Sunday, July 29, 2018

Identity and Belongingness: Five Anecdotes


"Well, I'm glad it won't be all white men up there," I said to A., upon hearing she'd been invited to represent teachers on a panel on equity in math education. A. seemed irritated. "You're all making assumptions because I'm a black woman. I grew up educated mostly with Jewish white men; I have white privilege from that. Why should I be the one up there?"

I had a sudden recollection of talking with my friend from high school after she won a National Achievement Scholarship, which was (until 2015) a National-Merit-Scholar-like award given to black students who scored well on the PSAT. We were both children of single parents who were receiving a wonderful education at a Quaker private school on scholarship, and she didn't like the idea that she was receiving an award I wouldn't have gotten with the same score, even though we had the same education. But even at that age, I knew we didn't really have the same education. It was harder for her; there were times she was treated more like an outsider. Besides, I told her, we both should try to exceed at science (my main interest then) and engineering (hers) partly to represent women well, but as a black woman, she would be a more important role model than I would be. Essentially, she was talking about her own personal history and identity, and I was talking about the history and identities of women in STEM fields. I can't remember if she was convinced, but she did take the scholarship money with her to Princeton. Did she really have a choice?

How and when does an individual represent a larger group? Is it how the individual identifies, or how others identify them? "Positionality refers to the tension between how we see our identities and how the wider society sees them," say my notes from a Robert Q. Berry III presentation. If that equity panel had been all white men, I'm pretty certain our wider society of math educators in the audience would see them very differently than we did with a majority of non-white panelists, before they even opened their mouths. And in fact, A. has passion and insights which the math education world needs, and which seem to spring from her lived experience as a black woman... partly because other people see her as a black woman.

Tension is right.


The girl is a biracial (black/white) sixth grader. She's in my math class, but she isn't. She isn't disruptive. She isn't rude. She isn't unfriendly. She isn't anything. Other kids like her all right. What's not to like? But when she's with them at the whiteboards, or at their tables, she doesn't say anything about math. Sometimes she doodles with the whiteboard marker. Occasionally she jokes around, in a gentle way. She seems to like me OK. But if I ask her what she's thinking about a math problem, she just sighs. On tests, she writes a little and then gives up. I can't tell if she learned anything.

We call a meeting with the girl and her mother. Her father has substance abuse problems and isn't available. Her mother is loving and concerned and brilliant and wants to work with us. She tells us how hard the father's problems are on the girl. She tells the girl how much it means to her for the girl to do her best at school, and how much confidence she has in her. The girl looks calmly aside and lets the mother do the talking. We make a plan for the girl to get help, academically and from the counselor. We all tell her how we are on her side and how much we think of her. We all feel happy about the meeting. The girl smiles.

Nothing really changes.


I'm in my twenties, in graduate school, in a meeting with my advisor and his computational chemistry group. This is his current research group, not his Golden Days group. The Golden Days group produced brilliant work, but more than that, he misses their esprit de corps (and tells us so often). Along with him, they were a bunch of brash, confident American and (I think) Australian guys who loved working together, spent all their waking hours in the chemistry building, partied together, and had jokes together. The youngest member of the Golden Days group, G., is still there, but the rest have moved on to be replaced with Chinese and Taiwanese men and me.

At this meeting, we're talking about the latest workstations. Workstations are hugely expensive computers that can do our calculations far faster and better than personal computers (which most of us don't own), and they have something else personal computers don't have: WINDOWS, which let you see and do more than one thing at a time on the same screen. It occurs to someone that maybe soon you can actually watch TV on the workstation while doing your research! This strikes everyone as hilariously inappropriate, and we are all laughing about it together. "You could check out football while you're running your Monte Carlo simulation!" says G., and we crack up. And then he adds: "Julie, you could watch soap operas!"

I stop laughing. I want to tell him that I've been watching football all year because I was curious about why people liked it so much and wanted to see how little knowledge it would take to fake my way through a conversation about it (answer: not much). I want to tell him that I never, ever watch soap operas, and that when most other girls I knew in high school watched General Hospital after school, when it was trendy, I refused, because the main characters got together through a rape story line that infuriated me as a feminist. But I don't say either of those things. I can hear the reaction in my head as if I did, though. It was just a joke. Lighten up. 

My boyfriend gets the brunt of stories like this. He tells me I am smart and belong there and should just act like it. He tells me my anger is off-putting, that men don't want to talk these things through with me because I get upset. Lighten up. I think he probably does listen, though. Later, after we break up, he goes to grad school and eventually becomes a science professor and researcher. Now he's a crusader for women in science, years after I stopped being one.

How strange that this stupid little remark about soap operas, with so much less apparent impact than other worse examples of exclusion or sexism from those times, bothered me so much I remember it decades later. I wonder how much of that is because I didn't find any productive way to react to the anger, and since I've been ashamed of that, I've tried to shame my anger out of existence. Proponents of mindfulness say to acknowledge a feeling and not judge it. Perhaps that's worth a try. Anger takes up less room by itself than with shame attached.


The boy is in eighth grade. When he started public school, he didn't speak English. Now he's a student at the top of his class and an Eagle Scout. He plays the violin two hours a night. Later, he will get degrees from UCLA and Harvard, and will rise to head a school district of almost 50,000 students. But all that is unseeably far in the future. Right now, at school, he is a kid who is so quiet he wonders if his teachers know if he can talk. They don't know anything about his home or his family. None of them look like him, and few of the other students do either. Later, he says he felt disconnected and disenfranchised.

