Friday, May 29, 2015

choral counting language question

I was recently watching a choral counting video of a colleague teaching a small group of Spanish-English bilingual second graders (I'll share the video if I get permission, but in the meantime, just imagine some cute kids counting up by 10s starting at 64). Students count pretty fluently from 224 to 294, and there's just the tiniest pause before they all say 304. It's common for students to hesitate at points like this where numbers "cross over" the hundreds (is there better language to describe this? I don't have an elementary background, so I might be missing a term here), because as @davidwees wrote about in his comment here, students who've been chiming in based on a pattern ("two-seventy-four, two-eighty-four, two-ninety-four...") might be tempted to say something like "two-tenty-four" instead of "three hundred and four." This momentary pause is such a lovely opportunity to dig into conversations about place value or to probe what students do or don't understand; I might say something like "I heard you pause just a tiny second before you said 304, and going from 294 to 304 is often tricky. Why do you think that might be tricky for some students?" or simply "How did you know it was 304?" or even "Why isn't it two-tenty-four?"

This made me wonder, and I'm super curious to hear how others who are more familiar with choral counts and/or more familiar with linguistics might think about this: how would such pause points play out in different languages? Here are two ideas I have:
  • In Mandarin, it's common to drop the -ty from tens (saying 24 as "two four" instead of "two ten four" or 86 as "eight six" instead of "eight ten six"), especially when counting, and also common to insert "zero" when there's a zero in the tens place (saying 104 as "one zero four" instead of "one hundred and zero four" or "one hundred and four"). There are some interesting quirks with teens, too, such that 114 would be "one one four" rather than "one hundred four"). Counting up by 10s starting at 64 might sound like "six four, seven four, eight four, nine four, one zero four, one one four..."). Now, I learned to count in Mandarin in an informal home setting rather than in school, so it's also possible that elementary school teachers require students to say the whole number ("one hundred one ten four" instead of "one one four") the same way that in English, we might ask students to say "one hundred [and] fourteen" instead of "one fourteen"). That, on the other hand, would require students to go from "eight ten four" to "nine ten four" to "one hundred one ten four" rather than to the more intuitive "ten ten four," which might bring its own set of challenges.
  • In French, the number seventy is said as "sixty ten," eighty is said "four twenty," and ninety is "four twenty ten" (but this pattern doesn't hold true in smaller numbers). So the same series would sound like "sixty four, sixty fourteen, four twenty four, four twenty fourteen, hundred four, hundred fourteen, hundred twenty four, hundred thirty four..." The really nice thing about French is that the 17, 18, and 19 at least are said as "ten seven," "ten eight," and "ten nine," which helps with thinking and talking about place value, even though 11-16 all have their own unique words (although it also means that 98 sounds like "four twenty ten eight").
So now I'm thinking about what choral counts sound like in other languages and where the tricky spots are, and how that influences what is tricky about choral counts for ELLs and teachers planning to teach ELLs. I imagine that it'd be helpful for them to know the number structure of their students' home languages to anticipate challenges and places worth probing in their choral counts, which would provide some really delightful opportunities to support both students' mathematical thinking and their language development.

Have you noticed this? What does it make you think? Can you give us an example in another language?

