I have an awesome friend and colleague at my school who is a geogebra master. He has started keeping a blog — Geogebrart — posting fairly frequently some stunning, jaw-drapping mathematical art he created using this powerful program.  Check this recent one out — which happens to be one of my favorites! Dualities!

Although I know most of the basics of Geogebra, I have not yet progressed to the stage passed “novice.” However I really want to get there, because this program is so freaking awesome.

When I was at TMC14 this summer, there was a sesh run by John Golden, Audrey McLaren, and Jedidiah Butler. They are like Jedi masters of Geogebra (though I know Audrey will play coy and say she isn’t…). When I was there, I learned about conditional objects, and it was awesome. (The google doc they used to help people out is here.) In about 30 minutes, with the help of John Golden and some kind people near me, I was able to make a rinky-dinky geogebra file which has a triangle on it, and has three points on the three different sides. When you drag each point close to where an altitude of the triangle would hit that side, I had something like “WOW!” or “YOU DID IT!” pop up! And if you got all three points close, something like “ALL THREE?! YOU’RE A SUPERSTAR!” show up.

Okay, okay, I wasn’t going to show you it because it’s sooooo dumb. But heck, whatever, here it is. Click on the image to check it out.

Okay, you and me, we both know that file is totally useless a teaching tool. And it is gross looking. By all accounts, I should not be excited by it. But the weird thing was: I was really proud of it, and I wanted to show everyone around me what I created. Even though I know it was simplistic and useless, I wanted to create a file that did X and I was able to do it! Although it felt dumb to get psyched about it, I was so excited that I could create something that would do what I wanted it to — that I couldn’t do before!

Today I was again inspired by my colleague and friend’s geogebra art, so I wanted to create some of my own.

I was quickly able to make this in 10 minutes [click on the picture to go to the file and mess around with the parameters! cool things happen!]

My goal was to define a curve parametrically and then have — at a ton of points on the curve — a circle to be drawn so it would look like a tube. That ended up looking only moderately neat. So I changed it so that as one traveled on the parametric curve drawing the circles, the radius of the circles would change (based on some formula I fed it). The reason this wasn’t so hard for me? I knew all the commands to do this except for the parametric curve command, which was easy to figure out.

But then I wanted to try my hand at something that would take more than 10 minutes and that would challenge me. I wanted to have something “show” a sphere via the animated drawing of “slices” (ellipses). It was inspired by this beautiful gif, but I knew that was going to be too hard for me to start out with. So I decided I would start out with a simple sphere with slices going horizontally and vertically, with no rotation.

After somewhere between 90 and 120 minutes, I did it! (You can click on the gif to go to the file and play around with some of the parameters.)

Although the image isn’t as cool as the one that took me 10 minutes to create, I’m way prouder of this. It is because it took a ton of learning and trial and error in order to figure out how to do this. The set of problems I encountered and somehow figured out:

• I know how to create a single ellipse in the center of the circle, but how do I make another ellipse a certain distance away that still only touches the edge of the circle?
• How do I make the ellipses “width” (minor axis) decrease so that it is fattest near the equator, and almost like a line near the poles of the sphere?
• Without manually typing a zillion ellipses, how do I tell Geogebra to create all the vertical ellipses at once, and all the horizontal ellipses at once?
• The way I was generating the ellipses resulted in a problem… once an ellipse “hit the pole”(became a point), it would turn into a hyperbola. So I needed to find a way to make sure that once an ellipse “hit the pole” it would disappear.

I figured all this stuff out! So even though the sphere doesn’t look nearly as cool as I’d like, I feel so much more accomplished for it than with the super-cool-looking circles of variable radii drawn on a parametrically-defined curve.

