One of the more fulfilling things I’ve participated in this spring is the launching of an Innovative Teaching Collaborative at my school. We’ve got a group of 20 or so faculty who meet monthly as a large group, and weekly as small groups of 5-6 people across each discipline. During these meetings we are talking about “innovative teaching” in general. Some topics we’ve explored so far:

• Hearing from a teacher who is exploring self directed learning in his advanced calculus and chemistry courses.
• Hearing from two teachers who have developed and are teaching a senior humanities course.
• Discussing essential skills/habits of mind across the disciplines of math, science, English and languages.
• Discussing assessment and the possible elimination of grades
• Discussing project based learning
• Discussing interdisciplinary curricula, like the fascinating Ninth Grade Program at Lawrence Academy.

All of this was inspired by a post about a similar group at Loomis Chafee run by Scott MacClintic. Almost as soon as our group started meeting, I contacted Scott about the possibility of us holding some sort of virtual joint meeting. Thanks to a fortuitous free day today, we were finally able to schedule the conversation.

Last night, when Scott and I were talking he came up with a simple prompt to get discussion going—”share something innovative you are doing in your classroom or are talking about with colleagues.”

The participants from my school started off the conversation and we discussed the following ideas:

• A history teacher looking to find a way to develop an international collaboration with a school in Israel and other parts of the Middle East for his History of the Middle East course
• A project I’m helping to lead where seniors in our tutorial program are experimenting with blogging.
• A biology teacher who is exploring misconceptions by having his students conduct “person on the street” interviews of other students and adults in our community with basic biology questions like “why do we breathe?”
• A French teacher who is having her most advanced students do semester long projects exploring complex ideas like developing a researching language instruction in middle school and then delivering a presentation for teachers our local district
• A Spanish teacher who had her most advanced students each choose a text and then lead a few weeks of class discussion on that text as a culminating project in the second semester.
• Getting rid of math placement tests and instead developing an integrated problem solving course that all of our new students would enter. After a month or so, students would then be placed in different sections based on the interest and ability they’d shown in this integrated class.

Our colleagues at Loomis shared the following ideas:

• Developing new ways of embracing online learning—stipending teachers to create review courses to bridge the transition between first and second year languages, or to cover foundational topics in algebra/geometry.
• Exploring ambient music creation in a music theory class and sharing works on Soundcloud.
• Working to prepare for the new AP curriculum in Spanish
• Developing an all school composting program that diverts 6 tons of food waste and turns it into compost for local gardens.

In this hour long meeting, I came away with more than half a dozen new ideas for my own teaching and school. Perhaps most surprisingly was almost every idea I heard from colleagues at my own school was new to me. Mostly it’s a reflection of how busy we all are and how little time we have to share.

It also makes me think how easy it is to tear down walls of the insular boarding school world. It is common knowledge that boarding school is a “small world”—even when I had been teaching only a few years, I seemed to “know” someone who was teaching at a dozen or more boarding schools around the nation. But at the same time, really didn’t know much about those people or the schools they taught at, since I never really took the time to build meaningful connections with colleagues outside my school. There’s something about being in a fully self-sufficient community that can lure you into the myth of intellectual self sufficiency as well. It’s my hope that tools like Twitter, Google+ and conversations like today can help us to overcome that myth.

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I’ve written in the past about the wonderful graphing program that is OmniGraphSketcher, and I have also proselytized about LinReg, a bare bones free graphing program from Pomona College that we use in our physics classes.

I love both of these programs. OGS makes beautiful graphs that you can draw by hand, which is perfect for drawing piecewise graphs for math and physics classes. It’s also super easy to simply paste in one’s data from a spreadsheet and graph it. Problem is, it’s somewhat expensive—\$20 for the educator version. Also, it won’t plot any sort of mathematical function other than linear regressions. It’s much more of a tool for drawing graphs than for analyzing data.

LinReg is great for students. It’s free, and it forces them into certain habits of good graph making that are essential—things like adding axes labels, units, and uncertainty to each measurement. But at the same time, LinReg can be frustrating and limiting—you have to type data in by hand, it won’t plot two sets of data on the same axes,  it’s a pain to clear a regression line you no longer want, and you can’t plot functions. Plus, it’s a program students need to install on their computer to use at home, which can be a barrier to entry.

Now, what would be better than LinReg? How about something that doesn’t need to be installed, something that creates reasonably beautiful graphs with a simple cut and paste of data (without all that excel chart junk), and something that on the backend allows for incredible power and flexibility. Oh, and it’d be great if it was also free, and allowed for easy syncing with services like Google+ and Dropbox.

