New Beginnings

So last semester, I had my first semester off of teaching in maybe 13 years. The last six years or so, I have taken every summer off, but I this is the first time I have been eligible for sabbatical. I expected it to be both relaxing and productive. I’m quite good at organizing and motivating myself to do research and other outside work during the semester and the summer, so I anticipated that this fall would be more of the same.

I couldn’t have been more wrong. Apparently, my life is held together by pressure, and without that pressure, I was entirely at a loss as to what to do. Don’t get me wrong, I got things done on my sabbatical project, especially if you are a representative of the university I work for. If you are, then I got tons done. Every day. Wildly productive. But for the rest of you, I struggled the whole time. Even simple tasks became nearly impossible.

So now I am back at work, and I’m both humbled and happy. I taught my first classes this week, and it was fun to prepare them. By the end of the spring semester last year, I felt burned out, and found I wasn’t all that happy teaching. But a break can be an amazing thing, and right now I feel grateful to have the familiarity, rhythm, and challenge of teaching. This semester, I am teaching Patterns and Functions (aka Pre-calculus), Calculus I, and two sections of a course I developed called Math, Art, and Design. Here’s what I’m doing.

  1. Patterns and Functions. We started by doing an exercise where students got into pairs and then determined which of three functions matched a description given in words. I got this from Approximately Normal. That went well, and then we backed up and talked about what a function actually is, giving examples and connecting the definition to the general notion the students had that functions had to do with dependency. Then we did just a couple of the “team graphing” graphs from Study of Change. The point that I made there is how easy the task is when we can clearly name what should be drawn (like “a stick figure”) and how we can use the language we’ll be developing in P+F to be able to name more things clearly, and thus to get better and this task. Perhaps we’ll revisit it later in the semester. Tomorrow we’ll talk about domain, range, and piecewise functions, and we’ll use Des-man from Desmos.
  2. Calculus. We started with simulating the spread of a disease through a dice-rolling lab from Gary De Young. Working through that took even more time than we had, and we’ll be finishing the activity up tomorrow. I wanted to start the semester by giving them a project we can keep working on throughout the semester, and also provide a way to start the course with a context, so that we can come back to it to make the ideas we learn meaningful. On Friday, we will finish that up, and then move into an activity with spreadsheets, because I want to get students using spreadsheets pretty heavily this semester.
  3. Math / Art / Design. This class is weekly and doesn’t start until this upcoming Monday. Unlike in previous years, we are starting by talking about perspective, so we are going to do a tape-drawing activity like this one.

So far, I’m happy. I feel more relaxed. I like my students. My difficulties with sabbatical really did humble me, and allowed me to see that it may not be the best idea in the world to push myself so hard that without the constant pressure I collapse. We’ll see how I do.

Classroom poster with students names and mastered math facts

Shame in classrooms

I ran across a blog post by Brené Brown earlier today. In the post she relates some of the kurfluffle around her comments to Oprah Winfrey about teachers and shame (original video now hard to find — its the last clip on this page). This all happened at the end of September, so I’m late to the party, but I had a few thoughts about what happened here . To my mind, the biggest place where Brown goes wrong is when she says that shame is a classroom management tool used in schools. By calling shame a tool, she implied that the use of shame was conscious, and since her whole thing is talking about how bad shame is, that’s a pretty damning way to call teachers out as being bad for kids. That turned her comments into a public shaming of teachers. Brown didn’t intend to shame teachers, but that’s the sneaky way shame works — it’s everywhere, and it is hard to avoid.

Shame is certainly present in every classroom — it is a nearly ubiquitous emotion, so it happens in classrooms like it happens everywhere. For the moment, let’s try to avoid shaming the emotion of shame — shame and the threat of shame are intertwined with nearly every connection we have with other people. That’s not good or bad, it just is. When our relationships are working, we are able to use subtle clues about shame in ourselves and others to figure out how to navigate the relationships without alienating ourselves or other people. But an excessive weight of shame, or shame that calcifies in certain areas can break our relationships, causing disconnection and isolation. In classrooms, both students and teachers can feel shame, and that shame is a signal to us that our relationships are either fractured, or in danger of being fractured. Teachers feel it when they are disrespected by parents, administrators, and other teachers. Students feel it when they understand that they aren’t worthy of connection with classmates, teachers, and specialist because they aren’t good enough. We have to watch for signs of shame in other people, and use it as a sign of relationship danger. Relationships can be repaired and shame can be healed.