***

Note: it’s amazing how “simple” this sphere image is once you figure it out. Once you create three sliders:

t goes from -5 to 5 [incriments of 0.1]
StepSize1 goes from 0.05 to 2 [increments of 0.05]
StepSize2 goes from 0.05 to 2 [increments of 0.05]

and you enter the following two (that’s it!) geogebra commands:

Sequence[If[abs(t – n StepSize) < 5, x² / (25 – (t – n StepSize)²) + (y – t + n StepSize)² / (1 – sgn(t – n StepSize) (t – n StepSize) / 5)² = 1], n, -5 / StepSize 2, 5 / StepSize 2, 1]

Sequence[If[abs(t – k StepSize2) < 5, (x – t + k StepSize2)² / (1 – sgn(t – k StepSize2) (t – k StepSize2) / 5)² + y² / (25 – (t – k StepSize2)²) = 1], k, -5 / StepSize2 (2), 5 / StepSize2 (2), 1]

Then you’re done! Well, you should animate the t-slider to make it cycle through everything without you having to drag the slider!

Seriously, two commands, that’s all it takes. But hopefully from the commands themselves you can understand why it would take me so long to figure out…

Sue VanHattum (of Math Mama Writes) is in the finishing stages of editing a rich collage of works that is aptly named Playing with Math: Stories from Math Circles, Homeschoolers & Passionate Teachers.

Truth be told, I tend to eschew reading about math education because most of what I’ve read feels dry and irrelevant to me. I tend to stick with who I trust when it comes to math education: my colleagues, whether they be in-person or virtual. And although I didn’t tell Sue this, because she was so kind to share an advance copy with me, I fretted about falling asleep while slogging to get through 67% of this book because of the subtitle. (I have never led or been to a math circle, nor do I work with homeschoolers.) I’m just an average joe teacher who keeps his sights on his classroom and his kids, and… well… that’s about it.

Now for the punchline: I couldn’t stop reading it. All 100% of it.

The book isn’t composed of traditional articles-as-chapters. Playing with Math is, rather, a collage. I was treated to bursts of math puzzles, activities, and games (the majority of which were completely new to me) wedged between short and medium-length vignettes from people who are working with kids on math. (There are almost 50 contributors to this book, some of whom I know!) I can see this book being a great present for one of my NYC colleagues, because as I was reading it on my laptop, I kept thinking how perfect this book would be for subway reading because each piece was only a handful of pages. A testament to the book is that as I was reading it, I wanted a zillion post-its and tabs to flag this or that.

Even though I haven’t been to a math circle nor am in any way involved with the homeschooling community, reading the pieces around those topics were interesting precisely because I know so little about them. But moreso, they got me thinking about ways I could differently think about my classroom and my kids. When it came to the math circles, it gave me ideas on how to let go and trust kids to take charge of their own mathematical learning more. And when it came to homeschooling (and unschooling), I wondered how much kids lose their love of learning precisely because of the structure of school. The author of the pieces did this by telling stories. Some were like video cameras, documenting and explaining the “teacher moves” in some particular math circle sessions. Some were powerful and wrenching first person narratives about mothers trying to help their children. And the teacher section was a curation of powerful stories of teachers like me, trying to be a little bit better each year. Some pulled lines to whet your appetite:

We began today’s math circle, the first of six sessions, sitting in an “ogre.” Not a circle, not an oval, but an ogre, the kids’ way of precisely describing the shape we made.

Peter Panov and David Plotkin can barely stay in their seats. They’re firing questions and comments and conjectures and quips at their instructor, Jim Tanton, as fast as he can respond. The whole class of thirteen-year-olds was giggling when I walked in. On the board is a list of some Pythagorean triples and a procedure for generating more. Tanton had just generated the triple (-1,0,1), and a general hilarity about the idea of a triangle with a negative side-length erupted. Now it’s as if he were dangling strings in front of a pack of puppies. They’re all worrying at the problem, tossing out ideas, wiggling in their seats.

Looking back now, I see how far off the mark we were. We should have advocated for our daughter to ensure she received an intellectually, socially, and emotionally appropriate education. But we were overwhelmed by the more-pressing problem of Ryan, so we missed her quiet desperation. I wish I had been more proactive and looked below the surface. I wish I had worked more closely with her teacher. I wish I had trusted my own instincts about my daughter’s needs and abilities.

