I’ve got my mindset on you, growth.

Last week Sara brought up growth mindset as an important part of talking to students dealing math anxiety and insecurity. When I first heard the phrase a few months ago, I thought it was maybe some hokey visualization mantra for aspiring CEOSs. However, I am now convinced that growth mindset is not hokey at all—it is a non-trivial concept that can really affect our students’ chances of success in mathematics, especially students from underrepresented groups. The idea is simply this: our mathematical abilities, and more generally other qualities like intelligence or character, are not fixed but may instead be developed through effort. If we believe this, we will be more successful at math (and other things) than if we don’t.

Dr. Carol Dweck is a former Stanford researcher who has done a lot of work in this area.  Dweck’s research shows that a person’s mindset can have a huge impact on their success, particularly in mathematics. She and her collaborators found that a growth or fixed mindset was a predictor of math grades for 7th graders, and that achievement was greater if students were taught that their abilities are not fixed and could be increased with hard work. There are important social justice implications for this research. Women and other underrepresented groups in math and science may have the mindset that their abilities in math are inherently lower because of gender or other predetermined factors. Researchers have now shown that even that idea can undermine the performance of these individuals and make them less likely to succeed in math.  Those of us who care about diversity and equal opportunity in math, science, economics, and all other areas need to pay attention to these studies.

Recent research shows that math anxiety is contagious; in particular that it can be passed from parents to children, not genetically but when anxious parents help their children with homework. The fact that math anxiety is learned is incredibly hopeful news, because it means there are ways to break this cycle and release people from math anxiety.  A recent study about bedtime math makes me excited that there are tools for parents to engage with their children about math in a positive, anxiety-free way, that actually help kids do better at math. Growth mindset, the theory goes, is another part of the equation.  This mindset reduces anxiety because making mistakes does not reflect upon your underlying fixed qualities as a person—it is just part of the learning process.

I learned about growth mindset this fall, talking with a colleague, and then in a little more depth at a great workshop I attended on Math Circles. Once again, I’m a bit late to the party—growth mindset has been in the news for several years. In any case, after coming into contact with these studies, I felt an actual responsibility to present the idea to my students. If I hadn’t heard anything about the implications of growth mindset, I certainly couldn’t expect that they had. And if just being told that abilities were not fixed could help them succeed, how could I not tell them that? Plus I am selfish—I want to teach students who feel that they can take responsibility for their own learning, and are less afraid of making mistakes! Who are less defensive and like to be challenged. Yes! Sign me up.

The day that I returned from the Math Circles conference, I had a talk with both of my classes about the question of fixed versus plastic intelligence and mathematical abilities. I asked them to vote on whether intelligence was fixed or could be changed through effort. Most thought it was fixed. Then I asked them the same question about mathematical ability. Even more of them thought that mathematical ability was fixed.

This disparity was weird to me: how can math ability be fixed if intelligence is not?  But nobody reading this blog will be surprised to hear about people putting mathematics in a special, frightening, alien category.  How this math fear began I have no idea. It could be related to our culture’s entwining of mathematics and genius, and then to our weird ideas about genius.  That is a whole different blog post, though.

After the vote, I talked to my students about the abilities of our brains to form new connections and the implications of growth mindset. I posted links to Dr. Dweck’s TED talk and two written pieces on the class website. The students were engaged in the discussion, asking me many questions I couldn’t answer. My current students are mostly Biology majors (I teach two sections of Biocalculus), so they were really interested in how the studies were designed, and different types of intelligence. Despite my lack of full knowledge on the subject, I had the sense that attitudes shifted that day. A student sent me an email that night thanking me. Later, when we talked about it in my office, she said nobody had ever discussed with her with the possibility that intelligence or mathematical ability could be changed. She was struggling, but this gave her the sense that if she worked hard, she would get it. Damn. Exactly what Dr. Dweck claimed! Awesome.