Teachers care about kids, that’s why they become teachers. They work hard every day to help kids succeed. They also face a job that is so difficult, so painful, so demanding of every resource they have. They do it without enough pay, and with a whole world watching to see where they are going to screw up first. In other words, teachers do their jobs in an environment that is a shame pressure cooker. When people are in that kind of pressure cooker, they will push their shame onto other people, and it is easiest to push shame onto weaker people. I do it as a teacher to my students, and I do it as a parent to my kids. I don’t do it because I’m a bad person, or because I am sitting around cackling and thinking up ways to torture students and children. I do it because I am human, and fallible, and, because, as Brown said in her mea culpa post “learning is vulnerable and classrooms are tender places.” We do need to raise awareness of the presence of shame in classrooms because awareness is really one of the only ways to combat shame. But we have to raise that awareness gently and carefully because any time we start to really see how shame operates in our lives it is easy to become overwhelmed.

I do think there are some institutional practices that increase students vulnerability to shame in schools. Particularly, I am thinking about public accountability in classrooms. In elementary math classes, you will sometimes find charts of times tables and other math facts, where you can see how each student is doing on proving their mastery. Short timed tests like “mad minute” are used in most elementary classrooms, and those are also a form of public accountability since all the students know who finishes on time and who doesn’t because they can look around and see everyone, including those kids that inevitably shouted out as soon as they were finished, before the timer rang, stopping the kid who still had half a page of problems left. Public accountability charts are also used for behavior management and in other subjects. For some kids, this is highly motivating. Students want to do well and the competitive spin of public results help spur them to work. But for other students, being at the bottom (or even in the middle) week after week is demoralizing and shaming. I think that in order to support all students, we need to have students set goals and chart their progress, but that this should be private. This won’t provide the competition that helps some students focus, but it will avoid the discouragement and feelings of shame and stupidity that other students experience.

Classroom poster with students names and mastered math facts

Here not only is your math prowess public, but it’s linked to ice cream

We can’t change the fact that in every classroom, the students know where they stand in academic rankings. All of the students know who is at the highest reading level and who is still struggling to read beginning books. They all know who finishes the math assignments before everyone else and who never finishes. We humans are constantly comparing ourselves to others, and figuring out where and how we rank. But when those rankings are publicly displayed, their importance is reinforced, even if the teacher is telling the students verbally that effort and progress are the most important things. We need to put the rankings away and to consistently remind students that they need to look to themselves to measure progress. They need to be better able to handle fractions at the end of the unit than they were at the beginning. They don’t need to be better than another student, they need to be better than they used to be. We also need to talk to students about how we handle that terrible feeling we have when we realize that we aren’t doing as well at something as we wish we were. We need to talk with our kids about how to manage the pain when we find out we sang the wrong note, or messed up all the problems, or our drawing wasn’t selected for the prize, or we realize that our friends are all reading at a higher grade level than we are. If we avoid emphasizing rankings and do some explicit teaching around how to handle the emotions that arise when we fail or don’t do as well as we would like, then we give students the tools to navigate pressure and criticism without falling into a pit of shame. Most teachers already do a lot of work to combat shame, but connecting the dots on the impact and mechanisms of shame can help teachers better see what they are doing and how to do it effectively.

Our successful schools

School closings rally

School closings rally (Photo credit: chicagopublicmedia)

I have been thinking a lot lately about the failure of our schools, particularly with regard to mathematics. It’s impossible not to notice what a terrible job our schools and students are doing at math, particularly when I am reminded about it once every hour or so by my twitter feed. I got on this kick because of the latest way we’ve been flogging ourselves, the PISA (Programme for International Student Assessment) results. When you look at the US in the rankings of countries, you won’t believe where we stood. We were, well, average. I know that here in the US we think that we live in Lake Woebegon where everyone is above average, but being average is just, well, average. The sky isn’t falling and our scores didn’t drop alarmingly. We’re all OK.

What if we aren’t all terrible at math, and what if our schools aren’t letting our students down? What if our schools are doing an OK job of educating students? Are they doing the best job possible with each individual student? Not likely. Are they moving lots of students through the pipeline, teaching them to read, do math, write, understand politics, know something about history, and even giving them a little art, music, and physical education? Yes, we are doing all that and more.