I waited eagerly for him to arrive the next morning, looking forward to the moment when he would put AAAAAALLLLLL those tiles together in neat rows by category, and he would have to exchange several times (not to mention his surprise at seeing all the units disappear when multiplying by ten). Instead Roland came in, shook my hand, and said: “My dad told me that all I have to do is add a zero to 8,696 and I’ll have my answer, because when you multiply by ten you just add a zero.” My heart sank. Oh no, Dad! You robbed your son of such a cool experience!

Several years ago, my school experienced a shortage of geometry books. There was talk of teachers sharing class sets and photocopying pages for students. I decided to try a different strategy. I took this as a professional challenge to see how long I could teach without a textbook. I knew whatever happened would be a growing experience for me as well as my students. Through no fault of the school library, two or three weeks stretched to seven. By that time, I was well into my “textbook-free” strategy, so I just kept the ball rolling … for the rest of the year.

I like stories, and that’s what this book is. Not disquisitions or pronouncements or shallow research studies. Stories. The authors bring to life their experiences and interactions with kids and their insights and their frustrations, and I started care about these people, their children, their classrooms.

If there is one theme that stood out to me, it is this: we need to work at undermining the constraints that we are confronted with (whether it be textbooks for teachers, or the entire school experience for some parents) to allow us to do what we all know is best for kids… playing and engaging with math in a way that tugs at internal motivation (curiosity, the excitement of discovering something) rather than external motivations (praise, grades). We need to continue to find ways for doing math to be beautiful and creative acts of passion and wonderment and joy. The contributors of Playing with Math are working on this, and I am inspired by their stories.

Sue speaks about the origins of this book here:

And she is having a crowd-funding campaign. “The book has been written, edited, and illustrated. The money raised here will allow us to pay the artists, editors, and page layout folks, and it will pay for the print run.” I contributed so that I could get a paper copy of the book and finally mark it up with all the post-its and flags I want!

About a month ago, I received a teaching award at my school. Technically, I suppose it isn’t an award, but a chair (“the William C. Stutt Chair for Math, Science, and Technology”). Fancy, right? I wasn’t going to blog about it, but it is something I want to archive and that’s the biggest (but not the only) reason I blog.

It’s given out every three years, and the last person to get it is one of my best friends at the school (who is also the person I look up to as a teacher).

When I was called up, there was a standing ovation from the faculty. Of course, let’s put the cards on the table here: there always is a standing ovation from the faculty when anyone gets an award. But I can’t help but admit I got a real glow-y feeling. I was overcome when I saw my parents there, a surprise! They popped out of the curtain and hugged me. I didn’t quite know what to say, so I babbled. All I remember saying is my teaching motto: “Try to suck a little bit less each day.” I posted this on facebook, me feeling babble-y, and a friend said: “You are amazing. Your comment to the faculty about trying to suck less everyday was perfect and came up again a number of times over the remainder of the meeting. I hope you and your parents had fun celebrating your awesomeness this afternoon. Also, please take that standing ovation personally. We could have gone on clapping forever. There was nothing perfunctory about it. Congratulations!” So yes, me all feeling warm and fuzzy.

I also posted this on facebook: “Although I’m not one who basks in honors and awards (I even skipped out on going to my college Phi Beta Kappa induction and a writing award in college), I do feel like teaching is a profession where you don’t get a lot of positive reinforcement for the emotional struggle that you carry with you every day. A few kind words from students occasionally, or a nice email from a parent, if that. 99% of what we do goes unseen and unacknowledged. It’s isolating and exhausting. So this award was a nice thing, something I can turn to when I feel like I’m emotionally drained and a failure. (Which is more often than not.) But more than that, it reminds me how important it is that we teachers give accolades and kudos to each other in a million unofficial ways, *everyday.* Because most all the teachers (especially the math and science teachers) at my school are pretty awesome. And every one of us are working to do right by our kids. And more than awards that get handed out once in a blue moon, we need to pay attention of the good that everyone else is doing around you, and acknowledging and huzzah!-ing those things. Yes, that’s what I see from this. Let’s prop each other up.”

