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.


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).

Liberation Math: Penultimate Week

In Liberation Math, the class, the students and I are wrapping up the semester by writing and writing some more. Students are (slowly) getting content up on a website the class has organized, and over the next couple of weeks our writings will be collected into the form of print zines focusing on different topics.

On Monday, we welcomed Shaunalynn Duffy, from Sprout & Co, who talked to us about what she does with Sprout and her educational vision. The topics ranged from science to math to music. As Shaunalynn said, the mission of Sprout is to turn science into a cultural experience, and that idea resonated with many of us in the class.

I have a few interesting readings, groups, and events along the themes of community, culture, education, and liberation:

Next Monday in class, we will:

  • Watch a couple of videos focusing on teaching math concepts, and have a couple of live shares
  • Get a flavor of some art from graduating seniors whose show got moved due to the events in the Boston area last Friday
  • Finalize our written works, pulling the whole semester together! This means that you should all be finishing up your writing this week!

You should also work on a summary account of this class as your final blog post (of course, you are welcome to continue your blog even after the end of the semester). Take a look back through what you have written. Look over your original memory that you shared, and think about if anything has changed (or not changed!) about how you view that memory, or how you view yourself in relation to mathematics. Ultimately, what has this class meant to you? This final piece will be due on your blogs by 5/9 (that’s the thursday after our “final exam”) so that I have a chance to read them over before I have to close out the books on this semester.

More Doing, Less Learning

A little over a week ago, I was fortunate to participate in Ladies Rock Camp Boston, which is a shortened version of Girls Rock Camp Boston, a summer camp for girls 8 to 17. Ladies Rock Camp lasted three days, and in those three days participants formed bands, learned a new instrument, wrote a song, got good at that song, and performed the song with our bands at a sold-out showcase at TT the Bears, a local club. It was an amazing experience in many ways, and I have been thinking about how amazing it was as a learning experience in particular. Oddly, it was amazing as a learning experience precisely because it wasn’t about learning.

We had instrument instruction each morning; my instrument was drums, and I learned the basics of a rock beat, how to hold my sticks, and other useful tidbits. But I found practicing with my newly formed band to be much more useful than than instrument practice. During band practice my band and I were writing a song and learning to play it simultaneously. We were not learning our instruments, learning to write music, or learning to be a band. We were just playing — actually being a band, rather than learning about it.

There were, however, knowledgeable folks everywhere. There were people who already knew how to play the instruments that we were struggling with, and they were ready to step in with help whenever we had questions. There were people who had written songs who helped us when we felt stuck or frustrated. There were people who had been or were currently in bands around to help solve problems, to cheer us on, and to guide us.

In the morning of the second day, during instrument practice, I felt like a total failure. I thought I’d never really “get” drumming, that I’d let my band down, and that surely I was the only idiot there that couldn’t coordinate my hands and my feet. Everything seemed to come easily to everyone else. But then I got back to practicing with my band again. Once I was doing instead of learning, everything came much more easily, and I was able to do what seemed impossible while I was learning.

Ladies Rock Camp did plenty of other things to help us transform ourselves into rock stars. We were instructed never to say “I’m sorry” after making a mistake but instead to say “I rock.” I never actually succeeded in doing that, but it reminded me to move through the world boldly, and to claim my own space. We practiced getting into our powerful rock personas, we screamed, we shared, and we did punk rock aerobics. We also had enough time pressure that we were willing to keep moving forward, even though we sometimes felt that we weren’t good enough. When you have to go from zero to performance-ready in 60 hours, you simply can’t wait until you are “ready,” and that was one of the keys that allowed us to move forward — we knew we didn’t have time to waste “learning,” we just had to make do and keep moving.

Typically, when we decide to learn something, we defer doing. First, you learn to play the drums, then you join a band and rock out. First, you take a research methodology course, then you design and implement a research plan. First you learn programming, then you write an app. But don’t the best experiences involve both learning and doing or creating? You learn to play the drums by being in a rock band. You learn research methodology by doing research. You learn programming by writing an app. You don’t wait until someone else certifies that you are ready — you just do it.