Remember the 90s and how Baby Eistein and similar products brought us the mistaken idea that if it is good to parent your kids, then its even better to parent them really really really well? If parents do flashcards, teach their babies to sign, buy the right educational toys, and twist themselves into the right knots, we will raise a generation of kids that is uniquely poised to become super-geniuses. Except that none of it really works. Yes, when a child is hungry or neglected, or when a family is living too close to the edge to provide a normal environment for the child, then the child’s brain will be impacted in a negative way. Poverty really does hurt kids. But that doesn’t mean that environments that are excessively enriched will produce geniuses. More is better when you don’t have enough. But when you do have enough, more won’t continue to produce improvements in results.

The same is true of schools, of math education. We need to have schools that are good enough. Schools should be full of teachers that care about kids, that have some training in both subject areas and pedagogy. Schools should have the financial resources and leadership to support teachers and families. But schools don’t need some kind of huge overhaul. There is no magic bullet of ipads or entrepreneurship that is going to change our failing schools into amazingly successful schools. Sometimes it seems like we have found the answer. Like giving kids computers. Like unschooling or hackschooling. Like teaching kids to code. There are a lot of good ideas out there, but we can do them all and still not get better results. That’s because our schools are already doing OK, and thus any new idea we cram into our full educational system will replace something else that was already good for a lot of kids.

Yes, we should continually look to improve the way we educate kids. From where I sit, I see that we should particularly pay attention to how kids learn math, what math they need  to learn, what math they might want to learn, and how to creatively help kids get more of what they need and want while we still have them in this amazing system that seeks to help absolutely everyone to gain skills and knowledge. But we will get a lot farther in that enterprise when we acknowledge that we are trying to solve a problem that is really hard, and that the people who are at the front lines of our educational system — the teachers, support staff, parents, administrators, and higher-education faculty — are doing a lot of amazing things and having a lot of success already.

Math is jarring

I ran across this video yesterday, by a math major, taking about other people’s reactions to learning that she is a math major.

I particularly like the analogy around 1:15 where she is talking about the jarring nature of switching from ordinary conversation to math. She likens it to being asked mid-conversation to compose a poem in Russian when you don’t know Russian. I think that is a lovely analogy. She notes that math feels this way when you are not used to it, and sometimes even when you are used to it.

I think this is an apt analogy, because academic math comes out of left field for most people. In math class, it isn’t that weird to have someone tell you:

The track at Made Up School is one mile long and features semi-circular ends connected by straight lines. Find the area enclosed by the track as a function of the radius of the semicircles. What dimensions allow the maximum area to be enclosed by such a track?

Say what? What does it look like? If the track is there already, how can we change it? Why are we doing this? And sometimes things get even worse:

If line segment BD is a perpendicular bisector of line segment AC, prove that triangle ABC is isosceles.

It just makes your brain hurt due to the sheer number of technical terms, and I have no sense whatsoever of this being a meaningful task that there would be a human reason for being able to do.

Notice that this is very different from other subjects that we study in school. In history, you might be asked:

In 1938, British Prime Minister Neville Chamberlain chose to adopt a policy of appeasement toward Hitler’s aggression against Czechoslovakia. What did this mean? (from this list of sample questions)

There may be some confusing terms in here. Maybe I’m not totally sure what a Prime Minister is, or who Chamberlain was, or where Czechoslovakia was in 1938, but I can get the sense of the question and have some idea of why I might want to be able to answer it, namely because I want to understand how the international world works.

Same with a thick subject like physics:

A battery is connected to a light bulb with copper wire to complete a circuit. The bulb immediately lights. Why?

Whoa, that’s intense. You are asking how a battery works. It may seem like a hard thing to understand or explain, but I can see why I would want to try an answer it — because I want to know how the world of electricity works.

I am not suggesting that there is no point to learning “higher” mathematics beyond arithmetic, but I am suggesting that those reasons can be obscure and subtle. We learn mathematics past basic computation because we want to understand the world, but it is an understanding of the world of thought, the world of algorithm, the world of logic, the world of abstraction. It is not the “real world” that we are seeking to understand, although higher mathematics often does have applications in the real world. Instead it is a fantasy world in which we ask “what if” and try to find a way to get consistent results. It is a world that is jarring precisely because it is so headily academic and is tethered to everyday concerns like a balloon that may slip away.