I got this story from a video of three Latino and Latina school district leaders speaking with a moderator about "seeking solutions to familiar challenges." Portland's superintendent, Guadalupe Guerrero, was answering a question about how his background highlighted issues that affect kids in our schools now. He added:

"We often think of students of color who are having challenges—you know—particularly those externalizers. We see it in the disproportionality of behavior referrals. But we don’t often think about the internalizers… How do we get to know, in the public school setting, who our students are? How do we ensure that they can point to at least one adult—imagine if we could create a round table or circle of people who are explicitly making sure that our students have that level of attention and support so that we’re not pushing them out?"

In the same video, Richard Carranza, New York City Schools Chancellor, says about his own time at school: “All of the teachers, all of the counselors, all of the support people that thought I was smarter than I was—so I was. So they believed in me, and they pushed me… It’s those little moments in time that build the resiliency of kids… I can remember thinking wow, if she’s really smart, if he’s really smart, if they think I can do it, maybe I can do it… We’re not educating widgets, we’re educating souls, and souls need to be told that they have a vision, and we have a vision for them.”


The letter is shoved under my door, typed on white paper and signed. I had no idea it was coming.

Dear Ms. Wright,

I wanted to thank you for being such a great teacher. You have made me more confident in my math skills ... and you have made me a better person in general. I look forward to your class every day and I look forward to getting to talk to you... I used to always dread math and try to avoid it at all costs but now I love it. It has opened so many more gateways for me. You have really boosted my outlook on school and changed what I want to do with my life. You not only made me proud of myself but you've helped my parent realize my strength and become proud of me for it. You're the best teacher I've ever had and I don't want to leave your class next year but I'm so excited to take my math skills to the next level and make you proud. Thank you for all you've done for me.

As I read this letter, I'm smiling, and tears are in my eyes. I feel wonderful. I feel validated. I am so happy she knows her power as a mathematician and loves math now.

It's bittersweet when I think of the other kids, the many who would not and could not write me such a letter.

But I'll hold onto the vision that maybe more of the math dreaders, the internalizers, and the externalizers will feel this way someday, too, if I keep becoming a better teacher.

Friday, March 31, 2017

Hidden Figures

Last Friday, every student at the arts-focused public middle school where I teach math walked to a nearby theater to see Hidden Figures, the 2016 movie about African-American women who were essential to the success of the early US space program, based on the book by Margot Lee Shetterly. We were accompanied by most of the school staff, including our principal and assistant principal, as well as dozens of family chaperone volunteers, a crowd totaling more than 400 people. Seeing the film together and talking about it afterwards with one of my classes were some of the most meaningful and fulfilling experiences I've had as a teacher, and I feel so much love and pride for my whole school community.

A lot of people made that field trip happen, but I was the main force behind it and grew almost obsessed with it, and I want to write about why. As a math teacher, I've become more and more focused on equity in math education and passionate about inclusion and empowerment, and this event sprang from that passion. I love helping my students love math and become confident and competent mathematicians and problem solvers, but I've never lost the memory of what it feels like to know others think you don't belong in math or science and to wonder if maybe they are right. I know that having good intentions and promoting logical thought isn't enough by itself to counteract the fog of racism we all live in, and that STEM fields are still unstable territory for girls and women in our culture. As a math teacher, I have the responsibility and the privilege to help make things better. Sometimes I manage it, and sometimes I still fall short of where I would like to be, but I have to believe that if I keep caring and educating myself, I will be more and more effective at raising everyone up as mathematicians, students, and citizens. I'm fortunate to teach in a time when we can collect and share resources to do that.

As soon as I started seeing information about Hidden Figures last fall, I was hooked. It sounded almost too good to be true — a fun, feel-good, PG film that brings STEM and civil rights achievements by Black women in the early 1960s into the light, shows the excitement of the intellectual accomplishments behind the space program, features stellar actors, and puts a mathematician center stage as a heroine — wow! I read a blog post by a fellow member of the "MathTwitterBlogosphere" (#MTBoS), Delaware physics and math teacher John Burk, with the subject "Let's Start a Movement for Hidden Figures," and thought a school field trip would be a fantastic idea if the movie turned out to be as terrific as it looked. Teachers in my professional learning community at school were intrigued with the idea and interested in working on it, and the school administrators, the school Site Council, the Equity and Climate and Culture Committees, and every other teacher who heard about it were positive and supportive.

As more previews were posted and early rave reviews came out in the media and from STEM educators, I got more and more eager to have my students see this story of courage and accomplishment. I made my screensaver into the picture of Dorothy Vaughan (Octavia Spencer) and her coworkers marching down the hall, posted about it on Twitter, and got an "I LOVE THIS!!" retweet from one of the actors in the picture (fun!!). The week Hidden Figures came out in local theaters, I showed all my math classes previews and a brief clip of the real mathematician Katherine Johnson (now 98) and told them about my own excitement. One African-American girl had seen it before the regular release date, and had watched the previews enough that she was mouthing along to all the words... about women doing math and engineering! A Black mom of another girl later told me the family went to see the movie and thanked me, saying, "I think it’s great that she’s being exposed to careers that require a good math knowledge." Several Black and multiracial girls told me about doing their Social Studies current event assignments on the real-life figures from the movie. White students and boys also went to see it and told me how much they loved it. After a snow-related delay, I finally saw it myself with my own teenage son and we both found it incredibly moving and thought-provoking (and the whole, full theater cheered at the end). As I had expected to be, I was struck by the main characters' heroism as individuals, but I was also impressed with how well the movie showed teamwork and support, how accurately it portrayed math as problem solving, and by the complicated story line about Dorothy Vaughan promoting her whole group's skills and successes and anticipating how the IBM computer would change their workplace.