Wednesday, May 27, 2015

books by and about women of color

I'm probably the only one who's noticed this or cares, but the "Recently Read" section of this blog is woefully outdated, mostly because I've given up on Goodreads. I hope to fix it this summer-- and by fix, I mean shift my entire blog to a new domain name/platform (any advice? this is all new to me!)-- but in the meantime, because a) I know a number of friends are currently building their summer reading lists and b) there can never be too much attention to books by and about women of color (see #weneeddiversebooks), here are a few I've read in the past few years that I think are worth your time-- some have become new favorites, others raised more questions than answers, and there are even a few that felt just "meh" to me, but I am including anyway because my personal taste should not limit your literary exploration :)
  • Everything I Never Told You, Celeste Ng
  • When the Emperor Was Divine, Julie Otsuka (I liked The Buddha in the Attic better)
  • An Untamed State, Roxane Gay
  • Gathering of Waters, Bernice McFadden
  • A Cup of Water Under My Bed, Daisy Hernandez
  • Land of Love and Drowning, Tiphanie Yanique
  • How to Leave Hialeah, Jennine Capo Crucet
  • The Round House, Louise Erdrich
  • Half a Yellow Sun, Chimamanda Ngozi Adichie (because I don't even need to mention Americanah, right?)
  • The Grass Dancer, Susan Power
  • NW, Zadie Smith (I liked White Teeth better, but there aren't as well-developed female characters in it, hence its exclusion from this list)
  • Free Food for Millionaires, Min Jin Lee
  • Twelve Tribes of Hattie, Ayana Mathis
  • In the Time of the Butterflies, Julia Alvarez
  • Salvage the Bones, Jesmyn Ward
  • Three Strong Women, Marie Ndiaye
  • Typical American, Gish Jen
  • Forgotten Country, Catherine Chung
  • Tiger in the Kitchen, Cheryl Tan
  • Krik Krak, Edwidge Danticat
  • You Are Free, Danzy Senna
  • Messages from an Unknown Chinese Mother, Xinran
  • Before You Suffocate Your Own Fool Self, Danielle Evans
  • Thread of Sky, Deanna Fei
And, just because these were too good not to recommend, here are some by men of color, about men and women of color:
  • In the Light of What We Know, Zia Raider Harman
  • John Henry Days, Colson Whitehead (also, everything he's ever written)
  • All Our Names, Dinaw Mengestu (also, The Beautiful Things That Heaven Bears)
  • I Love Yous Are For White People, Lac Su
  • On Such a Full Sea, Chang-Rae Lee
  • Open City, Teju Cole
  • This is How You Lose Her, Junot Diaz (of course)
  • We the Animals, Justin Torres
  • Say Her Name, Francisco Goldman
  • Cutting for Stone, Abraham Varghese
  • Learning to Die in Miami, Carlos Eires
If you have read or do read any or all of these, I'd love to discuss! 

Tuesday, May 26, 2015

engaging vs. motivating

Did you immediately read Vicki Hand's latest article when @tchmathculture tweeted it a few weeks ago, because everything she writes and the way she thinks about students and equity and space in the classroom is brilliant? If not, here's a short excerpt about the difference between engagement and motivation that will hopefully compel you to read the whole thing (bold formatting mine); it's not that long, and it's for Mathematics Teacher so it's an easy read too:
Teachers who organize contributions with students and utilize CI pedagogy are focused on engaging their students, rather than trying to motivate them. This orientation goes against the tide of improving student motivation, a concept we find to be problematic for two reasons. First, an orientation that focuses on motivation lays the blame with students for low participation and achievement. As a result, teachers try to change students, instead of the conditions in their classrooms that lead to participation gaps (e.g., prioritization of solutions over reasoning, lack of awareness of the multiple resources students bring to school learning, and stereotypical perspectives of smartness). The fact that participation gaps fall along racial, ethnic and linguistic lines tells us that we must pay attention to structures, both local and distal to the classroom, and not just individuals. The second reason we find an orientation around improving student motivation problematic is that (groups of) students are repeatedly labeled “lazy” or “unmotivated”. The positive and negative labels we assign to students’ motivation form the basis of stereotypes we have about different ethnic and racial groups.
We know, however, that children’s actions are always motivated by something. Their behavior often makes sense when it is placed in the broader contexts of their life experience. For example, consistently negative experiences in mathematics classrooms often lead students to seek ways to be valued or save face. They might try to gain their status with peers through impertinent or comical behavior, or escape notice by remaining silent. Often when students engage in this behavior, teachers feel the need to control them in some way. However, exercising control over students’ behavior tends to constrain students’ mathematical sense making and to deepen classroom inequities. 
No comment necessary, except that I want to email this latter paragraph to a million people, and especially to anyone who thinks that students need sticks and carrots to be "motivated" in math classrooms. This may also be a nice time to remind y'all of @delta_dc's thoughts on student engagement, and to encourage you to read the Morgan & Saxton taxonomy linked within if you haven't already :)