The little news blurb on our school website is here. Archived.

This January, for seven days, I taught a seven day course with a friend and fellow teacher. Our school eliminated midterms and instead instituted different programs for different grades. Juniors and seniors were given the opportunity to sign up for full-day courses designed and taught by faculty on topics of interest. Faculty were given the opportunity to design courses which got kids to think about topics in a different way.

My co-teacher and I developed a course that was designed to be interdisciplinary (we were working at the intersections of history, science, and philosophy), hands-on (students would be working in the laboratory), and rigorous (meaning kids would be expected to think and work at a high level).

Designing and teaching this class was one of the hardest things I’ve ever done as a teacher. And I don’t know — honestly, I don’t know — if we were successful or not. Even with the feedback we received. Thus even though it was challenging, I’m not sure I felt it was rewarding. In fact, the reason I’m writing this blogpost now, months after this, is because I was so exhausted with the whole thing I couldn’t bring myself to even think about it in a reflective or objective way.

The origins of the class go back to the previous year, when my co-teacher and I started trying to envision precisely what the big picture ideas were, and how we were going to get kids to go from point A to point B in their thinking. This also was coupled with the question: how the heck do you design seven days with the same group of kids, from 8:3o to 3:15. Seriously put yourself into our shoes for a second. Initially, it’s pretty exciting! All this time! Do what you want! But then you realize: you are going to have 12 to 16 kids in your charge, and you need to fill up that time with multiple activities! Quickly this went from exciting to daunting and anxiety-filling. For months, the co-teacher and I would have meetings, read books and articles, come up with ideas, refine our ideas, and throw out our ideas. Coming up with a lesson plan for a single day took weeks of work. The agony, the hours, the frustration… I don’t wish that upon my worst enemy. But we finished.

Our course abstract:

Can you imagine building a battery without the concept of electrons?  What would it be like to describe chemical reactions without discussing atoms?  Would you believe Einstein’s theory of relativity if no text book told you to and there were no way to test it?

In this course, you will have the opportunity to put yourself in the shoes of scientists who (in retrospect) revolutionized the way people viewed and understood the natural world.  By carrying out famous historic experiments, you will explore the process of creating “scientific models” and “scientific facts,” many of which we now take for granted as self evident. This course will be hands-on and interdisciplinary. In addition to lab work, we will read primary and secondary sources that will allow you to place science in historical context and understand scientific knowledge making as a process and a product of its time.

Our course objectives:

Through this course, students will explore:

• science in historical context
• how science is influenced by and a product of its time
• that the process of science involves models changing over time
• that what we take for granted is often messy, weird and sometimes illogical
• that science is a human endeavor
• that the making of science is a process
• how scientific “facts” get accepted/discarded –  that ideas are nothing without the acceptance of many people

and ask the big questions:

• What is an experiment?
• What is a scientific fact?

Anchor Texts:

Thomas Kuhn’s The Structure of Scientific Revolutions
Original papers by Robert Boyle and Alessandro Volta
Secondary texts

Experiments:

Originally, we planned to have a number of experiments: Proust, Boyle, Volta, Oersted, Einstein. However because we had a snowday (there went Einstein and the discussion of thought experiments), and because some of the experimentation took much longer than expected, we had to eliminate more (Proust and Oersted). Thus, we only ended up working extensively on Boyle and Volta.

Content:

One day was spent on a field trip to the Chemical Heritage Foundation in Philadelphia, but the rest of the days were spent having deep class discussions and carrying out two in-depth experiments in the labs. We did Boyle’s Law experiment, and they had to bend glass to make their own J-tube, and play carefully with mercury. (We inducted all our kids into the Royal Society, after reading bits of the original charter, and administering the oath that the initial founders took.) Our kids saw that our modern instantiation of Boyle’s Law (PV=k) was nothing like the original formulation (they only were given Boyle’s original paper to guide their research and help them figure out how to reproduce the original experiment), and they started to get at the idea that Boyle was looking at his experiment through a totally different lens (“the springiness of air”). My favorite part was when kids saw how their little sidebar about Boyle in their chemistry textbooks was just a black box for so much! And how it wasn’t just “one crucial experiment” that suddenly worked and changed our understanding. Mwahaha, the title of our course is precisely the thing we aimed to get our kids to debunk. 