It was around this point that my enthusiasm for a whole-school field trip moved to determination, especially when our principal went to see Hidden Figures and became one of the strongest proponents of a whole-school field trip, and when our school counselor encouraged me to follow through with planning and generously shared materials, tips, and work from her experience on similar trips. The whole thing took a ton of planning and support: honestly, I might have hesitated more if I had realized how much I was asking from our secretaries and counselor, especially. But I could not have asked for a more positive, can-do attitude from the whole community, and it made the whole thing stay fun and inspiring. Just about the entire staff worked on organizing permission forms and payments and sponsorships, even the school nurse, who wouldn't even be at our site that day. For our professional learning community (PLC) one afternoon, a dance teacher, a social studies teacher, the principal and I had one of my favorite school meetings ever, as we brainstormed about curriculum for teachers to use the afternoon of the movie viewing. Staff, students, and parents thanked me so many times for leading this effort that I lost count, and it was so encouraging to know they were happy about it. Even the National Council of Teachers of Mathematics president, Matt Larson, was talking about Hidden Figures on his NCTM blog the same week as our trip!

The day of the trip, I was a little edgy as our large crowd walked to the theater, wondering how it would go, but the kids were in excellent spirits and great fun, and I don't think I heard any complaints on the 3/4 mile walk, even though it rained all the way (our native Portlanders were completely unphased by that). We actually split across two theaters, and it was pretty awe-inspiring to see how many people were there in each. I loved seeing the movie just as much the second time. The kids seemed totally absorbed during the movie, though in our theater they were quieter and more solemn than I expected (except with Octavia Spencer's line to the white supervisor, "I'm sure you believe that."... that got a vocal reaction!). I was delighted and a little relieved when they clapped as the credits rolled. It seemed fitting that the sun unexpectedly came out for our walk back. Everybody seemed happy to have this experience together and I heard such a positive response to the movie from those who had seen it for the first time, both students and adults.

After lunch, we split up into our sixth period classes, and using our PLC's discussion guide which was a lot like this one (in which I edited out a few questions that originated elsewhere), we talked through these questions:
  • What did you think of the movie? Did you enjoy it? 
  • What questions or feelings did it leave you with?
  • Why do you think we spent school time seeing this movie?
  • What does the term “Hidden Figures” refer to?
  • In the movie, we saw many times characters were treated unfairly because they were Black. Which three examples stood out most to you?
  • What character attributes and/or actions did you admire most about Katherine Johnson (the mathematician), Dorothy Vaughan (the manager who taught herself how to program the computer), and/or Mary Jackson (the engineer)? [I also ended up asking them which was their favorite; as I expected, Mary, played by the glamorous and fun Janelle MonĂ¡e, was in the lead, but I was surprised that at least a third of them picked serious, ultra-competent Dorothy Vaughan.]
  • In the film, Space Task group director Al Harrison (Kevin Costner) was depicted as a heroic breaker of boundaries when he smashed down the “Colored Women” restroom sign. Unlike many of the dramatic moments in the movie, this incident was entirely made up. In real life, Katherine Johnson herself chose to use the restroom white women used. Why do you think the screenwriter and filmmakers added this incident? Do you think adding this incident improved the movie? Why or why not?
  • What was most striking to you about how men treated women in this movie, and how women treated each other? What aspects of how women were treated do you think seem similar to our times, and what seemed different?
Just two hours before spring break, almost all of my sixth grade class members were intent on considering weighty questions and issues about history and justice and racism and sexism. One girl wondered, "Why were these women 'hidden'? Why didn't we all know this history?" These eleven- and twelve-year-old students were so insightful, intelligent, and empathetic that I couldn't possibly do justice to the whole discussion here. I was so incredibly proud of them. I did write down their answers to the question, "What examples of characters being treated unfairly because they were Black stood out to you?" so you can get an idea of how observant and thoughtful they were. This list was generated not just by a few kids, but from a majority of kids in the room:
  • Dorothy Vaughan and her kids were chased out of the library by the security guard (and she pays taxes for libraries!).
  • Katherine Johnson's coworkers set up a "Colored" coffee pot for her... and it was empty.
  • The bathrooms and drinking fountains were segregated, and the courthouse and bus had "colored seats in back." Separate was NOT equal: the things labeled "colored" were dirtier and cheaper. White people acted like this segregation was right, using words like "your kind". 
  • Mary Jackson couldn't access the class she needed to because of the segregated night school.
  • Katherine Johnson's new coworkers assumed she was the custodian.
  • Paul Stafford kept telling Katherine Johnson that computers can't be authors of technical papers. [Actually, I think the way she was restricted to being a computer was more about her being a woman, but it wasn't completely clear.]
  • Mrs. Mitchell claimed to Dorothy Vaughan that "I don't have anything against you all," which was obviously false (and even the way she talked about black women as if they weren't people like her was insulting).
  • Dorothy Vaughan was doing a supervisor's work, but without the credit or pay.
  • When the police officer approached the women by their broken-down car, he assumed they were doing something wrong, and they were scared of what he would do.

We had been afraid that the hour would drag by as we teachers tried to convince kids to have a serious conversation when their minds would be on the Talent Show that followed it and, of course, on the start of spring break right after the show. Instead, we had a fascinating and vigorous discussion that I was sorry to cut short. Kids even asked if we could continue it the Monday after spring break!

