with negative orientation”). This seemed to make the task something other than modeling.

I said all of that and she replied that, given her student population, she had to offer them some of that information because the task might be too foreign and unmanageable otherwise. I’m sure she knows her students’ capabilities better than I do. Whatever our students’ capabilities right now, though, I hope we’re all moving in the same direction, towards the same definition of modeling.

Posted in 6-12, Arth, Karen, CMC-North Tagged: , | 1 Comment

[Karen Arth] Making Functions in Algebra Active and Interesting

By Dan Meyer Published December 3, 2012

Karen Arth is a strong communicator, full stop, and for that reason alone I’ll be sure to look for her byline in future conference schedules.

She also works for CPM, so if that’s curricula you use, it’s likely you’re already familiar with the function activities she described in her talk. I’ll explain them briefly and share her resource [PDF]:

Function Machines

You give student four functions on cards. Ours were:

  • f(x) = sqrt(x)
  • k(x) = -x/2-1
  • g(x) = -(x-2)^2
  • h(x) = 2^x-7

Then you give them an input and tell them what the output should be after you compose all those functions.

For example, Arth gave us 6 as the input and told us that 11 had to be the output of the last function.

We shuffled cards around and finally determined that f(h(k(g(6)))) = 11.

These functions aren’t chosen arbitrarily, though, and that’s where you take your small-group instruction and whole-group debrief. k(x) changes the sign of the input and creates non-integer outputs if the input is odd, for instance. So Arth asked us why we made the choices we made, and our responses revealed a lot of mathematical understanding.

Silent Board Games

“Teachers love these,” Arth said. “Because students love them, because they’re silent, and because the hand that goes up is not necessarily what you consider the A student.”

Arth drew a horizontal x-y table on the board. There were two complete x-y pairs. The other pairs were only half complete.

Students were to raise a hand when they thought they knew a missing part of the table. Arth would then invite the student up to write the missing part. She’d do nothing if the number was correct and she’d erase it if it was incorrect.

Beneath that table on the board was a blank for “English Sentence” and then “Math Sentence.” It felt a bit like Wheel of Fortune, where you know the missing pieces and then decide to solve the puzzle.

Here, again, Arth emphasized process. “What was your strategy? What were key moments for you?”

One participant said that once someone else completed the x-y pair (0, -1) she knew an important fact.

Personally, I was confused throughout much of this activity. If Arth said the function was linear, I didn’t hear it, which meant I had to figure out what function family we were even looking at all while my fellow participants were having no trouble completing the puzzle.

I’m interested in hearing from people who have tried this activity in a CPM class. Do struggling students find it as easy to participate as Arth suggested?

Number Walks

Arth took us outside and laid out a large x-y axis on the ground. Several of us stood from -4 to +4 on the x-axis. Arth then said, “Take your number and add two.” As we all formed the line y = x + 2, she had us turn and link shoulders, creating a visual of the line.

Other people took their turn. f(x) = 2^x was a real treat to watch, as the people standing at negative numbers struggled, first, to figure out if they should step backwards, after which they did their best to shuffle forward just an inch.

Treasure Hunt

You can find this resource on page ten of the handout but my group never managed to make sense of the instructions, which probably means they’ll need to be clarified or simplified for students.

Resources

Featured Comment

Matt Vaudrey:

I’ve done the Silent Function Game with my students (though I didn’t have the English Phrase and Math Phrase, which I will add). I added a few components to make it work well for any class:

–Each person can only get up once.
–You can erase or write, but not both.
–The last person writes the function rule (y=2x-7).

Then I sit in the back and make menacing looks at students who make noise.

Posted in 6-12, Arth, Karen, CMC-North Tagged: , | 3 Comments

[Lisa Nussdorfer] Using the iPad in the Mathematics Classroom

By Dan Meyer Published December 3, 2012

I try to attend every iPad session I can. The technology interests me itself but the massive interest of teachers at these conferences interests me even more. Lisa Nussdorfer’s session would have broken through the fire marshall’s maximum three times over if she had let everyone in the room who wanted to attend. What accounts for that interest?