Further, I would argue that the best experiences also involve a community where there are plenty of people around to answer questions, to help us to figure out how to get back to that rock rhythm after a transition, to give us feedback about our research plan, to suggest a way we might overcome a programming problem. The best communities contain plenty of people who are just a bit more advanced than us, so that we can ask questions and get advice without being completely intimidated. We need to be able to ask the really “stupid” questions — the ones that reveal us to be beginners struggling just to keep the beat.

But as I struggle to integrate these lessons that I’ve learned with my own practice as an educator, I come across one hurdle that looms over all others — compulsory education. Ladies Rock Camp worked well for a population of women excited to rock. Yay! If 40 women showed up who wanted nothing to do with rock-and-roll and instead wanted to better understand the history of musical theory, the model probably would need some tweaking. Still, if we gathered a group of people together that shared a common interest, we could find a way to engage those people through creating something, and have them learn the skills they needed along the way.

But what would happen if we gathered together a group of people that wasn’t interested in anything? Or, thinking about it another way, what if we regularly gathered groups of people together around a topic that might be of little interest, but we compelled the group to gather and learn about the topic anyway. Oh right, we already do that and it’s called school. In school, we gradually erode people’s natural interest in the world and replace it with an interest in grades and other arbitrary rewards. We spend 12 to 16 years training people to engage in meaningless learning exercises for arbitrary rewards, and then we wonder why students don’t emerge from the experience more capable.

In my professional life, I only a few students each year who actually want to do math. I teach a number of students who enjoy and are reasonably skilled at getting right answers to questions. And those students can learn by doing, but the doing is “doing math assigned problems” and their enjoyment hinges on having and “expert” (me) judge their performances favorably. The rest of my students don’t want what I have to offer, which makes it challenging to engage them through doing and creating. I try to avoid forcing students to do meaningless work, and instead provide them with opportunities to create and do, but it is challenging, and I need to find better ways to support them so that they can see themselves as creators even in a subject as seemingly dehumanizing as mathematics. I’m very glad to have another model of this kind of education, and hopefully that I can find ways to bring the spirit of Ladies Rock Camp and Girls Rock Camp to my own teaching and learning.

PS Because I can tell you are all dying to see the results, you can see the video of me and my band playing at the showcase. I’m hoping to volunteer with the girls’ camp this summer to steep myself a little more in how the whole thing works!

Who Benefits from Educational Intervention?

Last Monday in Liberation Math, the class, we had a great conversation about Logan LaPlante’s “hackschooling” TED talk. It’s a great talk, and I highly recommend checking it out. LaPlante is a winner and is doing great things with his education. He makes it all look easy, but extraordinary resources go into making his hackschooling education possible. His family skis a lot, which tells us that they have an income that makes that possible. He has an extensive network of opportunities, which means that his family is aware of the opportunities, that they are available to people like LaPlante, and they have the time and funding to allow him to access all of those opportunities.

LaPlante is a kid who is likely to be successful no matter what system he finds himself in — he’s charming and well connected. We can’t use him as a test case if we want to figure out if hackschooling is useful for a broad range of students because he is not a typical student, and he’s certainly not one of the millions of students struggling to successfully complete their education. Of course, it could be argued that if LaPlante was forced into a soul-crushing educational system that he might have begun to have difficulties. A bad education might have hurt LaPlante and decreased his chances at success in education and in life.