I went to the Talent Show full of fondness for my own class around me, the wonderful performers who included my present and past students, and the adults that support them in their arts and in their studies. Da Vinci teachers regularly go so far beyond classroom teaching. Our drama, dance, visual arts, music, and writing teachers do incredible work to bring our students' art to the wider community and bring professionals to the students; a math teacher is one of the Talent Show organizers and a mentor to the student rock band; a language arts teacher organizes a yearly trip to the Oregon Shakespeare Festival in southern Oregon; our science teachers have kids work on community projects; our social studies teachers have them engage with the broader world as citizens in so many ways; our librarian just spent many lunchtimes and a Saturday with kids doing Oregon Battle of the the Books; the list just goes on and on. I volunteer to assist with some of these events, but usually I feel like my community contributions are limited to my classroom role and General Adult Help. It was inspirational to have my own idea and to be supported and given the freedom to carry it through, which was an experience that makes me appreciate my school and its administration and staff even more. And although it's nothing new for me to get resources and ideas from my online math teacher community, this was yet another time that they helped me create something better than I ever could have on my own.

I was already elated with the entire day, but it got even better when the assistant principal thanked me from stage during the Talent Show for organizing it, and the whole school cheered — what an incredible feeling! From what I heard from teachers at lunch and after school, they also loved the whole event, both the movie and the discussion afterwards in their classrooms. As if all that weren't enough, one of those teachers sent the most amazing email the next day to me and the counselor and the secretaries. Here's the part that brought tears to my eyes: "Of everything I've been involved in at dV, this was the most rewarding and was true community building from the heart. [...] Overall, what amazing kiddos we have!! Thank you guys was so meaningful."

So I started spring break with a full heart, in the best way. It felt good to contribute to community building, good to do a little bit to move society forward, and absolutely fantastic to have 389 kids see this movie and love it. I am so proud of our kids and our community, and grateful to Hidden Figures that we are all more educated on this story than we were last year. What will happen next with our community discourse on race, gender, workplaces, schools, justice, film, math, science, technology, engineering, heck, the whole world? I don't know, but I can't wait to see where we go.

Saturday, May 7, 2016

How a Farflung Internet Conversation Helped Me Make Sense of My Students’ Sense-Making

A shorter version of this article originally appeared in The Oregon Mathematics Teacher (TOMT) in May/June 2016, and is reprinted here by kind permission of TOMT and the Oregon Council of Teachers of Mathematics. The writing of this article was supported by the Writers’ Retreat facilitated by the editors of TOMT and funded by the Oregon Council of Teachers of Mathematics.

Math educators almost universally place high value on reasoning, sense-making, and logical thinking. The first two Common Core Standards for Mathematical Practice begin “Make sense of problems” and “Reason.” We promote math education as a means for students to learn how to analyze, understand, and improve the world around them. But what are we to conclude when students respond to a mathematical absurdity not with skepticism and reason, but with hapless and nonsensical attempts to impose classroom math upon it?

An online blog article by math educator and researcher Tracy Zager titled “Making Sense”raises this question with a fascinating example, a video filmed by Robert Kaplinsky, a district math specialist in California, who also wrote about it at his website. In the video, eighth graders were presented with a question popularized by Katherine Merseth: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?” Astonishingly, 75% of the students performed operations with the numbers in order to come up with a numerical answer. Robert added that a group of sixth graders had done even worse, universally coming up with numerical answers rather than pointing out that the problem does not make sense as written.

I’m sure I wasn’t the only person to watch this video and immediately question whether it was truly representative of American students. Maybe, I thought, Robert’s group of students was particularly intimidated, or unusually bad at math. But further reading showed his results were actually quite typical. The original researcher whom Merseth wrote about, Kurt Reusser, also found in 1986 that three fourths of schoolchildren did not make sense of the question. More recently, Robert’s comment section had comments from several teachers who tried the experiment with their students, and Tracy wrote about elementary school teacher colleagues who tried similar corresponding problems with their younger students. In many different groups, with varied teachers, a majority of students failed to make sense of the problem or object to it. Robert added in a comment on Tracy Zager’s post that when he reran the problem with a whole class, even after thorough group discussion of all the individual answers, many students still chose the nonsense answer. Tracy admitted that she had thought younger children would do better: “I had assumed kids went with nonsense by middle school because of what we’ve taught them about math. Now that I have preschoolers and 1st graders going with nonsense too, I have to revisit my assumptions.”

What is going on? “America has produced a generation of students who engage in problem solving without regard for common sense or the context of the problem,” concluded Merseth in 1993, before blaming societal attitudes about math, limited curricula in which “stress is on computation and procedure, not understanding and sense making,” and poor teacher preparation. Jo Boaler, author and professor of mathematics education at Stanford University, has also criticized our culture’s emphasis on computation and procedure. In her video “Why We Need Common Core Math,” she states, “If you ask most students what they think their role is in maths classrooms, they’ll tell you that it’s to answer questions correctly. They don’t even think that they’re in maths classrooms to learn or to explore the rich set of connections that make up mathematics. They think they’re there to get questions right.”

On Robert’s site and in Tracy’s blog’s comment section, teachers struggled to make sense of why students weren’t making sense. Here were some of their thoughts:
  • Some believed the interaction between the teacher and student was key: “The student’s main objective is not to make sense of the question, or even to answer it correctly, but to give an answer that will satisfy the teacher,” said a commenter named Dave on Robert’s site. “Students believe that an honest answer (‘I don’t understand the question’) would not satisfy the teacher, so they make random guesses instead.” Commenter Pierre added, “Students are conditioned to assume that the teacher is teaching something sensible.” “It does appear that some students are overriding their common sense… and my hope for this video is that it will help us all reflect on why that is,” Robert Kaplinsky responded to Dave.
  • Commenter MG on Robert’s post blamed overtesting, in part: “When we base schooling around testing, students learn to look like they’re smart in order to fool the test […] without understanding.”
  • Annie Fetter of the Math Forum wondered in Tracy’s comment section if drawing a picture would help kids “step back and do the sense-making that they won’t do if they think it’s math.”
  • Commenter Joshua (of ) on Tracy’s post theorized that “students seem to be triggered to think they are answering a ‘math-class’ question (MCQ). MCQs have […] exactly one single correct answer, […] can be answered using the tools you have been taught, […] require a mathematical object as an answer […], are asked of students, not by students, [and] they require entering into an idealized world where you only have the information given to you within the question, no more and no less.” Joshua’s ideas seemed to correspond with some shared by Dan Meyer in his well-known TED talk “Math class needs a makeover”: “[W]hat problem have you solved, ever, that was worth solving where you knew all of the given information in advance; where you didn't have a surplus of information and you had to filter it out, or you didn't have sufficient information and had to go find some.”