Of course, this is just a fantasy, right? No such magic graphing tool actually exists, does it?

Check out Plot.ly.

Here are the instructions for making a graph.

All software instructions should be this easy.

Adding axes labels, a title and annotations is are one-click operations.

Here are the instructions for doing a linear (or polynomial) regression:

But we’re just getting started.

Create a new window and click on script to be able to create complete graphs using python and the numpy package, which means the sky is the limit with what you can do with this graphing package—just check out this gallery.

Plot.ly seems to be the holy grail of science graphing for schools. It’s a tool that is simple enough to create beautiful graphs with no fiddling, while at the same time allowing for lots of customization. And with the scripting power of python it’s awesome tool for introducing students to the power of computational thinking. It provides a perfect ramp from creating the simplest plot to the most complex data visualizations that will grow with students as they grow in sophistication. Saving to Google Drive and Dropbox are also built in. Works on iPad, too. With Plot.ly, Desmos, and Wolfram Alpha, I can’t think of anything you can do on a nSpire graphing calculator that you can’t do on an iPad with these free webapps.

And if you needed even more reason to check them out, their support is equally amazing.

Check it out.

My school has the most amazing dance teacher. Dance is a relatively new program at the school, but I’m constantly amazed by the incredible work his students are doing and all ways he’s added to our our school community—he’s one of the driving forces behind the manliness curriculum and man talks we have regularly on the freshman boys dorms. He’s also one of the most positive and upbeat people I know. That he can do all of this as a first time dad of a 3 month year old simply astounds me.

Today, Avi approached me with this incredible idea he and his students are developing for arts weekend—a “day in the life of the school,” and in one of the scenes, a student gets really upset upon receiving a 96 on a science test. The student then begins to dance around the room, playing with science, and not focusing on grades, and then the student suddenly sees the real joy in learning. Avi wanted me to give him some ideas for some visually impressive demos that would help communicate the ideas of playing with science and be visible to an auditorium that seats 400.

So here are a few of the ideas we came up with:

• Dancing around with a large hydrogen balloon and then blowing it up
• Making a rainbow on an overhead projector
• Using a CO2 fire extinguisher to propel a human riding on a large cart across the stage
• Dropping a tennis ball on top of a basketball and launching it into the audience.
• Using a vortex cannon to shoot smoke rings out into the audience
• Using the swinging tray to swing a wine glass filled with colored water in a vertical circle
• Launching styrofoam peanuts into the air with a Van de Graaf generator
• Making giant soap bubbles
• Floating objects using the Funflystick mini VDG.

Most of these are the classic cannon of physics demo days I’ve seen in the past, so I’d love to know if you’ve got any fresher ideas for other impressive demos that might work well in a dance performance like this.

This really has me excited and I want to see if I can find some students to create a special program insert to explain the demos that Avi chooses to feature in the performance.

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Today I had the pleasure of traveling up to Saint Ann’s School in Brooklyn (my first trip to to see a dear internet friend, Paul Salomon, and meet a few of his amazing colleagues.

One of the great things about visiting a school like St. Ann’s is just how different it is from the schools I’ve taught in and experienced—leafy campuses of hundreds of acres or more with a bounty of space (though we often still find ways to complain about a lack of space). Saint Ann’s packs an entire 4-12 program into a 13 story building with two elevators. Assemblies meet in odd classrooms and the lobby, and the entire building feels like a beehive of activity and engagement.

On of the very best things I saw is that every teacher in the department teaches in both the middle and upper school. It’s not at all uncommon for a teacher to teach 3rd grade math, 6th grade math, and functions (Algebra II). This seems hard wired in the DNA of the faculty, and as one teacher explained to me it comes from wanting students to be exposed to the entire intellectual enterprise of a department from the very beginning. Having visited and worked in an number of K-12 schools where it was difficult (if not impossible) to form inter-divisional connections, this seems like a pretty ideal solution, but it also clearly calls for a truly extraordinary skill set in in their teaching faculty and makes me wonder if this can be taught—can an extraordinary high school math teacher learn to 3rd grade with no previous experience? And vice versa?

Paul Salomon’s Functions Class

My day started with Paul Salomon’s Functions class. He began by taking his students out to this incredible sculpture:

It’s a meta meta node, and this photo doesn’t even begin to do justice to the impressiveness of this sculpture and the effort that it took to produce it. So here’s a video from Justin Lanier:

Paul starts the day by asking his students to think about a few questions:

• How many nodes are in the sculpture?
• How many holes are in the sculpture?
• How many unfilled holes are in the sculpture?