I am thoroughly convinced that a growth mindset is helpful to students. What else can I be doing to foster this attitude in my classroom? Is talking about it once enough? What parts of my teaching are already in line with this philosophy, and what should I be on the look out for? One important aspect of encouraging growth mindset seems to be praising effort instead of intelligence. This is something I’m really trying to keep in mind this semester. Of course, I always encouraged students who asked questions in class, whatever they asked, but now I make a point that this is part of the hard work of learning. Another great idea from Maricela Montoy-Wilson is to “normalize the struggle”.  Make it clear that struggling is not failure–it’s the right way to learn.  This semester I have made a special effort to discuss what a credit hour means and how much time they should expect to put in to the class to succeed.  I also point out that they shouldn’t be discouraged if they can’t breeze through the homework—that work is really how they are learning. You’re supposed to struggle, it doesn’t mean you’re dumb or doing something wrong. Struggle is learning.

Another recommendation I read for bringing growth mindset into the classroom was to explicitly welcome mistakes, and to share your own mistakes and experiences with developing your abilities through effort.  Mistakes, got that covered.  I will also tell my students about my lengthy and ongoing battle to learn to play the accordion.  I had what might be called “music anxiety” for many years: although I really love music, for a long time I thought that I was inherently unable to play an instrument or sing. Like, I would panic if I was pressured to sing along with something or do anything remotely musical in public. About 5 years ago I decided that fear was major dead weight in my life and I resolved to learn to play at least a little, no matter how long it took. I got an accordion and took some lessons and it was really slow going at first, but now I can play some songs pretty well. I even sing along. I love it! Now I can’t believe I waited until I was 30? Why don’t we really talk about this earlier in life?

This photo illustrates growth mindset overcoming serious music anxiety.  It also illustrates three mathematicians (Amy Ksir, me, and Renzo Cavalieri) playing Christmas Carols on accordions in public.  I know, you wish you were there, or that I at least had sound for you...

This photo illustrates growth mindset overcoming serious music anxiety. It also illustrates three mathematicians (Amy Ksir, me, and Renzo Cavalieri) playing Christmas Carols on accordions in public. I know, you wish you were there, or that I at least had sound for you…

In fact, my accordion story is something that I already tell—I talk about how math is like sports, or like music, something you have to practice, and tell them that I know it is hard. I tell my students the same thing my best professors told me: You can do this, you just have to challenge yourself, put in the effort, and work through the discomfort of confusion. The research on growth mindset is really just a supplement to what many math teachers and professors have been saying all along.  Now there is science saying the same thing, and that believing in this possibility alone can make a big difference.

I’m really curious to hear from people who have also been thinking about this. Have you incorporated growth mindset into your courses or thinking about your own math life? Did you already know a lot about it, or is it relatively new to you, too? I’d love to hear about it in the comments.

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Math Therapy for the Unqualified

I’m sure you can all relate: while making small talk with a stranger, eventually they’ll ask what you do. “Professor” is never a sufficient answer – they want to know what subject. And once you say “math,” the potentially interesting part of the conversation is immediately over. Maybe it’s that I’ve got a nonthreatening face and a Midwestern female inability to tell people to stop talking to me, but nine times out of ten I’ll get to hear all about every traumatic mathematical experience this stranger has ever had. And boy, do they have a lot of ’em.

Eventually, after tenure, I’d like to write a pop math book for these people. They’re (usually) not stupid by a long shot; successful in their field despite their past difficulties with calculus or statistics or algebra. Many of them are ashamed of their mathematical failings and have a strong urge to someday revisit mathematics and try to overcome their old obstacles.  They just want someone to assure them that yes, math is hard sometimes, but you don’t have to be a wizard to learn it. Persistence is enough for most people, and if they didn’t have the grit when they were 18, they’ve probably got it now.

This math therapy is a little dull, but it’s easy to put up with, and (mercifully) over quickly. But this isn’t the kind that I want to devote the bulk of this article to. The other kind – the kind to scared, anxious students – is much more serious, and with terrifying consequences. I’ve been trying to write this post in my head for at least a month now, and I’m still not sure my ideas are fully-formed. But this work and these thoughts have been taking up a pretty big chunk of my mental energy lately, and I hope others can contribute to and benefit from this post.

Since my very first days working in the tutoring lab in graduate school, students have asked me for as many pep talks during office hours as actual mathematical answers. Sometimes it goes beyond light encouragement and into discussions of seriously heady issues. Depression and anxiety are endemic in our students, but many are secretly dealing with deaths in the family, difficulty with their relationships, identity issues, chronic illnesses or disabilities, financial problems, domestic abuse, and suicide.