These are expensive tools, after all, and the question in the back of my mind at all times is, “What problem are they meant to solve?” If “iPad” is the answer, then what is the question?

Is the question:

  • “How can we incorporate technology students love into our own practice?”
  • “How can we lighten the administrative load in our classrooms?”
  • “How can we enhance and expand access to the very math our students learn?”
  • All of the above.

In every iPad session I’ve attended, I tend to feel like the rest of the participants and the speaker have all settled on the question or questions, while for me that question is still open.

These sessions must be hell on presenters, who have to differentiate a population as diverse as any you’ve had in your classroom. We had participants with every level of experience from those who didn’t know that double-tapping the home button gets you a list of open apps to those who were reliably using box.net to distribute and receive class assignments. There were teachers managing one iPad, six iPads, and thirty iPads per classroom.

So Lisa Nussdorfer covered some of her favorite apps and fielded questions for the remainder.

She highlighted ShowMe, Educreations, and Explain Everything saying, “You’ve heard of Khan Academy. These give you the ability to be Khan.” (A participant called out Doceri as another alternative.) Later she said, “I wanted to be Khan briefly but whenever I show a video to a kid they’re kinda like ‘Eh.’ It didn’t quite have the response I was expecting.”

Another participant said, “I don’t use [those screencasting apps] because they don’t look good and I don’t sound good on them.”

Nussdorfer called Algebra Touch the coolest math app out of all of them. It’s a useful illustration of algebra skills, but a participant pointed out that “Algebra Touch does the work for you.”

She pointed at another app and said, “This is free but you have advertisements so that’s kind of a problem.” I’m hoping we’re all on Nussdorfer’s page with this one. Advertisements in front of kids should be a non-starter.

The questions from the crowd took us in some interesting directions. One asked Nussdorfer why she wasn’t recommending Fuse, Houghton Mifflin Harcourt’s Algebra textbook for the iPad.

Nussdorfer replied: “I got the sample of it and thought, ‘It’s a textbook.’ I thought it was like a textbook. If you’re going to spend money on an iPad, you should get something more than just a textbook. And it’s sixty dollars.”

Another participant called out the low resolution of writing on an iPad, relative to writing on a whiteboard or paper (see above, where the word “Doceri” takes up half the width of the screen):

I’ve found the iPad device to be not a very good device to write on, at least with math. You just don’t have a fine enough stylus or control over your writing. It’s like one of those big pencils for first graders.

As I said, I always find these sessions interesting and I’m glad I attended. At the moment, the question that interests me is, “How can we enhance and expand access to the very math our students learn?” But not just on an app-by-app basis. I’m not necessarily in the market for a list of apps mapped to content standards. I’m mainly curious how an investment of tens of thousands of dollars to put digital, networked devices in every student’s hand will result in more interesting math being taught to more students. That question, for me, is still open.

Resources

Posted in CMC-North, General Interest, Nussdorfer, Lisa Tagged: , , | 10 Comments

[Kyndall Brown] Access, Equity, and the Standards for Mathematical Practice

By Dan Meyer Published December 3, 2012

Kyndall Brown deserves credit for bravery. He opened up CMC-North 2012 on a controversial note, tackling race and equity in math education and allowing an enormous amount of participation and interjection from his audience.

His top-level point was that the Standards of Mathematical Practice, if implemented with fidelity, lead directly to access and equity for underserved populations. As he involved his audience in making that point, it became clear that the teachers of CMC-North are all over the map when it comes to their beliefs about and proposed solutions for inequitable access to a good math education.

First, he established his credentials (African-American male, raised in Los Angeles, taught in Compton, passionate about extending access); he established a mandate for equity (referring to NCTM and NCSM’s equity principles); he established a lack of equity (referring to data drawn from California’s Basic Educational Data System showing lower quantities of advanced and proficient minority math students).