And quickly students were off, calculating and asking questions of each other. I saw a number of different approaches, from taking a single plastic node and counting the holes on there, to more elaborate arguments based on the tiling of rectangles, triangles and pentagons that make up the nodes. After about five minutes of figuring, and students reaching some success with their calculations, Paul encouraged students to consider writing up their work for the end of the week.

For the remainder of class students worked collaboratively on a set of 6 problems from an NYC math tournament. In watching Paul’s students, I noticed that they jumped right into to problem solving. They didn’t seem to be daunted by the fact that the problems were unfamiliar, different from each other or part of some math contest. Paul was able to hang back for quite some time before students even began to ask him questions. I also noticed that Paul’s students weren’t the least bit daunted by vocabulary—some didn’t know the meaning of isosceles or what exactly a chord was, but Paul was quickly able to redirect the question to another student willing to take a stab at defining the term and students readily moved forward from there. And before you knew it, the bell was ringing and Paul closed out class by reminding students that if they found a particular problem interesting, to write up a solution and submit it to him.

After Paul’s class I ran into Justin Lanier, one of the Math Munch triumvirate, and we sat for a chat where I got to pepper him with all the “so…no grades” that all Saint Ann’s teachers must hear whenever they go out. Justin did share some interesting thoughts about how Saint Ann’s managed to transform from a heavily tracked math curriculum into the middle and high school to its present curriculum which emphasizes themes and differences in particular teachers’ preferences rather than particular ability groupings.

After meeting with Justin, I went to see Paul Lockhart teach a 6th grade math class. They were working on a series of Sangaku, small geometrical puzzles that students in the class had developed. When I came in, this drawing was on the board

After settling the class a bit, Paul called on a very enthusiastic student who wanted to present a discovery he’d made about this particular figure. The student drew a box around the triangle and then of dotted lines as shown below:

He cited the fact that the class had previously discovered that a triangle takes up half its box, and then began to count off the triangles, noting that each pair (1&2, etc) must have same area there were two boxes left over outside the the big equilateral triangle, and that these two boxes must have the same area as the inscribed square. “Genius!” the student said to the class and then he started to return to his seat.

Before the student sat down, Paul began to gently prompt the student to go back though his explanation making sure that every student agreed with each assertion in the student’s argument, asking the class and individual students “do we agree with this?”, while students offered their own arguments and explanations to support this student’s conclusion. And all the while, Paul was offering little narrations about the beauty of mathematics and the process they were following. Perhaps the most remarkable thing was that after about 10 minutes into this process, the students gently realized that the student’s result was non-obvious, and therefore interesting, Paul said, but also not really helpful in finding a complete solution to the the problem. Here’s where Paul helped the class to see that sometimes interesting results don’t lead to solutions, but that you can’t always see that from the beginning, and so you should always be willing to seek out these interesting results.

After that Paul brought up another student’s approach, which was to call the side of the triangle 1, and work to find an expression for the side length of the square, which they call $a$. The students seemed to know instantly from a previous investigation that the height of the equilateral triangle was $\sqrt{\left(\frac{3}{4}\right)}$ and a couple where tripping over their tongues to blurt out the answer in terms of the “square root of three fourths”, but Paul backed the class up and pointed out that “square roots of three fourths” sounds ugly, and a move mathematicians often make is just to give something ugly a name, like, $h$. It’s at this point that the class make’s a big insight and sees that the smaller equilateral triangle atop the square is just a scaled version of the larger triangle, and the scaling factor is $a$, and it was easy for the students to then see that the height of this smaller triangle must be $ah$ And from there, he asked the students to write out an expression for the the height of the larger triangle which they explained as just the height of the square plus the height of the smaller triangle, and so they could write as

$h=ah+a$

At this point, I was pretty stunned by how fluent these sixth graders were with fluent with scaling factors, how easily they could go from describing how to find the height of the triangle in words to writing an equation for it—it was some of the highest caliber mathematical thinking I’ve ever seen.

At this point, Paul is continuing to add thoughtful narration as the students work on solving the problem. He points out to them that they have now switched from being geometers who are trying to find relationships about these shapes, to algebraists who are are now trying to take an implicit relationship between a and h and “untangle” it to find an expression for h. And he turns the problem over to them again to think about how to go about solving this equation for h.

A student tries what seems like a solution derived from a previously learned mnemonic, and she writes up $h-a=ah$ on the board, and Paul helps the class to discuss how this is a perfectly legal algebraic move, but it doesn’t seem to be bringing the group closer to getting an expression for h alone.