I Wonder What Would Happen In My Classroom?

If you’re like me, as you read about this problem and how students reacted to it, you started itching to go ask your own students how old the shepherd is! After all this online discussion, I decided to try to investigate my students’ sense-making with a nonsense question. These were my goals:
  • Compare the performance of my sixth graders at an arts focus middle school (da Vinci, Portland Public Schools) with that of other groups of students. Although math is not generally the main interest or passion of my students, they often seem to me to be more willing to be original thinkers and to question authority than many sixth graders, so I thought perhaps they would be less likely to comply with a nonsense answer against their intuition.
  • Give a nonsense prompt that seemed very likely to generate multiple attempted numerical answers, in order to help ensure that no individual felt exposed or stupid when the answer was revealed in a whole-class discussion. I decided to change the question to one that I felt was somewhat less obviously nonsensical, to be cautious on this point. I chose a fraction problem because we had recently finished a fraction unit.
  • Generate a prompt which was close to one that could be solved, to see if students would make suggestions of what missing piece of information would help.
  • Examine the role of drawing pictures, as suggested by Annie Fetter.
  • Debrief in phases: sharing anonymous individual answers, discussing in pairs, then having a whole-class discussion. Ultimately, I wanted students to leave class understanding the actual answer (not enough information), knowing what motivated me to ask the question, and becoming aware that giving a numerical answer was not at all unusual. I also wanted to elicit their thoughts on why it is so common to try to answer a messed-up question with a specific answer, and how teachers can build a space where students feel safe and confident critiquing the question. 

With these goals, I prepared this Google slide for one class (Group A) and another just like it except without the cue to draw a picture for two other classes (Groups B and C):

and provided it to my students to “solve” on individual whiteboards, after giving them two key verbal instructions: that it was essential they work silently and individually, and that if they had questions for me, they should write them down as part of their work, but that I would not answer questions while they worked individually. Each class (Groups A, B, and C) had 27, 23, and 21 students present, respectively, sitting at tables of three or four students.

A followup slide for a “pair share,” which happened after I collected the individual whiteboards, included the question (“It takes…”) preceded by the prompt: “Now discuss the problem with the person next to you (or the people across from you if you’re at a table of 3).”

Following the pair share, we had a group discussion, during which students shared out their table’s thinking, then I showed them the Robert Kaplinsky video and summarized the research results. After our class discussion, I made sure to obtain the first two groups’ agreement not to spoil the surprise by discussing the problem with other classes (and I believe they did not).

What Actually Did Happen In My Classroom

Initial Student Reaction

Our school has a large emphasis on performance arts, and my students tend to be extroverts, so I expected that watching their faces as they processed the question would be entertaining, and I was not disappointed. In Group A, many students looked worried, confused, and doubtful, and a few looked indignant, within a minute or so, before they picked up the pens to draw or write. Several insisted on calling me over to whisper to me (I responded as described above). To my surprise, the expressions on the students’ faces were very different for Group B as they absorbed the question: they looked much less worried and more confident, and many looked amused. Fewer called me over to question the problem, and most were quicker to write down their answers. Group C’s students’ confidence seemed between Group A’s and Group B’s, and the few students who called me over seemed, by their body language, more ready to tell me off.

Effect of the Pictures

For Group A, the students who were prompted to draw a picture, students were generally willing to do this, but I saw little evidence the pictures helped them realize the problem was nonsense: in fact, if anything, my impression was that my insistence on a picture made some think it must be a problem I thought they could solve.

Overall Results for Individual Answers

Unfortunately, these were not perfect research conditions: I don’t have a background as an educational researcher, and with back-to-back classes it was difficult to keep perfect records. Nevertheless, the results I was able to record were quite interesting and suggestive to me.

First of all, a significantly higher proportion of da Vinci students identified the “Cocoa at Joe’s” question as a nonsense question than students in previous formal and informal studies tended to do for the “How old is the shepherd” question. In Group A, more than half clearly stated in some form or another that it was a nonsense question, and why. (My notes seem to say it was 15 out of 27, they but are not completely clear.) In Group B, a few students looked worried about it, but virtually everyone ended up answering that the problem did not make sense or there was not enough information. In Group C, 12 out of 21 students present clearly indicated that the problem could not be answered and/or was nonsense. Here are some examples of ways students expressed this thought on their whiteboards:

(More photos of answers of this type are available here in the "Not Enough Info" folder.)

A few other students in Groups A and B, as well as three in Group C, were clearly uneasy with the question and refused to commit to a real answer, writing “IDK” (I don’t know) or similar statements.

There seemed to be no clear correlation between sense-making and general math achievement; if there was any trend, the students who seemed most confident it was a nonsense question tended to be from both the high and low ends of the class, grade-wise, more than from the middle.