Friday, May 22, 2015

disentangling the dilemma

Wow! I was going to write a comment synthesizing the very generous thoughts and ideas in response to my last post, but the meta-comment got to be almost as long as the original post, so... here I am. Several categories of possible actions jump out to me:
Representing the incorrect idea: By not putting this idea on the board, I'm subtly signaling that it's wrong, or in some way not worth recording. This occurred to me-- which is why I attempted to record the idea of a unit rate by writing 1 hour under 18 mph (even though as @mathwater points out, 18 mph / 1 hour is not mathematically accurate because mph already includes the 1 hour). But, I didn't record the whole idea. Joshua suggests one way to do it that makes visible how the student is adjusting the denominator: she is using 1 hour (which we calculated above, although herein lies the error; the unit rate is necessarily different in each of these scenarios since we're traveling the same distance in different amounts of time, so we can't actually use the previously calculated unit rate) + 0.5 hours (which we could calculate by dividing the 1 hour rate in half) = 1.5 hours. Putting this on the board would have helped everyone understand what this student was saying, which would be useful regardless of what I planned to do next.  
Then, there's addressing the incorrect idea: It seems there could be value in moving on more quickly than I did, or in sitting with it longer than I did. Mylene would have moved on to minimize embarrassment and put some space between the idea and the student before returning later, Christopher would have pushed for a more conceptual justification (which presumably could have led the student to recognize her error, although I'm not sure I would have been able to facilitate that skillfully), and mathwater and David both suggest turn-and-talks for students to engage further with the idea. It makes me think that, had I represented the incorrect idea on the board, I could have elicited the second idea, and then asked questions like "what is similar and different about these two strategies?" or "how do both strategies use previous information to support our thinking?" (although, again, the first strategy used previous information incorrectly, so this would have to be clarified in some way). And I could even have, per mathwater's suggestion, asked "how could you use ideas from the first strategy to help you arrive at the second strategy / at the correct answer?" The value of any of these options, of course, is that the student's incorrect idea is treated as valuable and used as fodder to help others arrive not only at the correct answer but also at the understanding that the journey there matters, which Michael also emphasizes when he suggests returning to the student to ask how she revised or clarified her thinking after hearing others' ideas. Max even suggests an extension-- "for which question would this strategy be viable?" which Mylene also asks, along with "what might be some other plausible but invalid strategies?"  
As Joshua points out, however, sometimes as a teacher you just need to continue the original line of discussion because the intended instructional goal is different, and further unpacking the error would be a distraction. This was certainly the case here; my goal was to demonstrate a Number String (this was within the first 20 minutes of a 3-hour session on Number Strings) so I wanted participants to be thinking about the structure of the activity rather than the mathematics. So, I didn't want to deviate too much into a discussion about rates. However, because this moment presented itself, it does feel important to address it in some way, rather than sweeping it under the rug, both for the sake of how the staff member actually feels and for the sake of demonstrating how I might address it in a classroom. 
If I wanted to explicitly affirm the mathematically useful parts of the first idea but also make very clear that it is ultimately incorrect in a way that doesn't get us lost in the weeds, perhaps I could have said something like "[name] looked at previous problems to help solve this one, which is an important strategy. In this case, however, we can't use the previously calculated rate, because if we're traveling the same distance in a different amount of time, our rate will be different. What information from the previous problems do you think we COULD use to help us out instead?" 
Explicitly addressing the culture: David shares how he tells students that mistakes help us learn and proposes an additional debrief in which the participants talk about the usefulness of this example in helping us think about teacher decision-making, Joshua suggests that modifying the task/problem itself could actually help shift the culture, and Max wonders whether it's okay to actually just flat out say "that's wrong!" so that we can talk about why without tiptoeing around the incorrectness. This seems worth doing, and I appreciate Mylene's reminder that this is a demonstration of intellectual courage and autonomy-- which are much more valuable than just being right all the time.
I think what I wish is that I knew the right way to say something like "hey. It sounds like you felt embarrassed when you realized you'd shared an incorrect answer. Why did you feel embarrassed? What were the conditions-- your expectations of what we were doing, expectations or pressures you felt as the person volunteering a response, norms we've come to assume in math classrooms, features of your own identity or mathematical history, the power dynamic between you and me, perhaps-- that made you feel that way? Can we talk about those things, and how they influence the way students feel in our classrooms? And how it might influence our practice as teachers to be conscious of that?"

...So if you know how to do that, let me know :) Otherwise, thank you for engaging in this thought experiment with me-- if you've got additional ideas, please continue to comment, and I do hope this sparks some of you all to share your video too!