Our second experiment was building (well, improving upon) the first voltaic pile. Again they only had Volta’s original paper to work from, they were given many materials that Volta mentioned in his paper to play around with and test (e.g. lye, silver, zinc, tin, coins, leather, cardboard, salt water, etc.), and they were working to win le Prix Volta (a real prize Napoleon and the French Academy of Science offered for research in electricity, after Napoleon saw Volta’s original battery demonstrated). This contest was good to talk about collaboration and competition in science, but my favorite part was having kids read a challenging history of science article about what actually was behind the creation of the battery (a torpedo fish!) and what sorts of things had to have happen for there to be the physical and intellectual space for Volta to even have the conditions for him to come up with his Voltaic Pile. That the battery is historically situated, and tools, ideas, and people had to come together in a specific way for the battery to emerge and look the way it did. I also really liked that students could understand that there could be an explanation of electricity that didn’t center around electrons.

That dovetailed really nicely into how we were talking about Thomas Kuhn. We used Kuhn’s Structure of Scientific Revolutions as our core text that they were reading extensive bits here and there each night, and although I was worried it would be too abstract for them, they grappled with it and came out victors. And I think (hope) it was a real mind-blowing experience when they realized that “old” theories weren’t “bad” because those scientific practitioners who adhered to them were dumb (or at least, weren’t smart enough to see the Truth with a capital T). And listening to them discuss Kuhn, grapple with the idea of Normal Science, and start to see glimpses that (1) science isn’t accumulative in the simplistic way that textbooks tend to say it is, and that (2) we always are looking at data, theories, experiments, observations through specific eyes, and what we see is dictated by the paradigms we accept.

Images: Here are images from the Symposium, without student faces in them. (Hence, we don’t have the majority of my favorite pictures.)

Now that I have gotten your attention, I’m sorry. I don’t have the best activity ever for an Algebra II class that involves cupcakes. But fine, you want cupcakes. Here.

Now for the reason why I lieeeed to you. You know it’s gotta be big, and important. It’s this. I need you to read this, and take a moment, and actually consider it.

We have a math department chair opening at my school, and you or someone you know might be the person who would be perfect for it.

So I have a lot to say. I should probably note at the top that everything I’m saying is my own opinion, and this post doesn’t come from my school or my department. Just me. Now to the other stuff… I am not someone who wants to go into administration. And my colleagues also love being in the classroom full time. We tend to love our little classroom universes, and even though we engage in the bigger picture of the curriculum-at-large, our primary interest is being intellectually stimulated by classroom teaching. So we want to find someone from the outside who can see the bigger picture and wants to shepherd a bunch of thoughtful and awesome-face teachers as we push forward into our next step.

If this even remotely sounds like something you’ve been toying around with, keep on reading.

For some background. I teach at Packer, a fantastic independent school in Brooklyn Heights, New York City. The school is a Pre-K through 12 school. There are so many wonderful things about my school, I don’t know which to list. It is not religiously affiliated, but we are housed in an old church — and there is a chapel where we have meetings, and this chapel has beautiful stained glassed windows. The architecture is Hogwartian. There are about 80 to 90 kids per grade, and class sizes tend to be around 12 to 16 (though sometimes things go under or over). The school underwent a comprehensive renovation of the “Science Wing” and this summer it is going to renovate many of the Upper School (high school) classroom. The kids all have laptops, and all the rooms currently have SmartBoards, but next year they will be upgraded to Sharp LCD boards (and some will have ENO boards). When it comes to teachers being able to get “things” they need to teach, we do. Similarly, I have never been turned down for any professional development opportunity I wanted to pursue, and have always been fully funded. There is a commitment to teachers on that front.