I think that if we all realized that we currently have enough math to understand our worlds, we’d all be a lot happier. The math most of us use in life is more straight-forward than it is portrayed in school, and  you may, right now, be as good at it as you need to be. Or you may find that you have some math-related problems in your real life that always frustrate you. That might be because they are really hard problems, and would be hard even for someone with advanced mathematical training. For instance, if you want to figure out a system of bonuses for your employees that reward certain types of job performance, then you probably will want to use some math, but the problem won’t be simple, and math will only be one part of the solution.

I also love the end of the video above where Sarah emphasizes practice, and the fact that mathematical skills can be developed. Absolutely true. You probably already have most of the math skills that you need, and if you need more, practice is a good way to get more. Of course, one of the big troubles that I see is that K-16 math classes don’t give people skills they will need after school, and it is actually quite hard to find needed and useful math skills if you aren’t in a STEM field (see, for instance, Audrey Watters on the difficulty of learning to code).

Writing a Paper and Asking People to Read It

Original image description from the Deutsche F...

(Photo credit: Wikipedia)

Last week I had a paper published in the journal Rationality and Society, “Division of Labor in Child Care: A Game-Theoretic Approach” . You might like the paper — it’s pretty interesting actually. But the first thing that’s interesting about this paper is that it did not occur to me to announce it here. For heavens sake, why not? My excuse is that I assume (erroneously) that you might be interested in my thoughts on education, but not in my thoughts on applied mathematics. Really that doesn’t even make any sense, and it’s not the real reason. I hope that you will find something in this paper that gets you thinking. In the paper I use game theory, a theory from mathematics and economics, to model an imagined situation in which two parents are caring for a child. The model itself is like taking the whole complex story of how real parents live and work with small children at home, and taking most of the story out, leaving just one aspect of the situation intact, in order to see what mathematics might say about how such parents would behave. The paper provides a great example of using mathematics to explore human relationships and building models of the real world in mathematics. It shows off what math is best at — abstraction and simplification. Writing it allowed me to explore sociology, gender, and economics, and it could provide a window on those vistas for all of you as well. I want everyone to read it that is interested in math education, applications in math, gender, and parenting.

Young couple with baby.

Young couple with baby. (Photo credit: Wikipedia)

So why didn’t I think to announce it here? Because I don’t want you to know that I want you to read it. I want this paper to be read without having to take the risk of actually asking people to read it. It is embarrassing to  release something you created out into the world. Asking you to read this makes me imagine you reading it, thinking about it, and judging it. That puts me in the realm of self-conscious emotions — embarrassment, pride, shame, guilt. I can imagine lots of reasons why you might find the paper lacking, and being able to produce quality work is important to me, hence the potential exists for me to feel shame. So I am caught in the middle between two desires that pull me in opposite directions. On the one hand, I should ask a lot of people to read the paper, and hopefully some of them will, and some will even talk to me about it. But I should also bury the work and never mention it to anyone. Best is if you just stumble across it in a journal, and email me to tell me how much you like it and what kind of interesting conversations it spawned in your family or in your classrooms; then I never have to imagine you reading it with a frown on your face. But that’s not going to happen, so I decided to share it with all of you in the hopes that you will read it and talk about it with me and with other people. (And FYI, in case you are wondering why I would say all of this,

I find that being honest and open about my fear, especially fear of shame, allows me to manage that fear. I still feel exposed, but when everyone know about that feeling I can more easily manage it since I don’t have to hide it.)

Math is [fill in the blank]

Today and tomorrow, I’m trying to get people to fill in this sentence: “Math is [fill in here].” Feel free to give your own answer in the comments below or on twitter. Once I get the blank filled in, I’m trying to prompt people to ask why or probe a little deeper. In part, this is preparation for a talk tomorrow that is on the liberation math class. The class discussed what we should do last week, and most of us liked the idea of creating a taste of the experience of the class. So we’ll talk about what math is, why it is that way, get people to contribute some brief memories and talk about what themes and stories we notice, and then talk about hidden curriculum and how we are working to reclaim power, starting with having people ask questions about pictures drawn from It will be fun!

The talk will be on Wednesday, 3/27 from 3:45-4:45, in Cambridge, MA (porter square) at 1815 Mass Ave, upstairs in room 2-078.