Thursday, May 21, 2015

dilemma: defusing the culture of correctness

I want your help. I've been thinking lately about facilitating adult learning (because, let's be honest, when am I ever not?) and how to communicate/share/model/instill particular mindsets or orientations around the learning and teaching of mathematics that are different from participants' default setting. Specifically, how do you address breaches of the culture you're aspiring to create? So, I'm making my practice public-- an instance where I wish I'd responded differently-- and hoping to hear some alternatives that you generate. Please?

I’m demonstrating a number string for a group of staff members (including some who are joining virtually via webcam/phone) who support secondary math teachers, many of whom I’ve never met before but would be working with for a full school year. These staff members have never taught or seen number strings before even though many of them are former math teachers—this is part of their initial learning experience. In addition to creating a mental image for these staff members of what a number string is, I also wanted to demonstrate the type of learning experience we should be providing for teachers; in contrast to the typical “I lecture about a strategy and give examples/non-examples and then you apply it” professional development session, this is a demonstrate-then-debrief approach. One more layer: I wanted our work together over the full year to gradually chip away at the cult(ure) of correctness that often pervades math classrooms (fishing for or steering people towards the right answer, treating wrong answers as dangerous, only valuing people who give right answers), especially because I knew that many of these staff members—despite having taught math themselves—still didn't see themselves as “math people” and still saw the discipline of mathematics as a difficult-to-access temple that not everyone could enter.

Prior to the clip below, we've worked through finding the speed when traveling a distance of 36 miles in 2 hours and in 3 hours, and I've asked whether we'd be going faster or slower if we traveled that same distance in 1.5 hours. Someone has explained why we're going faster, and now I'm asking for a calculation of the speed. Here's what's on the board:

During the demonstration, a staff member acting as student gave a strategy for finding speed that wasn't mathematically viable, thinking it was a viable strategy (not trying to throw me off or test me). Making an actual mathematical mistake can be embarrassing for an adult who obviously knows how to calculate speed, and it could be particularly embarrassing because she was trying to not just give me the standard algorithm; she was trying to offer an alternative. I wanted to respect her contribution and validate what was there, but not spend a ton of time digging into the exact misunderstanding both because I wasn't sure in the moment exactly what she was thinking (“she’s confusing rates with numbers and we can’t operate on rates the way we can on numbers, like the classic miles per gallon vs. gallons per mile calculation!” was about as far as I got; this is an area of growth for me because I think a teacher more fluent in talking about rate and ratio would have been able to understand this more quickly and handle it more adeptly), and because I suspected she’d quickly recognize and be embarrassed by her error. So the approach I took was to identify something that made sense about what she had said—the idea of a unit rate—and move forward to consider another strategy. However, you can hear her laughing and being embarrassed and whispering as she tries to figure out why her method doesn't work… for the next several minutes. And she brings it up in the debrief later. Take a look:

Why does this feel like a dilemma? The most superficial response here is that she and her seat-partner were now distracted and not paying attention to what I’m trying to demonstrate, but more importantly, I've allowed her to reinforce the cult of correctness. Hearing her be so vocal about her embarrassment could (hypothetically—I can’t prove this is how anyone actually experienced this particular moment, but it seems worth discussing nonetheless) reinforce for someone else that being wrong is embarrassing (rather than an important and inevitable part of learning) or that being right is more important than contributing mathematical thinking to the conversation. Someone who shared her inaccurate idea but hasn't yet figured out the flaw might feel even more embarrassed; I’m stupid both for being wrong and for not figuring out why it’s wrong. And, while she’s being vulnerable in public, it’s not exactly by choice; admitting an error because it’s been made visible is very different from exploring possible errors or areas for growth that others haven’t seen, and in fact can sometimes be a defensive posture: If I tell you what I did wrong before you tell me what I did wrong, then you can’t really criticize me.

What to do instead? 
There are two places I think I could have handled this differently: a) when she actually made the error and b) in the debrief. In the moment of the error, I could have stopped to dig in further, even though it would have taken us away from the number string, because it would have allowed us to engage in some important mathematical conversations. In the debrief, I could have explicitly addressed her reaction and used it to make a broader point about the type of culture we want to create in a professional learning community and why that culture is important. I don’t know if I necessarily should have done the former, and I’m not sure I know how I would have approached the latter. That’s why I’m curious for your feedback; how would you have addressed this in a room of teachers? When would you have said/done something, and what would you have said/done?