Details of Individual Answers

As I expected, many of the students who gave a numerical answer answered 1 ¼ (the sum of 1 and ¼, the two numbers given) or 4 (the number of ¼’s in 1; the quotient, although they probably didn’t think of it that way). I was surprised by how many people answered 1, including one girl who said she thought it was a riddle: perhaps in their attempts to make sense of the question, they decided only one serving was described, so it must be one person. Here are some pictures of work from people who tried to explain their numerical answer:

(More photos of answers of this type are available here in the "Numerical or No Answers" folder.)

I suspect several of the people who avoided writing down a specific answer or line of reasoning (despite my encouragement) didn’t think the problem made sense, but were too confused and underconfident to write that down.

Results for Answers After Table Discussions

In all three groups, during table discussions, every single student became convinced it was a nonsense question. Group B seemed to spend the most time talking about why they thought I asked a nonsense question. In Group C, I was interested to hear several students loudly and confidently referring to algebra (this idea may have spread from one table to another as they overheard it, or maybe several people came up with it). “It’s algebra!”, “n,” and “random number” were some sentence fragments I wrote down.

After discussion at their tables, many students volunteered to summarize their tables’ thoughts. The people who spoke for the tables were mostly medium- to high-achievers in math class, but many others seemed intent on listening to their answers and clearly agreed. A few typically high-achieving speakers had begun with a numerical answer and seemed eager to share why they were now confident it was a nonsense question. Here was what the groups said when they reported back (comments recorded in chronological order):

Group A:
Our table agreed that if you knew how much they used in one day, then you could answer, but [that was missing].
You were trying to make us look at the question more than the other information.
It seems like they took out a sentence. It seems like algebra – looking for missing things, and how they fit together. Some of the things we need to answer the question aren’t there.
I changed my mind from the discussion. It says “how many people ordered,” and you only have information about one cup.
It doesn’t give you that little teaser! Any number could be right. There’s not enough information.
The question doesn’t make sense!

Group B:
You can’t do the problem. There’s no information about how much milk or cocoa. You wanted us to think of how equations can be wrong or [how there can be] not enough information.
You’d need to know how much was used. At first I thought it was a riddle.
I thought it was n. You have the information but not all of it, so it’s as many numbers as it is. [Another student from that table interjected: It could also be n x ¼ and n x 1 and add together.] It reminds me of [Dan Meyer’s] TED talk – you give them partial information and make them ask for more. [My mind was blown by this, since I wasn’t thinking of the TED talk at all, and wasn’t even certain which periods I’d shown it to. I asked, “Who thought of the TED talk while you were working on this?” and four people around the room raised their hands.]
There’s not enough information. At the end it doesn’t say who ordered.
We all agreed there was not enough information. [We thought] “Maybe she wanted to test us.”
We could not do it! Somebody got some numbers, but I said, how could you do it??

Group C:
I wrote one person, but I realized there’s no way to answer, so now I say x.
We all could have realized it wasn’t multiplying fractions or anything. [There’s] no answer, so we could call it x.
At first we looked for an answer but [… it] could be any number.
I think it’s also a letter because we’ve been studying algebra.

I asked Group C, “Why do you think I gave you this problem?” Four new students answered (this time across the achievement spectrum):

To show sometimes problems have algebra letters.
In algebra, there’s not always a specific number answer.
Maybe the problem was written wrong. [Interestingly, though, this answer was from the only student in this class who reported that an elementary school teacher had given his class a similar (nonsense) problem.]
Maybe whoever wrote this wanted us to think about algebra.

I believe I asked Group B a similar question, although unfortunately I didn’t write down my question(s), just their answers. One student stated firmly that people “lack common sense.” Another said that people “talking among themselves helps make sense.”

How They Felt

I asked students how they felt about this question when they were working on it on their own, and being middle schoolers, they were eager to share.

In Group A, a student reported feeling “annoyed,” and about three-fourths of the students agreed. Another student reported thinking it was funny; three or four people agreed. One added, “I thought it was one of the funniest things a math teacher’s ever done.”

In Group B, when asked to describe how they felt while working on this problem, one student replied, “That was amazing!” Students volunteered words to describe other ways they felt while they were working on the problem by themselves, and we got a count for each adjective of how many people felt that way. Out of 23 students, five felt “frustrated/GRRR,” nine felt confused, four felt content (which puzzles me!), six said “funny,” two felt surprised, and eight felt stupid (“because I couldn’t get the answer”).

In Group C, I solicited a list of words from students to describe ways they felt while they were working on the problem by themselves, and we got a count for each adjective of how many people felt that way (out of 21). Eleven people reported feeling stupid, seven felt confused, five felt indignant or annoyed, three felt determined, and two felt amused.

Take-Aways and Conclusion

I probably question and second-guess every instructional decision I make as a math teacher. Do problems that are challenging or that do not have a particular right answer help students develop as critical thinkers, or are they just discouraging “trick questions”? Does group work help students communicate ideas and come to a deeper understanding of math, or is it just pointless off-topic chatter or copying of work without comprehension? Do my students need more time for practice of procedures and less for conceptual understanding? Should they respect my authority more, or continue to be allowed to question me and the tasks they are given?

I’m going to continue asking those questions of myself, and I am sure the answers will fluctuate depending on students’ needs. Nevertheless, this activity and the results from it, especially compared to the results obtained in other settings with similar problems, gave me a renewed appreciation for the role of non-“math class questions,” group work, conceptual understanding, and student questioning in developing mathematical reasoning and sense-making. I work hard at promoting critical thinking and conceptual understanding, and believe elementary school teachers in my district do too. I think this emphasis contributed to how well the da Vinci students did at identifying the nonsense question. Students also clearly progressed in their thinking in the group discussion, partly because they were comfortable with this format, and they developed sophisticated ideas together of what the activity was about. Finally, I allow and sometimes even encourage students to criticize textbook problems that don’t seem to match real life (though often we solve them anyway). This used to feel indulgent, but now I feel it is important for my classroom culture and their critical thinking.

