Inspired by (Blatantly Stealing from?) Dan Meyer

Yesterday a colleague and I were discussing interval and set notation. Should we introduce it in the first year of our integrated math curriculum was our initial question, but we quickly moved into more general questions about these notations, what they’re good for, how much we expose students to them, and how much we expect students to use them.

And then, inspiration struck, and I could hear Dan Meyer in my head saying, “If set notation is the aspirin, what’s the headache?” (If you’ve never heard Dan ask this question before, then start here and read his collection of posts on questions like this.)

So, why do we need set notation in, say, an Algebra 1 or Algebra 2 class? My first response was, “Hey, I’ve seen it come up in some SAT II practice tests.” So we looked, and we found these:

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This problem has issues because it states that x is “equal to” the set instead of being “an element of” the set.

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This problem uses set notation in a way that’s fine but is, in the words of my colleague, just gift wrapping, i.e., completely unnecessary. I’m not sure what the purpose of including the notation in the question is. Are they trying to throw weaker students off? 
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This example uses the set notation just fine, but again, would anything be lost if we didn’t include it? So, “if you want to do well in the SAT” is probably not a sufficient headache to motivate the distribution of this aspirin.

My next idea was that set notation somehow simplifies communication of what we’re dealing with. After thinking about this for a few minutes, though, I don’t actually think that’s true. Whether a student writes “x = 3” or “{3}” makes no difference in a student’s attempt to clearly communicate her solution to an equation, and at the high school level we’re talking about, their ability to communicate this information is far more important to me than their ability to correctly apply a complex idea (sets) to a simple situation (the solution is 3).

Don’t get me wrong – there are ways in which ideas related to sets are very important to understand at this level. We want our students to know that two line segments cannot be called equal, why this is true, and why it matters. I’m just not seeing a reason to go beyond this set-based idea to actually introduce set notation and insist on its use in ways that don’t reinforce the important concepts that underlie the notation.

So what’s the right way to extend Dan Meyer’s headache-and-aspirin analogy? I think it’s to say that we have developed the bad habit of mathematically over-medicating. We all know the stories of people who demand antibiotics to fight off viral infections and what kind of problems this can lead to. I would argue that we do the same thing in high school math. We have taken problems that can be handled with a cup of soup and a good night’s sleep, and we’ve chosen to address them with IV medications (which is sometimes watered down, making it even worse). So I’ll keep looking for the headaches that need to be cured, but I’m going to start being a lot more aware of of the ones that can be taken care of with the remedies we already have available to us.


Reflections on a Device-Free Day, Part 2

As I mentioned in my last post, our school went device-free for twenty four hours earlier this week. In addition to experiencing life without technology and reflecting on the role it plays in my teaching (and other areas of my professional life), I spent some time considering the implementation of the day itself. In thinking about what worked, what didn’t, and how we might do it better next time, it occurred to me that our collective experience of this day was a microcosm of how to, and how not to, implement change in schools.

I will say right up front that my thinking about this topic of school changed is heavily influenced by Grant Lichtman, Julie Wilson, and Jonathan Martin, among others. These are people who are thinking about what conditions must be present for effective change on both the macro and micro scales, and their work has given me a lens for thinking about change that I am involved in, whether it’s as a leader or a participant.

While I had a great time going device free, my experience does not seem to have been shared by the majority of faculty and students. Responses to the day ran the gamut from complete abstinence to not even acknowledging the day. It would be easy to say that students were on their devices because the faculty weren’t setting a good example, or that the faculty were on their devices because they didn’t think the students wouldn’t be able to stay off them anyway. Both of these conclusions are too simplistic and miss some much bigger lessons that we can learn from the day. Here, then, is my assessment of the critical conditions of effective change and how we fared under each of these criteria.

  1. Know what problem you’re trying to solve – change for the sake of change won’t get much traction; without a problem to be solved through change, there’s little reason for people to embrace it. In this case, the “problem” was clear: we have all integrated technology into our lives to such an extent that we aren’t even aware of it at times. This can have both positive and negative consequences, and a day free from electronic devices is a good way to identify those consequences.
  2. Make a compelling case for change – even with the problem identified, one has to convince others that a change is needed to address the problem. The device-free day was first mentioned in a talk given by the headmaster, and was followed up with e-mails to the faculty and the students, but this didn’t go far enough in making the case for why we needed to do this. People didn’t see a compelling reason for being device free, so they saw no reason to participate in this event.
  3. Obtain stakeholder buy-in – even when the case for change is compelling, people will only commit to it to the extent they feel ownership of it. The school’s Academic Council was consulted about the timing and calendar, but there was no broader effort to get input from faculty or students into the device-free day. The best chance for success depended on people being committed to this, but with no say in the mechanics of the day there was no reason to feel that sense of commitment.
  4. Don’t offer a way out – when change is presented as optional, those opposed to it will opt not to participate. The device-free day was presented as entirely voluntary. Students were told there would be no policing of this, no steps taken to address those found using devices when they shouldn’t have been. Given the option to avoid the challenge presented, many people took it.