I never realized what a charmed life I led as an undergraduate until my students started pouring their guts out in my office.

Mid-semester student survival kit?

Mid-semester student survival kit?

We’re a little past the mid-semester mark now, and I’ve had a good half-dozen come by with some kind of distress or another. Most of them are just pushing up against the limits of what their previous study habits (or lack thereof) can handle. Some just want to vent, some want help. This is when I break out all the growth mindset talk I can, and offer tissues and pamphlets from the counseling center when appropriate. But I’m never sure that what I’m doing is actually helpful. So I went looking for resources.

There’s not a lot out there for faculty on how to deal with students in crisis. While I was first pondering this post, a great Vitae article came out called, simply, “We Are Not Prepared for Students in Distress,” by Katie Rose Guest Pryal. Her article gives advice from a former professor and dean who is also trained in counseling. Two main themes are “Be Gentle With Yourself” and “Know Who to Ask for Help”, which, coincidentally, are also numbers 2 and 5 in the Top Ten Things I’m Terrible At. I’ve probably re-read this weekly since it came out.

In researching more on the topic, I found another post from my new favorite blog, Tenure She Wrote. The post, called “I’m Your Professor, Not Your Therapist,” is from another new faculty member who, like me, is not naturally comfortable with other people’s tears. The comment section is a wealth of tips, from the simple (keep tissues in your office) to the complex (don’t burn yourself out trying to help your students emotionally). Also: beware of crocodile tears!

The comments on that article also treat something I’d never really thought about, namely that female faculty bear more of a burden in this arena. We’re expected to be caring and nurturing, therefore more students come to us, where we feel more pressured to help them – to the detriment of our research programs. To my knowledge, there’s nowhere in my tenure dossier to list the number of tears dried or anxieties eased, and I need to remember that when students start getting too needy.

All this was complicated enough, but then a ProPublica article, “When Students Become Patients, Privacy Suffers,” came out last week. The first half of it describes a 21-year-old student who was briefly hospitalized for mental health issues after seeking help from her campus counseling center. The center notified her parents, even though her record noted a “broken relationship” with them. Her parents flew down to campus, where the student felt pressured by the school to go home. There, her family prevented her from continuing therapy. She ultimately graduated, but was only allowed to return to school if accompanied by her parents. The school was trying to do what was best, but for students from a toxic home, I can’t imagine a worse ordeal.

I’ve pushed dozens of students towards on-campus counseling centers over the years, and many of them are terrified that their families will find out. Going to therapy may be on the level of a regular dental cleaning for some, but for others it’s the equivalent of being thrown into the stocks in the town square. School counseling facilities are the only option for students who can’t afford private treatment, or who don’t want their parents to see that they’ve used their health insurance. I thought FERPA meant the worst that could happen is that they use up their allotted sessions too quickly. Now I’m not so comfortable.

I’ve asked around campus for more help. I want specific guidelines about when I’m required to report a student’s issues to the school. I also want the counseling center’s policy on reporting health issues to students’ parents without their permission. I’d like a list of best practices for teaching students on the autism spectrum. And possibly most critically, I’d like to know what to do when I suspect a student might be a danger to themselves or others.

This column will likely be the first of many over the next few years. One of the reasons I love my job is that I like helping people, especially with math. I like seeing students overcome their anxieties. I like trying to be the person who I would have wanted to talk to when I was younger. I’m glad my students see me as a resource both for their classwork and for how to be a grown up without going crazy. But I need to make sure that I’m actually a force for good in their lives, that my advice is valid. And more importantly, I need to not focus on their problems at the expense of my tenure obligations.




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Like a book club with math instead of books

I love reading and the idea of book clubs, and I have always wanted to be part of a really stimulating and fun book club, with bright, unpretentious people thinking hard about good stuff. I haven’t found the book club of my dreams yet, but I may have found the math-instead-of-book version: Philadelphia Area Math Teachers Circle (PAMTC). Math Teachers Circles are communities of teachers, professors, and education professionals who meet to work on interesting mathematics problems. The model is similar to that of Math Circles for students (see Adriana Salerno’s report about her math circle here). Someone presents a problem, and the participants work on it in groups, everybody has fun and learns new math. Since the participants are teachers, there is some discussion of how to connect the problem and ideas to the classroom, and it is possible to serve wine (all the more like a good book club!).