I also realized that our students need us to convey that mathematical reasoning and sense-making is not about getting procedural answers as fast as possible. When students use math outside of the classroom, they must make decisions about what types of problems they can solve and what information they will need. We do them a disservice if we only give them “plug-and-chug” problems. For Groups B and C, I recorded that 19 out of 44 students (43%) felt stupid when they could not solve the nonsense problem, including many who had correctly answered that there was not enough information. I hope they finished class realizing that the same critical thinking that made them so uneasy that they were “doing it wrong” was actually a strength.

As Dan Meyer says, “Math makes sense of the world. Math is the vocabulary for your own intuition.” Our students all deserve to have that experience.

Wednesday, May 4, 2016

Online Networking for Math Educators (or: How to MTBoS)

This article originally appeared in The Oregon Mathematics Teacher (TOMT) in March/April 2016, and is reprinted (with minor edits) by kind permission of TOMT and the Oregon Council of Teachers of Mathematics. The writing of this article was supported by the Writers’ Retreat facilitated by the editors of TOMT and funded by the Oregon Council of Teachers of Mathematics.
Over the course of just four days last August, my math teacher colleagues and I had lively, fun conversations in which we exchanged curriculum and lesson ideas, played math games, shared examples of math in the news, recommended teaching materials, and discussed classroom culture in math class. Several of us talked about ways to teach order of operations and the distributive property. I learned a math version of hangman, and we had a great time playing several rounds. I heard about a fascinating article about the discovery of a fifteenth type of pentagon tiling that covers the plane and a neat online “population mapper” that shows different areas of the United States which have equivalent populations. A teacher shared a National Council of Teachers of Mathematics free lesson on considering percentages and nutrition while picking out food from a McDonald’s menu, and another recommended an excellent (and cheap) whiteboard cleaner. Finally, several of us talked with some experts in math trauma and math classroom culture to try to analyze what steps we could take to improve our students’ experience with and confidence in math.

Wouldn’t you want to have discussions like these with your fellow teachers? You can! Every conversation mentioned in the previous paragraph happened over a few days on Twitter. The network called “MathTwitterBlogosphere” (or MTBoS, sometimes pronounced “mitt-boss”) is open to any teacher who wants to join the online conversation or just listen in. The site, set up by teacher volunteers, has good advice on how to search for people and resources that interest you. The MathTwitterBlogosphere is a positive, welcoming community of creative math teachers from all over the world who share activities, resources, ideas, feedback, and encouragement. As one participant, Anna Blinstein (@borschtwithanna), summarized, “It's the place for connections, rich discussion & exchange of ideas. People here support, question, and challenge me to be better.” You can find discussions at times that suit you (Tuesday at 9 pm? No problem!), and connect with specialists who have expert knowledge, with teachers who have experience with the same ages, classes, lessons or teaching ideas that you do, or with people who teach math to students older or younger than yours. You can find and solve puzzles and problems that remind you why you love math so much. If more of us in Oregon join the MTBoS, we can use it to share Oregon-based lessons and tasks with each other and coordinate in-person gatherings to share ideas, expertise, support, and professional development.

Among all the social media sites out there, Twitter is uniquely suited to act as both a discussion platform and a set of links to other websites with valuable math teaching content. You can read anyone’s tweets by going to (for example, for OCTM’s account), but setting up your own free Twitter account will be far more convenient for browsing, even if you never tweet anything yourself… although I hope you’ll be inspired to do that, too, before long. I recommend setting up an account just for interacting with math educators. If you have or want other Twitter accounts, keep in mind that it will be far easier if each account has a unique email address associated with it (other people on Twitter won’t see it). Most MTBoS participants recommend picking a user name that corresponds somehow to your real name (as you’d introduce yourself to teachers at a conference, for instance, @TracyZager) or is otherwise memorably connected to your personality (like @Veganmathbeagle). Lots of names are taken, so you might need to use a middle initial or add something like “math,” but choose a shorter and easier to say user name when possible, and you may want to avoid special characters like “_” that are harder to access from phones. For your profile (brief autobiographical information), it’s helpful to add an image and identify yourself as a math teacher; consider adding your grade level and geographical location, too. You can change any of these items later.

After you have a Twitter account, you can set up your “timeline” (the tweets you’ll see on your Twitter home page) by following other accounts. To do this, search for a person’s or organization’s name or Twitter user name, click to go to their Twitter page, and click “Follow”. I’d suggest following OCTM (@OregonMath), NCTM (@NCTM), and Explore MTBoS (@ExploreMTBoS) to start. As you follow more accounts, Twitter’s recommendations of other accounts to follow will be increasingly useful.

When you feel ready, retweet some tweets you like, tweet your own ideas and questions, or reply to a tweet in your timeline by clicking on the little left-pointing arrow at the bottom. Accounts mentioned in your tweets or retweets will get notifications, although they won’t always respond. If the first character of a tweet is “@” (as it is for most replies, the tweet shows up in the timelines of only accounts mentioned in the tweet and accounts following both the author and a mentioned account. These tweets are not private, though – in fact, you can see any public account’s replies by clicking on “Tweets & replies” from their Twitter page – and in the MTBoS, at least, it’s considered perfectly fine to join a conversation whether you’re mentioned or not. You can also click on any tweet to see more of the conversation. If someone wants to tell you they especially liked what you had to say, you might get a notification that they “favorited” your tweet (by clicking on the heart).