This analysis might lead one to believe that our Device-Free Day was a waste of time. Beyond the tremendous personal value I found in the day, I actually think it was institutionally significant. On a small scale, it showed us how change will be received, so we’ll know better next time how to proceed in implementing change. As long as we can learn something from the day, and I think it’s clear that we can, this event can significantly influence how we proceed as we continue down the path of growth as a school.

Reflections on a Device-Free Day, Part 1

Our school had a Device-Free Day this week, a twenty-four hour period from 6 PM on Monday to 6 PM on Tuesday when all faculty and students were asked to refrain from using cell phones, tablets, and computers. There were two stated goals for this day: to connect with people in person instead of through technology, and to reflect on the role that technology plays in our daily lives. Twenty-four hours is a lot of time to reflect, so I’m going to break this into two posts. In this post I’ll share my own experiences and reflections on the role of technology in my life as a boarding school teacher. In a future post, hopefully tomorrow, I’ll touch on the larger lessons we can learn from how the day was implemented.

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I should start by saying that I was wildly enthusiastic about this day, much more so than many of my colleagues and approximately all of my students. If nothing else, I will take any chance I can get to remove myself from e-mail. That being said, much of what I do that isn’t in the classroom, and a decent amount of what is, is done on a computer. Without the ability to do this work, I was forced to decide what needed to be done before 6 PM Monday, what was so important that I would need to do it without technology, and what could wait until Tuesday night or Wednesday. It made Monday a little stressful as I kept thinking of other things to do – like remembering to wear a watch on Tuesday so I could time a quiz without the timer on my phone – but once 6 PM hit I was ready to go.

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I enjoyed the day quite a bit, and I was actually a little sad to see it come to an end. My work prioritized and completed in advance, I was able to dedicate a lot of time on Tuesday to the things I never get around to during the week. I read a couple chapters of Nate Silver’s The Signal and the Noise, and I went back through some math history that pertained to the work we were doing in class. I rearranged my bookshelf, finding a couple things I’d forgotten I had in the process, and I sorted through a backlog of conference handouts, meeting agendas, and business cards that had accumulated over the last couple years. For me, it was a fabulous day, one that I’ll have to make time for more often.

The day also emphasized some things that I’m aware of but don’t think about that much. I graded some quizzes on Monday night, but I couldn’t record the grades in my electronic grade book so I was unable to return the quizzes on Tuesday. I had to hand write my notes for class, something I haven’t had to do since I started teaching with a tablet PC and OneNote almost a decade ago. I use electronic textbooks, so my students couldn’t do any homework without printing pages out in advance (and wasting a bunch of paper in the process). A colleague had to take her son to the doctor and needed me to fill in for her with community service, which had to be handled through a quick conversation at lunch rather than a text or e-mail at a more convenient time. None of these challenges was significant because none was a challenge for more than 24 hours, but taken together they paint a clear picture of how much technology has integrated itself into my life as a teacher.

I don’t think there’s necessarily anything wrong with this. I am a better teacher because I am able to provide my students with more and better resources. I like that my students can access their textbooks from anywhere in the world and are spared the future chiropractic visits that would come with lugging around a full backpack every day. I also appreciate the time and convenience that texting colleagues and an electronic grade book provide. Stepping away from all these things for twenty-fours has helped me think about the ways in which technology might be keeping me from doing my best work, and it’s reminded me of all the ways technology has replaced things I used to do. If nothing else, reevaluating how I spend my day to keep what’s valuable while recapturing lost activities that are important made the Device-Free Day a productive one for me. As for the institution as a whole? I’ll tackle that question next.

Integrated Math, Part 1 – Don’t Grant the Premise

As I’ve mentioned before, a couple years ago we made a decision as a department and school to replace our Algebra 1 through Precalculus courses with a sequence of integrated mathematics. Since September, a colleague and I have had a one-course teaching reduction so that we could develop these courses. In addition to developing the curriculum itself, we’ve been working on administrative details, including a clear, concise justification of why we’re making this curricular move. We feel pretty good about the justification at this point; in one concise, well-written page (for which I deserve none of the credit), it describes the reasons for moving to integrated math in a way that we can easily convey to prospective students, their parents, and other interested constituencies.