He asked us, “What are the reasons for that inequity?” and the crowd’s answers varied. The first participant suggested that a math education wasn’t valued by those minority populations. Another participant suggested the language of our tests leads to inequity. Another suggested that parental background played a role in perpetuating inequity. Still another suggested that Asian populations score so well on these tests because Asian parents berate their students if they achieve only modest results. Some of these suggestions are, of course, baseless and highly counterproductive. Brown didn’t confront any of them directly and referred, instead, to a federal 2007 report that attributed the inequity in districts with high minority populations to their:

  • high class size,
  • low availability of advanced math classes, and
  • low access to quality math teachers.

He illustrated and confirmed each of those inequities with data displays.

So basically, the crowd whiffed it, and we should all take a second and feel humbled by that.

He drew from Haberman’s “Pedagogy of Poverty” (1991) which revealed that the dominant pedagogical practices in urban schools include:

  • giving information
  • asking questions
  • reviewing tests
  • settling disputes
  • punishing noncompliance
  • giving grades
  • marking papers

Which is nothing like what you find in the Standards of Mathematical Practice. As an introduction to “culturally relevant pedagogy” (Brown’s proposed solution) he pulled an urban district’s assessment:

It costs $1.50 each way to ride the bus between home and work. A weekly pass is sixteen dollars. Which is the better deal, paying the daily fare or buying the weekly pass?

The assessment graders said that African-American students had “strange” answer patterns. The graders assumed a five-day work week with a single trip to work and a single trip home, which would recommend not buying the weekly pass. But African-American students knew their parents worked more than five days in a week and also assumed the bus pass would be shared between family members, which would highly recommend the weekly pass.

So Brown asked us to define “culturally relevant pedagogy” in the math classroom. The audience’s responses, again, were unpredictable and occasionally troubling.

  • “Ask students to explain their thinking.”
  • “Tie a song that students like to formulas they have to memorize.”
  • “Ask them to make up their own problems.”
  • “Create contextualized problems that students are truly interested in solving.”
  • “Create real-world problems, but not just, ‘My students like basketball so I’ll do a problem about basketball.’”

A teacher expanded on that last suggestion rather artfully, saying that our attempts at creating real-world math problems can “lead to the essentialization of our students, boiling black people down to basketball and Mexicans down to soccer.”

To the question, “Isn’t this just good teaching for any student group?” a participant responded that “culture” isn’t the same as “ethnicity,” implying (I think) that “culturally relevant pedagogy” is important even if you’re teaching a bunch of white kids.

Brown then answered his own question by drawing from research, Delpit’s in particular. In culturally relevant classrooms, students feel “academic press” where the content is made clear and there are high expectations. Students are held accountable for their performance and given assistance in their achievement. There are social supports and strong social relationships. Students can build confidence and feel psychologically safe. They feel they can take risks, ask for help, admit errors, and experience failures.

Brown’s point, again, is that the Standards of Mathematical Practice, if implemented with fidelity, will lead to culturally relevant pedagogy. But as one of my companions said, “That’s a huge ‘if’.”

Brown pointed to the third practice, “Constructing viable arguments.” Brown claimed this practice required students to attend to issues of language, reading, writing, listening, speaking, and making sense of a text. He said it would require collaboration because you have to argue with somebody. This isn’t true, though, in the case of tasks that ask students to respond to the argument of a talking head in a textbook. There’s no one there to negotiate with or talk to.

He said that the fourth practice, “Modeling with mathematics,” required teachers to know their students’ lives, which, again, seems far too pat, given the uncertainty of implementation. Teachers can and will satisfy the modeling practice without knowing anything about their students. There isn’t a guarantee that teachers will ask students to model anything more than the prefabricated contexts in a textbook.

None of this is to reject Brown’s point entirely, rather to say that Brown implied too strong a line between the standards as written on the page and the experiences of minority students in the classroom. In reality, that line is mediated constantly by publishers, assessment authors, and teachers. He laid out a strong mandate with citation after citation but I would have traded a few of those citations for several more examples of faithful and unfaithful implementation, for several more images of what’s possible with the Standards of Mathematical Practice.

Posted in Brown, Kyndall, CMC-North, General Interest Tagged: , , , | 1 Comment

CMC-North 2012

By Dan Meyer Published November 27, 2012

We’ll be recapping CMC-North 2012 this weekend. Get crazy.

Posted in CMC-North | 1 Comment

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