The fact is that educational innovations and interventions work — nearly all of them work, which doesn’t mean that we should implement every educational fad. We spend a lot of money on educational “fixes” that don’t really change things very much, often at a high cost. And the impact of the intervention depends a great dean on the population you start with. Suppose we take 100 well-resourced students like LaPlante, 80% of whom were going to be successful without intervention. If we do an intervention with these students that shows a 25% rate of improving student outcomes, then we see the following:

100 students, 20% struggling, intervention helps 25% of students

You are a student in this group who is doing fine or better. Was it the intervention that made the difference for you? To calculate the probability that the intervention moved you from struggling to fine, given that you are a student doing fine or better, we have to take the number of students who are fine in the end but would have struggled (5 students), and divide by the total number of fine or better students (85 students). This gives a 6% chance that the intervention is what made the difference here. We have to provide this intervention to 20 students in order to move a single student from struggling to fine. If this is an expensive intervention, that may be impractical for the results that we get, but will be entirely worth it if you are that one students and you possess the resources for change

Now imagine a population of students that has far fewer resources and experiences greater struggles and a greater likelihood of failure. Let’s say this new population of students has only 20% who are going to be successful with no intervention. We’ll imagine the same intervention that helps 25% of students:

80% struggling, 25% intervention success rate

In this new situation, if you are a student doing fine or better, what is the likelihood that your performance is a result of the intervention? Here again, we take the number who were struggling and are now fine (20 students) and divide by the total number who are doing fine or better (40 students).  This gives a 50% chance that your positive outcome could be credited to the intervention, a big difference from the previous population of students! Here we need to provide the intervention to just 5 students to move a single student from struggling to fine, giving a much greater efficiency. Perhaps this hypothetical intervention now looks great, but if it is an expensive intervention, requiring a lot of human capital, we still may not be able to provide the intervention to a broad range of students who need it.

How effective does this intervention appear to be? If it is implemented with the first population, then 85 students are doing fine or better, so it appears to have an 85% success rate. But if the intervention is implemented with the second population, then just 40 students are doing fine or better by the end, so it appears to have a 40% success rate.

Population General Educational Intervention GraphBut of course I made up the numbers about the population and the effectiveness of the intervention out of my head! We can model the situation in general with a population in which F out of 100 students are fine and we implement an intervention which is I percent effective. In this general situation the success rate will be F+(100-F)*I/100, which we can graph F on the horizontal axis and I on the vertical axis, coloring each point in the plane with the success rate as below, where lighter colors means a higher success rate. You can see that both the base success rate in the population and the effectiveness of the intervention constrain the overall outcome. For very successful populations, nearly any intervention will appear to be successful, but not all interventions that “work” with a naturally successful population will work with a struggling population.  And for an intervention that is nearly 100% effective, you can achieve amazing things with nearly any population. The trouble, to my mind, is that there aren’t any interventions that are 90% effective or better. Even the best interventions are going to be unlikely to get much over the 25% mark. Of course you have to keep in mind that this entire scenario is simply a “toy model” — it’s not reasonable to measure effectiveness simply by reporting a percent or lump students into two categories of “struggling” or “fine.”

Educational success is not always easy to achieve. Interventions, even when successful, aren’t going to solve everyone’s problems. When we see amazing educational success, we need to ask ourselves who is successful and why they are successful. Our educational system holds out the idea of advancement for all, but the reality is often that the greatest advances are made by those who were already set up for success. I’m really trying to wrap my head around these things and connect them to other ideas, so I’d love to have your thoughts!

What is a Course?


maze (Photo credit: woodleywonderworks)

I can see the end of Liberation Math, the course. As we near the end, I want to both slow down to enjoy the last bit, and to ask “What next?” As I slow down, I start to ask myself this course actually is. Specifically, what is this course Liberation Math, but more broadly, what is a course in general? Is a course a collection of goals and outcomes that we meet in order to be successful at the course? Are courses stepping stones to a degree or a specialization? Are they administrative units that are collected in order to provide certification and signal productivity? Are courses events along a possibly transformational route to a degree? Do we need courses, or can education occur outside of them? Do courses help us focus, or do they constrain us, or do they do both?

I have a few readings and videos to share about courses and education.

From your point of view, what is a course? What are the benefits and drawbacks of the way our schooling is divided into these discrete units? What has this course been to all of us involved in it? What is the best way to wrap up the experience and move forward? Can we carry something with us that breaks out of the bounds of the class?