What we don’t have is a similarly strong and concise justification for colleagues in the math department, teaching candidates, and math teachers from other schools who might be interested in what we’re doing. This hasn’t been a pressing concern, but it is something I think we need. We don’t yet have complete buy-in from our department, which isn’t surprising since we’re talking about something that hasn’t been taught before and is still in development. As we continue to talk about it, it’s time to put something in writing that will serve our needs now and for the next few years. I was having trouble figuring out how to structure the argument until recently, when the  College Board’s AP Calculus listserve took care of it for me.

A few weeks ago, there was a question on the listserve about course sequencing. The question, which is frequently posed in math departments around the country, is which is a better sequence of high school mathematics courses: Algebra 1 – Geometry – Algebra 2 – Precalculus, or Algebra 1 – Algebra 2 – Geometry – Precalculus. There are benefits and drawbacks of each option, the most signficant of which revolve around when students forget the algebra they’ve learned and how much of that algebra will have to be reviewed, or retaught, later.

The debate over when in the sequence to teach geometry makes an implicit assumption: high school mathematics should be taught using these four courses to lead to calculus. While mathematicians are taught to look for fallacies in arguments, we seem to miss this one quite often. To argue over when to teach geometry is to grant the premise that these four courses are the best way to prepare students for calculus (and, more broadly, to be mathematically literate adults). If we refuse to grant this premise, the answers to the question suddenly become very different.

So let’s not grant the premise, but instead start with the questions of how do we best prepare students to study calculus, understand what it means to think mathematically, be mathematically literate, and see that there is much more to mathematics than solving equations. These were the questions that we started with and have kept in mind. In addition, we  ignored “the way things have always been done,” what textbooks are available, or what we’re most comfortable with (but, because we’re a college prep school,  we did pay attention to the SAT, ACT, and the college process). By not granting the premise, we came up with what we believe is a strong sequence of courses that pursues these goals. Over the next few weeks I’ll start to share some details of how we’re building these courses, what content we’re including (or excluding), and why we made the decisions we made. I hope people will tell us that they think and (especially) what hidden assumptions exist in our own work that we might have missed.

The Day My Students (Almost) Invented Calculus

One of my favorite things to do in Precalculus Honors is ask my students, “How do polynomials behave?” They started working on this deceptively simple question in early November and are still plugging away at it. In the process they’ve had to come up with a definition of polynomial, decide what we mean by “behave,” figure out a way to tackle the question, and then look for patterns from which they can develop conjectures that they can make convincing arguments for. In other words, they’re being mathematicians, not just doing math. Some people just stare at me in disbelief when I tell them that my honors class is in its seventh week of working with polynomials, but if didn’t spend this much time on it or do it this way, then we wouldn’t get classes like the one we had today.

A couple of groups in the class had been working on the patterns they were seeing for turning points on a graph, which are things like those labeled “A” and “B” in the graph below:


They had looked at a lot of graphs and were pretty convinced that for a polynomial of degree n, there should be n-1 turning points. (The truth is that there will be at most n-1 turning points, but I wanted them to explore their conjecture themselves, not just wait for the right answer from me.)

While this conjecture made sense to them, it led to some challenges in addressing curves like this:


This curve has no apparent turning points, but they knew that it needed to have some in order for their conjecture to hold, so they spent a lot of time trying to figure out where the turning points were. They come up with something like this:


When they looked at points “A” and “B” they knew that they were not maximum or minimum y-values, which they had figured out happened at turning points, but they felt like there was a distinct change in the shape of the graph at these points and so these must be the turning points.

As they were presenting their work to the class, there was a lot of discussion about these non-extreme-value turning points. Everyone was having trouble figuring out exactly where they were or how to define them. One member of the group said that she wanted to think of them as the places where the rate of change changes, but she knew that wasn’t correct because she knew the rate of change (slope) was continuously changing. This led a classmate to suggest that maybe they could pick a couple of points that were close together, find the slope between those points, and use that to get a sense of the rate of change. Prompted by another classmate, he went on to say that by doing that in a couple different places they’d be able to compare the rates of change, and by keeping the points really close together they should get at least a decent idea of what the rate of change was.

In other words, he was suggesting approximating the rate of change by calculating the slope of the secant line between points that were close together.

In other words, he was about 20 minutes from developing an intuitive definition of the derivative.

I do a lot of student-driven inquiry work with my classes because to me it’s like a road trip – we have a general sense of where we’re headed, but sometimes we take a detour because something interesting catches our eye. This was one of those times. What I really wanted to do was let three or four students starting playing around with this question of approximating the rate of change. To do so would have been an amazing, fun experience that would have let them continue to develop as mathematicians. Unfortunately, it also would have meant that I wouldn’t have gotten to something like sequences and series, plus I would have stolen the thunder of the AP Calculus teacher. Still, it does make me wonder – how flexible could our courses be so that we can follow a line of inquiry without worrying about going past what’s on our syllabus?