The school is in the middle of an ambitious 5 year strategic plan, which includes a special component involving math and science excellence.  For me, the most exciting thing about the strategic plan is that teachers are thinking more and more about the importance of the process of acquiring knowledge. For me, that’s exciting because I have been wanting to move towards a more “how do we do math?” approach rather than “here, let me show you how to do math, now do some problems.”

Now to speak specifically about the math department, and why I think it’s worth considering. The math department head is in charge of math in grades 5 through 12 (middle school is 5-8, upper school is 9-12). That would mean being the head of 13 or so teachers.

We’re a really well-functioning department, where everyone gets along and are friends with each other. When we’re feeling wonky, I might be in the office with TeacherX , and we’ll close the door, put on the Sound of Music, and we’ll spin around in our chairs. (Because we both love the Sound of Music.) And every single time anyone is going to the photocopier, they ask if anyone else in the office needs something copied. And we all buy diet coke and chocolate share it with each other. We do site visits to other schools to see what they are doing. And teachers of the same class meet regularly. We share materials all the time. We pose puzzles to each other. And we bounce ideas off of each other.

What I’m trying to say is: that would be a concern of mine… coming to a new school and not knowing how the department is. I can say that we is aweeeesome.

I personally see us at a crossroads, and one where someone could come in and do some great work to take us to our next step.

We’ve come a long way in coming up with a solid and coherent curriculum. We have been trying to push our curriculum to get students to articulate their reasoning more… We have made “writing in the math classroom” a goal of ours for the past two years. And although we’re all very busy, we have made a goal to visit each others’s classes a number of times (I think 8?) before the school year ends. (That reminds me… I need to try to a few observations soon!) And we’re now in the process of thinking: how do we get problem-based learning in our classrooms?

And this is the crossroads we’re at. How do we bring our teaching, and our curriculum, to the next level? (I think this is a question the whole school is asking, because of the strategic initiative.) For me, that means learning to focus on letting go more, and developing curricular materials which continue to push students to focus on the fundamental ideas and less on procedures. It means getting kids to do the heavy lifting. It means trying to deconstruct a curriculum so I can figure out what the essential mathematical idea is, and then find ways to really bring that to the forefront. That’s all for me. Different teachers are at different places in their career and have other ideas on what they need to do to get to the next level. But the takeaway for you is that we’re interested in the craft of teaching, and looking to forge forward as a department.

That isn’t to say that everything is all roses all the time. What place is? And better yet, what place filled with teenagers is?

But it’s a place which I’ve been happy and proud to call home since I’ve started teaching. (It is suppose it’s actually a second home to me, since I spend so much time here!) The school took a chance on me — a young kid with only student teaching experience — and gave me a place to grow professionally. I was allowed to experiment with standards based grading (this is my third year doing it in calculus). I felt like I needed to switch one of my courses last year because I was feeling stale with it, and just plain tired, and that happened. I asked for funding to go to multi-day out-of-state conferences and I have always been approved.

The school is going through changes, as we work towards the strategic plan. And I think our department can, with someone with passion and vision and a strong work ethic, help us take our work to the next level!

Our department head is leaving because of reasons unrelated to her job here. And this timing of this is — at least for independent schools — late in the game. That is why I want to reach out to you guys. A perfect audience of math teachers! If you can see yourself or someone you know in a place like this, working with meeeee!, get into gear and apply!

We want someone awesome, and I’m 200% sure that the teachers in the department will do everything we can to support whoever we hire in their new role. You won’t be walking in alone, but rather with the support of everyone in the department who wants you to succeed, and will do everything we can to make that happen. We are a department and we look out for our own.

Because of the lateness in the hiring season, please please please don’t wait a few days before getting around to it. It is (in my opinion) a one-in-a-career opportunity, but the window is not going to be open for long. We are going to be working on this hire ASAP. 

The job posting and instructions about sending your information are here.

I’m turning off comments.