The last big piece of Twitter know-how you need concerns hashtags (words or phrases starting with “#”). Hashtags are just markers that let people search for tweets about a particular topic or event. They can be used to flag people’s attention: if you add “#mtbos” to a tweet, for instance, people may find it even if they didn’t previously know about your account. At math conferences, people may add a hashtag like “#NCTMAnnual” (for NCTM’s annual conference) to their tweets, so that other people can find them.

Hashtags are also used to mark tweets that are part of “chats,” which are scheduled (and occasionally unscheduled) conversations open to anyone who wants to join. You can read tweets in a chat by entering the hashtag (for instance, #elemchat) as a search term in Twitter. You must click “Live” to see all tweets. Scheduled Twitter chats about math focused on a particular grade level range or subject area include elemmathchat, msmathchat, alg1chat, geomchat, alg2chat, statschat, precalcchat, and calcchat.  Some more general chats are probchat (about complicated problem solving; currently on hiatus), slowmathchat (various topics, with tweets spread out over a week or so; also on hiatus), spedmath (Special Education), edtechmath (educational technology), and the British chat mathstlp (Twitter Lesson Planning). About the last site, co-organizer Jo Morgan (@mathsjem) reports, “It's bloody brilliant. We're working together for the sake of our students. I cannot think of a more effective use of social media.” This comment could really apply to any of the chats listed here!

Chats are an excellent way to introduce yourself to the online math community while learning a lot, but busy chats can be confusing to follow at first. Sometimes people use Storify ( to save a series of tweets in an organized way, so that people can go back and read the “story” later. Two examples I especially like are and You need a free Storify account to make one, but anyone can read a story at a storify link.

If you’re feeling a little panicky at the idea of doing all the things in this article, please remember, take your time and move at the pace that works for you! I’m pretty certain none of the math teachers on Twitter participate because they’re expected to be there or get paid for it; we just do it when we can because it’s fun and valuable. If you do get to the point that you feel comfortable sharing your online contact information, let others know you’re online by adding yourself to the MathTwitterBlogosphere (MTBoS) directory at Let’s get more pins on that map and find each other! I hope to see you there and on Twitter soon!

Monday, September 21, 2015

"What three words come to mind when you think of school math?"

This post is a duplicate of a post I made to my classroom blog.

In a survey at the beginning of the year, I asked my 125 Math 6 students to tell me what three words came to mind when they thought of school math. Almost everyone answered, and this Wordle, which I also shared at Back to School Night, shows all the words that were listed by two or more people. The larger a word, the more students listed it. (The colors are random.)

To my teacher mind, some of these words describe associations I'm pleased to see incoming sixth graders having with math, some are merely neutral or factual, and some are associations I hope will change. I'd class them like this (listing the words in each category in order of frequency):

Positive (8): fun, challenging, interesting, learning, challenge, exciting, yay, smart
Neutral/Unclear (22): hard, multiplication, numbers, difficult, homework, easy, complicated, addition, subtraction, division, fractions, shapes, adding, equations, algebra, practice, math, complex, school, long, work, ok
Negative (9): boring, confusing, ugh, scary, irritating, annoying, weird, meh, stressful

I think it's great for students to find math challenging, as long as they're not discouraged, so I put "challenging" and "challenge" into the positive category. But I'm actually uncertain whether to classify "hard" and "difficult" as neutral or negative. Hard work can be intensely satisfying and can lead to great learning, but when "hard" is one of the three primary associations eleven- and twelve-year-olds have with math, well, I worry that they're feeling overwhelmed. At the other extreme, "easy" can be OK if it is a word used by a happy and confident student, but it might also be contributing to why "boring" appears so many times, so it reminds me I need to provide challenges in math class for all kids.

By the end of the year, I'm hoping "fun" and "challenging" replace both "hard" and "easy," and that the rest of the negative words are wiped out!

As for the students' description of the subject matter of math, arithmetic looms large, which is not unexpected for students coming out of elementary school. I do find it interesting that many mathematicians describe math as the study of patterns, yet not one student listed that word, even though they've undoubtedly looked many times at patterns in math class. Could it be that they believe "real" math is the symbols and arithmetic, not the patterns and relationships? If so, I'd like to change that so their view of math is more expansive.

Finally, I would love to see the word USEFUL showing up by the end of the year. Middle school math is arguably the most useful math students learn, but I hope they will realize how powerful it is now, not just later in life.

Saturday, July 18, 2015

The Perniciousness of Negative Numbers: Are Our Children Safe?

What follows is a paper I just wrote for a class called History of Mathematics for Middle School Teachers. The actual name of the paper was "The History of Negative Numbers in Mathematics and Education," but that's not exactly click bait, so I livened it up here. Most of the paper is about how mathematicians sort of pretended they were ignoring negative numbers for a while, then publicly freaked out about how ridiculous they were, then finally came to love them. I'm sure many of us have had an in-law relationship like that, so maybe you can relate. The end of the paper is an appendix with some thoughts about activities for students that reinforce the methods and philosophies described in the rest of the paper.

Monday, May 25, 2015

List of List of Websites for Math Teaching Resources

In less than a year on the "MathTwitterBlogosphere," I've collected tons of great resources in Diigo lists, which I've now exported to blog posts here. Some ideas and lessons on these lists are ones I've used already in my middle school math classes; others are on my summer research list.

Here's how I've sorted websites (sometimes arbitrarily):

Math Websites: Specific Lessons
Math Websites: Classroom Culture & Mathematical Practices
Math Websites: Professional Development and Teaching Ideas
Math Websites: Meaty Problems
(Not Just) Math Websites: Games & Puzzles
Math Websites: Educational Technology

And here's one more I forgot to add originally! This is a list of sites I made for my students so they could investigate math in different contexts that interested them:

Math You Can See: Art, Nature, Patterns, Society, and More