Week 6: Difficult Feelings and Getting Curious

This week in class, we worked together on using a google spreadsheet to find the amount of money in Fry’s account when he started with $0.93 and left the account for 1000 years. Spreadsheets take time to learn to use, but they give you a lot of power to do repetitive tasks. You can do non-mathematical things with them as well. For instance here’s a blog post about things a  literature professor does with a spreadsheet, and even a short music video created with a spreadsheet.  and even a music video in spreadsheets (no, I have no idea how they did this!)

We also worked with calculating interest by converting a percent to a decimal, and then multiplying by the principal (the balance in the account). This video covers this procedure (I like the guy who did this video a lot, and he has lots of videos about math in the real world). We did this in a google spreadsheet in class (feel free to open this up and tinker with it, or go to File->Make a Copy and create a copy for yourself). I have a challenge out to everyone to try to create (1) a spreadsheet that will show your balance over time for a savings plan in which you deposit $50 a month and early no interest, and (2) a spreadsheet of the same situation, but with in interest rate of 1% per year. Feel free to post a link to your spreadsheet with a solution, or describing where you are getting stuck (you’ll need to go to File->Share to make the spreadsheet publicly viewable).

Difficulty and Curiosity

What I really want to examine this week is difficulty. I taught two classes on Monday, this Liberation Math class and another class on Math, Art, and Design. I felt bad after each class, and that’s usually a good signal to me that something is happening that I should be paying attention to. In Math, Art, and Design, I felt like students weren’t interested in or connected with what we were doing together in class. In Liberation Math, I felt like the class was a bit frozen, and like I was floating out there alone. At the end of the day, I worried about the classes and my teaching, wondering if it was something that I did or if there was something going on with the students, and spinning out lots of possible scenarios. This morning, I finally remembered remembered that feeling bad and worrying doesn’t actually help anything, and that the proper response to things going badly is actually curiosity.

As I have blogged a bit about before, I think a lot of hiding and dishonesty goes on in a classroom setting. Teachers often hide their true motivations from their students, and they hide the emotions they have while they are teaching. Part of this seems entirely as it should be — it’s not my students’ responsibility to take care of me emotionally, and although I try to be open about my struggles as a teacher, I am careful to deal with my difficult emotions on my own before I bring them out into the light for examination. On the other hand, I think we teachers do our students a disservice when we pretend that we are free of emotion, especially since those emotions have a way of leaking out even if we think we are clamping down on them. Students also pretend a lot of different things in order to get by in a class — they may pretend to like a class, to like a teacher, to understand things that they don’t get, to have studied more than they did, or to have done work that they didn’t do.

The trouble with all this pretending is that we misunderstand each other. We teachers are not all that good at reading students. As Andy Hargreaves says in “The Emotional Practice of Teaching” (p. 839), “Teachers frequently misconstrue their students’ exuberance for hostility, bored compliance for studious commitment, embarrassment for stubbornness and silent respect for sullen resistance. This misunderstandings seriously interfere with teachers’ ability to help their students learn.” I recently ran across another paper by Hargreaves (here) that may draw out this issue even more usefully. Plenty of potential also exists for students to misread teachers, and when I look at students that are struggling in a class, I often feel like I see them writhing under my steely glare, when really I just mean to give them a look of reassurance.

This is why I say the proper response to feeling bad about a classroom experience is to get curious. I have such a limited view of the classroom and there so so much going on at any one time that I actually have no idea what is actually happening for the students. It may be I was having a bad day and the difficulties were all in my mind. It might be there was something I was doing that was decreasing the safety and connection for the students, who responded by protecting themselves. It could be that the students are feeling their own collection of difficult emotions about their performance in and connection to the class. It could be that I engaged students in a difficult task that decreased their sense of safety in the class. It could simply be that it is almost spring break and everyone is simply tired and overwhelmed.

I have a few students in mind that I want to personally connect with, but I thought I’d also throw it out to all of you. What do you think it was? Some of you were in one of these classes and might have an idea of it felt on the ground. Others of you have been students in many other classes and have felt the rise and fall in the mood of a class. As a student, what have you noticed that either increases or decreases your sense of connection to a class, your classmates, or a teacher? What shuts you down? What increases your safety? What makes it more likely that you’ll talk in the class?

P.S. I’ve cooked up a few tasks for this week for those of you participating in the Liberation Math class. Anyone can check them out and play along, particularly if you are interested in a discussion of the positives and negatives of group work.