"Pancake sort operation". Licensed under CC0 via Commons - https://commons.wikimedia.org/wiki/File:Pancake_sort_operation.png#/media/File:Pancake_sort_operation.png

Pancake image by Derrick Coetzee, “Pancake sort operation”. Licensed under CC0 via Commons

It seems that the best problems for Teachers Circles are “low-threshold, high-ceiling” problems (in the words of the National Association of Math Circles, and someone I talked to during the meeting). These are problems that everyone can get started on and figure out some examples, but which generalize to any degree of difficulty, and often have extensions that are unsolved. At this month’s meeting of the PAMTC, my awesome Villanova colleague Katie Haymaker presented the pancake sorting problem. Basically, you are a waiter with n pancakes of different sizes stacked on a plate. You want to arrange them from largest to smallest, but your only tool is a spatula. The one move you are allowed is picking up some pancakes from the top of the stack and flipping them over. How many flips does it take to sort the stack? The number of moves it takes a maximally efficient algorithm to sort the worst stack of n pancakes is called the pancake number of n. These are known for small n but open for n=20 and beyond. Katie and I heard about the problem in a talk last year by Ivars Peterson, and she thought right away that it would be great for a Math Circle. She was totally right.

Katie had developed a couple of very simple slides about the problem and talked to the participants for just about 5 minutes, explaining the idea, then challenged everyone to find the pancake numbers of n=3 and n=4. The groups got right to work and spent an hour or so figuring out the first few cases, working mostly on their own with some facilitation from Katie and the other organizers. There was no lecture, no explanation of the right techniques—just questions to help organize and spark ideas. It was a room full of people talking to each other and doing math essentially for fun.

Katie Haymaker flips some pancakes at PAMTC.

Katie Haymaker discusses pancake flipping  at PAMTC.

After working, people shared their process, insights, and results. The teachers had developed some nice ideas about upper and lower bounds for the pancake numbers. Katie gave them a handout that she’d made explaining the problem and some more resources and directions. The meeting ended with one of the organizers, Josh Taton, leading a short activity on evaluating curricula.

Having spent a lot of time planning activities for my math classroom (and having them not always be as fun as I’d hoped), I was surprised by and excited about how well it all worked at PAMTC—kind of a dream of what math class could be like if everybody wanted to be there. Why did it work? First of all, I think this was just a good group, with some veterans scattered around the tables keeping things running smoothly. The organizers deserve a lot of credit for creating a good atmosphere, providing good food so that people could relax and do math at dinner time, and working with the presenter to make sure the problem was presented in an accessible yet exciting way. Katie chose a great problem and did the work of turning into an activity, by creating a handout and providing little disks to use as manipulatives (physical problem solving props).  The teachers mentioned that the manipulatives had been really helpful.

One very helpful additional way to make Teachers Circles work is to find a way for teachers to get professional development credit for attending. The other side of this is the importance of professionalizing Math Teachers Circles for professors–finding a way for professors to also get credit for their work with the Circles when applying for tenure and promotion. After talking to friends who have been involved with Math Circles and Teachers Circles, it seems that not every department values faculty members’ contributions to Circles. I would love to (and plan to) start a Math Circle someday, and I hope that it will be perceived as worthwhile by my department.  NAMC is actively working to raise the profile of Math Circles and to help make sure that contributions to Circles are professionally valued.  They also have a ton of resources on their website (including Circle in a Box) making it much less time consuming and so theoretically possible for a pre-tenure professor to run a circle.

However, when I was at the Math Teachers Circle last night, I wasn’t thinking about professional credit, or that it was 5 pm, after a long day of teaching and everything else. I forgot about my big shopping bag of quizzes and homework to grade later, and that I still needed to calculate and report midterm grades for my students, as well as figure out what I was going to teach the next day. I was just digging the math. It would take something pretty cool to get me and 25 others, who all worked with people all day, to spend two hours sitting in small chairs in a middle school classroom, talking to each other about pancake numbers. I guess I can report that Math Teachers Circles are actually that cool.

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