My Christmas Present from my Students (and My PLN)

I have really come to love Twitter for the way I can use it as an on-going source of professional development, and Wednesday’s AP Physics 1 with Precalculus Honors course is my latest, greatest example of why.

It all started Tuesday night when Frank Noschese (@fnoschese), a physics teacher in New York, retweeted something from Monica Owens (@MonicaKonrath), a physics teacher in Indiana. First of all, I’ve never met either of these people, but without them Wednesday’s class wouldn’t have happened. Anyway, here’s her original tweet:

Now, our plan for Wednesday, which was our class’s last meeting of the term, was to give a couple quizzes, work for a while on momentum problems, and then continue our open inquiry on polynomial equations. This activity sounded way more fun, and I really liked the idea of looking at texting while driving right before the students leave for break, so we decided on the fly to do this instead. The results were even more amazing than we’d hoped.

We started with a simple question: “If you sent a text message while you were driving, how far would you travel while sending the message?” They looked at us for a couple seconds, then turned to their groups, and what followed was music to our ears:

  • “We’re going to need to make some assumptions.”
  • “If the velocity is constant then we can ignore acceleration.”
  • “Let’s run some trials to see how long it takes to send a text.”

It really is quite a nice gift to hear that the students have internalized at least some of the things we’ve been working on all year. Because things were going well, my teaching partner suggested we add a second question: “Suppose you collide with a stopped car while you’re sending that text. How much force will the seat belt exert on you?”

This is a nice little momentum/impulse question that extends the life lesson we’re trying to get at with this problem, and it was once again a joy to see that they immediately recognized it as such and went to work on it. When all was said and done, we had a nice collection of whiteboards to share. Here are a couple of them:

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We finished up by trying to give the students some context. First, we did some quick unit conversions so they could see that the force of the seat belt is the equivalent of having 1000 pounds or more sitting on your chest. Second, we took a little field trip to the street that borders the school, where the speed limit happens to be the same as the one assumed by the white board above. We measured out 80 meters so they could see how far that really was. We then noticed that in this distance there were seven cars parked by the side of the road, any of which a driver might hit while looking down. We also saw two pedestrians, one jay walker, and an illegal U-turn, which were just additional opportunities for a terrible accident.

It really was a lot of fun, and we think pretty powerful, to tie the physics we’ve been doing to a real world issue that teenagers (should) hear a lot about, and it was a great way to close out the month. Thank you, PLN, for bringing me just the right activity at just the right time. Happy holidays to all.

An Unintended Benefit of an Interdisciplinary Course

I co-teach an interdisciplinary course in AP Physics 1 and Honors Precalculus. Combining these two courses, which we did for the first time last year, was a no brainer: much of the math we do in the precalculus course ties directly to the math needed in physics; we tend to do some physics applications in a precal course, like projectile motion and simple harmonic motion; and there’s a big emphasis in setting up and solving word problems in both courses (although in physics they’re just called “problems”).

There are some other ways to connect the courses, too. Physicists and mathematicians work with vectors with different levels of rigor and formality, for example, which allows us to highlight the difference between “doing math” and “using math to do science.” Similary, what we can know or prove in math is very different than what we can know or prove in science, yet there are similarities in the rigorous structures the two disciplines use to collect evidence and draw conclusions. Seeing and understanding these parallels seems to help the students understand their work in both disciplines better than they otherwise would.

Today, we discovered a new benefit of having this interdisciplinary course and using team teaching for it. Our students recently took a quiz on solving non-right triangles, and even though they’ve been quizzed on this before, many of them are still making the same mistake: failing to check for multiple solutions in the SSA case. They also took a physics quiz that had an object moving down an incline plane before launching off the end of the ramp and falling to the ground. They did the first part of the problem just fine, rotating their coordinate axes to be parallel and perpendicular to the incline plane. When they got to the second part of the problem, which dealt with projectile motion, many of them made the same mistake: they left their axes rotated but solved the problem as if they weren’t.

Were we teaching these classes separately, we each would have addressed these mistakes in our own ways and moved on with class. Because we’re co-teaching and looking at student work together, we realized that the mistakes on both quizzes really stem from the same issue. In both cases, the students needed to recognize what kind of problem they were dealing with and then proceed based on that recognition. We were able to talk this through with our class, helping them see that this was less an issue of knowing specific content and more an issue of how to approach and solve problems. Had this not been an interdisciplinary course, it’s likely that no one, including the teachers, would have made this connection. Score one for interdisciplinary learning.