He asks them what makes this equation hard to solve for a, and the students recognize that the two a’s in the equation make it difficult. He asks them to think of how you might combine the a’s, and launches into a mini discussion of what happens when you multiply two things together, like $\left(a+b+c\ldots\right)$and $\left(x+y+z\ldots\right)$ and the students quickly talk about writing out all the combinations, and are even able to say how many total combinations there will be. Paul then points out that sometimes we encounter a bunch of combinations, and if we look carefully, we can see how to re-write them as the product of simpler things. From there the students realize you have h a’s and are adding an a, so you could write it as:

$h=\left(h+1\right)a$

Never once did Paul or any of the students say factor, FOIL, distribute or any other math procedure laden term.

From there students worked to solve the equation and come away with with

$a=\frac{h}{h+1}$

Narrating again, Paul notes that our work as algebraists is complete, and we can switch to being arithmeticians, now that we’ve found a beautiful, simple expression for the length of a side.

Students were quick to say that

$a=\frac{\sqrt{\frac{3}{4}}}{\sqrt{\frac{3}{4}}+1}$

Students then get into a discussion about how you might be able to write this in different ways, as $\frac{\sqrt{3}}{\sqrt{3}+2}$, and wether you could ever write it in such a way that no square roots appeared. It lead to a good discussion about why the answer had to be an irrational number.

This is getting long, so I won’t get into the amazing discussion at the end of class that the students had about another student’s Sangaku:

Incidentally, I just discovered Sangaku for myself in writing this blog post. These were sacred mathematical tablets created during the Edo period Japan (1700s) that presented a sort of geometrical puzzle, and were solved by entirely different, non-western mathematical techniques, since the Japanese did had not seen calculus. I discovered that a Princeton Physics Professor is collecting them and published them in a book.

What do you remember from sixth grade math? I don’t remember much. Maybe I was introduced to irrational numbers then. If so, I’m sure that soon after it was drilled into my head to always rationalize my denominators to get rid of those radicals in the denominator (though I never knew why, and better math teachers that me now say this practice is dumb and outdated). But I bet if I had been introduced to sacred Japanese math puzzles in 6th grade, I wouldn’t have forgotten them, and I would have developed a much greater appreciation for the beauty of mathematics and the uniqueness of culture at a much younger age.

Mathematical Art

The last class I visited was Paul Salomon’s Mathematics and Art class were he did a very cool exploration of Platonic solids, why there are only 5 of them, and then a cool exploration of the various Archimedean solids, prisms and anti-prisms (the Freedom Tower is a great example of an anti-prism). None of these topics were ever in my high school math curriculum, and Paul’s students are now working to use everyday objects to build Archemdian solids of their own.

Conclusion

Overall, I was wowed by how comfortable Saint Ann’s students seemed with struggle and how immersed they are in the world of mathematical thinking. I can immediately see two major contributing factors to this. First, Saint Ann’s has no grades, and I can only imagine how empowering this must be for helping students to take risks, to learn from feedback from the teachers and peers without having to worry about the judgment of grades and ranking.

The other factor might be that St. Ann’s Math department has developed a K-12 vision of what Mathematics education should look like (it is beautiful-jump to page 36 to read it), and then taken it upon themselves to see that every member of the department has a role in making that vision a reality at every level. It’s simply not possible for a teacher to just teach high school and grouse about how the teachers in the lower levels aren’t doing X or Y to prepare kids for high school math.

It is incredible just how present this vision of the beauty of math was in each go the classrooms I visited, and I could easily see how this atmosphere could lead to creating something like MArTH Madness MArTH Madness.

Many thanks to Paul Salomon, Justin Lanier, Paul Lockhart, and all the rest of the Saint Ann’s teachers who gave generously of their time to allow me to visit.

My school, St. Andrew’s School in Delaware, is looking for both a full time physics and a full time math teacher for 2013-14. St. Andrew’s is an outstanding coeducational 100% boarding school situated on 2500 acres of farmland, 1 hour from Philadelphia.

Here are a few other tidbits about my school:

• The culture of the school is unique, and really must be experienced to be believed. 300 students and 80 faculty living and working together to find deep joy in learning, working to make the world a better place, and to create a compassionate, caring community that rejects the culture of pettiness and cynicism that infect the traditional high school.
• The school draws students from all over the world and is 100% need blind in its admissions. Nearly half of the student body receives financial aid (this is almost unheard of in the independent school world).
• The school is led by a visionary headmaster, Tad Roach. To get a sense of what an incredible educational leader he is, I encourage you to read a few of his chapel talks.
• The faculty are incredible. They are experts in their fields, deeply committed to the craft of teaching, collegiality and continuous improvement.
• You’ll have a great opportunity to shape the future of the science curriculum at our school. We use a modified version of the modeling curriculum in physics and chemistry, along with standards based grading, and very open to your ideas for how we can continue to experiment and improve science education for our students.
• Similarly in math, you’ll have a great opportunity to help re-imagine math instruction at the school. We’re in our 3rd year of having all incoming students start in a integrated problem solving class that stresses collaborative work and has really changed the way students view mathematics, and are now looking to further incorporate this approach into upper level courses.
• Clasess are very small, usually around 12 students, and a typical load is 3 or 4 sections.
• Both the math and science departments have tremendous resources. The physics department has a special endowment to bring leading scientists to campus to deliver a lecture and speak to classes (past speakers have included Brian Green, Bill Phillips, Janna Levin and this year, Jill Tarter. We have resources to bring teaching experts to campus to observe classes and offer coaching, like Rhett Allain, and to provide for almost any professional development opportunity you can imagine.
• As a small school, we’re also very flexible. If you have interest in teaching another subject, this can likely be accommodated in the future. In my time here, I’ve taught multiple levels of physics, computer science, mathematics and worked as a college counselor.

In short, the community is truly inspiring. If you’ve never considered boarding school teaching, you’ll be amazed by the connections you can form with students and the things they can accomplish when you encourage their interests outside the school day. Just last week, students led a 2 hour discussion of climate change and the economic and environmental impacts of the Keystone XL pipeline one evening, in order to prepare for sending over 90 students to the Forward on Climate Rally this past weekend in Washington DC.

Here’s the link to the job description. Please feel free to ask questions (confidentially) in the comments, and I’ll try to answer them or connect you with those who can.

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We’re studying logarithms in my Honors Algebra II class, and when looking at a logarithmic function for the first time, a very astute student asked “does it have a horizontal asymptote.” And my quick response was a slightly dismissive “no.” But the student was insistent, and pointed out that the slope seems to keep getting smaller and smaller, so shouldn’t it eventually reach zero, and won’t that mean that there is an asymptote?

I like this question because, if I let it, it pulls me away from simply repeating the same information I was handed back during my math education, and really gets me thinking about the nature of the logarithm and the meaning of horizontal asymptotes.

What I “know”…

So here’s what I “know”—the logarithm is just the inverse of the exponential function, and the exponential function doesn’t have any vertical asymptotes—you can always exponentiate a larger number. Thus, it should be that when you invert this function to form the logarithm, there shouldn’t be any horizontal asymptotes. As you take logarithms of larger and larger numbers, the output of the function should continue to increase.

A comparison: $\frac{-1}{x}+3$

Let’s compare the logarithm to a function that we know does have an asymptote, namely, the inverse function. Just to make it a bit easier to visualize, I want to work with 1/x reflected over the horizontal axis and vertically shifted by 3, $\frac{-1}{x}+3$

In many ways this function looks similar to the logarithm—it is increasing, and as it grows, its slope decreases. But this particular function has a horizontal asymptote at $y=3$.

What about the rate of change?

Things get more interesting when I look at the rates of change for these functions.

For the logarithm:

$\frac{d}{dx}ln(x)=\frac{1}{x}$

for the inverse function:

$\frac{d}{dx} \left(\frac{-1}{x}+3 \right)=\frac{1}{x^2}$

As we can see from the graphs, both slopes decrease as x increases, and the inverse decreases at a bigger rate. Does this difference explain this behavior?

It does seem to tell me that just because the rate of change of a function goes to zero at infinity can’t be proof of a horizontal asymptote. But could it be that any function whose rate of change goes to zero faster than a particular rate (say $\frac{1}{x}$ will have a horizontal asymptote?

I’m flummoxed. How would you explain to a student why a function whose rate of change is always decreasing never reaches a maximum value, particularly when every similar function he’s previously encountered like this does have a horizontal asymptote?

Cal Newport, author of How to Be a High School Superstar and more recently, So Good They Can’t Ignore You, just did an interview with Steve Hargadon for his Future of Education Podcast, one of my favorite podcasts.

The entire interview is excellent, but this one tidbit near the end of the interview grabbed me. It is one of the best bits of advice for a high school student I’ve heard. Here’s Cal’s elevator speech for current high school students on what his latest book is about.

There’s not one thing that you are hard-wired to do that you have to discover before you’ll be happy. You’re young—if you’re going to college, spend your next years exposing yourself to things, but more importantly, learning how to do hard intellectual things. Keep in mind that to build a life you love, what matters is how good you are at things, not whether or not you’ve found the one sort of true match you’re supposed to do. There’s no one true passion out there waiting for you, you have to go develop passion, and getting good at things is how you’re going to do it.