To begin with, an oft retweeted and favorited thing at TMC was this (and heck, I probably retweeted and favorited it):

I heard #tmc14 described this way yesterday. 150 teachers who all believe they can change the world. Here they are. http://t.co/Xa2bz01o9r

—

Glenn Waddell, Jr. (@gwaddellnvhs) July 24, 2014

I *strongly* don’t put myself in that category. That isn’t part of who I am. I don’t teach to save the world. I don’t see myself as changing students’s lives, nor is my goal to have students come back to me and say “your class changed my life.” I don’t blog to change math teaching. I don’t have grand ambitions or even care to think on such a large scale. It’s a nice sentiment, but it’s just not me. I’m like a hobbit, happy and content with my little corner. Working at things with a small scale. Getting an ah hah moment, or altering the way a student sees and understands mathematics, or helping a fellow teacher out with this or that. This is what I enjoy doing.

With that said, a lot of my thoughts around the mathtwitterblogosphere (MTBoS) and TMC in particular have come out of two things:

1. A post by Mo titled “I am a fraud” and as a follow up a post by Lisa Henry titled “Hi My Name is Lisa” which resonated with me and with many others. Some key lines:

Mo: They were so honest, so completely naked, and I, wanting to join in “fit in” offered some of my fears but then as I awoke today I feel dirty. My heart is heavy, because I lied. Well I didn’t completely lie I just shared certain fears and strengths that manipulated people to see me the way I wanted them to see me. We were all skinny dipping but I had a flesh colored bathing suit on “with painted on abs”… And as I enter my 9th year of teaching, I could be entering my last year. There is a high possibility that I could be going into sales and this conference confirms my movement into that field, because I feel so inadequate….. so beyond inadequate.

Lisa: I am not the best math teacher. I am not an amazing math teacher. I have a LOT of work to do to improve. There. I said it. I wrote it in my blog and I am not taking it back. It is there in print. Ever since I have been involved with the Math Twitterblogosphere (MTBoS for short if you are not familiar), I have felt this inadequacy. I see what other teachers are doing in their classrooms. I have tried some things. Even blogged about what I have tried. But for the most part, I haven’t changed a whole lot in my teaching since I started Twitter almost 5 years ago.

2. A flex session held by Lisa Bejarano which was about how we leave TMC and change things — both in ourselves, and in our schools. I have personal notions about barriers involved changing things in my school, nor do I have the presumptuousness to say that I am The Person To Effect Change or that I Know The Right Ways. Because I too feel very much like Mo and Lisa in that I’m not “there” yet. I’m not a master teacher. However when it came to our discussion of how we change things in ourselves, Lisa B. threw up this great chart on the projector:

At first, I was skeptical. I have this bad habit of seeing charts like this and immediately dismissing them. They take something complex and box it in to something simple. But you know, the more I thought and looked at it, the more it made sense to me.

For example, I’m teaching geometry next year, and I have tons of resources, a vision for what I want the course to look like, I think I have the skills needed, I have the motivation/incentives to do well. But I don’t have a way to go through the massive amounts of ideas and resources to actually move forward. I don’t have an action plan. So I’m in the middle of false starts. And it feels that way.

But what I want to focus on is missing *skills — *and the resulting anxiety. I was thinking about the times in my classroom when I didn’t just take a little step forward but dove right in to make a big change because I had a bigger vision I wanted to accomplish. I can think of two:

1. Implementing standards based grading in calculus (four years ago)

2. Running a class entirely through group-work (two years ago)

Although I advise people to take baby steps, and change slowly, those times I didn’t take that advice were the times I grew the most as a teacher. Those were also the times I felt the most anxiety. Why? Because I hadn’t yet developed the skills I needed. I didn’t know how to organize standards based grading in a way that would work. I didn’t know how to make sure I would catch the conceptual as well as the procedural in this system, nor how I would get synthesis of skills. I didn’t know how to make sure students worked well together. I didn’t know how to create actvities and lessons so that students would have to rely on each other to progress. But you know what? Without jumping in, I never would have gotten the skills needed.

Let me tell you: those were high anxiety times. They required a lot of emotional energy and a ton of time. But they were also times of immense growth for me.

Now to the “me” part.

I don’t feel like I’m a master teacher or close to it. And I feel confident with that assessment of myself, because who knows me better than me?

As part of that, I also often feel inadequate, and sometimes like a fraud. If I look only at the world of my school, I think I wouldn’t feel this way. I’d feel fine. I’d actually have very little incentive to change, because it’s a lot of work for no extra rewards and I am doing well by the kids (see the chart above). But because there is this much bigger world, which I am exposed to (namely the math-twitter-blogosphere), things are different.

I am constantly exposed to many things online. A lot of them are resources and lessons, but sometimes there are ideas about good teaching that I wouldn’t have access too. Like the importance of mathematical discourse (talking, writing), or to question the nature of assessments and what grades mean, or the importance of having students see each other as mathematical authorities. I am not exposed to these ideas in my school constantly, so I would not think that these are things I believe in. But being bombarded by all the stuff out there online and at TMC, and seeing what resonates with me (or what inspires me to change), is helping me (probably subconsciously) evolve my personal, theoretical framework about teaching and learning (thanks Dan Meyer).

And that is where the anxiety comes in. Because now my bar about what is good teaching has been thrust upwards. And now I have to work on reaching it. It isn’t that I feel competitive with others, but that I feel competitive with myself [1]. I have a drive to be my personal best, and to do the best by my kids.

So because of this exposure to great ideas for the classroom, and bigger ideas about what makes an effective classroom, I get caught up in feeling like I’m not doing a good job, and the anxiety hits me. And sometimes this nadir will last for *months*. I don’t feel like I’m doing a good enough job in the classroom. I haven’t given any formative assessments. Kids aren’t engaging in real mathematical discussion. I haven’t improved at all from the previous year. Heck, maybe I’ve even de-evolved. And I get in this cycle of anxiety. It sucks.

But it’s a double edged sword, because it is this anxiety that drives me, that pushes me. I enjoy the intellectual challenge it gives me. And at least for me, it’s this anxiety and this feeling of inadequacy, coupled with my own personal desire to better myself, that provides a productive tension. I recognize in myself that I need those lows and those feelings of anxiety in order to get better. It’s part of my own personal growth process. And as much as I wish I could be confident and grow without the feelings of inadequancy, I’ve come to realize that’s what works for me. At least for me.

I end with a tweet from Jami Danielle who pretty much summed this up for me, and makes this whole post just a bunch of verbal spewage (didn’t I say that at the beginning)?:

[1] I suspect, though I do not know, that all this talk about “inadequacy” and how it resonates with so many people in the MTBoS and at TMC is tied up in some sort of cycle like this. Because of this, I don’t think it’s something to be “fixed” (e.g. how can we make it so when people come to TMC they don’t feel like crappy teachers?). At least I wouldn’t want someone to “fix” it for me.

]]>

***

I’m going to make a short post inspired by Twitter Math Camp 2013 (TMC13), rather than TMC14. Both @calcdave and I led morning sessions for precalculus teachers. Through that morning session, some nice end-products were created — an organization for the curricula, actual classroom activities — and you should feel free to check them out here. [1]

@calcdave and I brainstormed how we could get people in the morning session to know each other, but make sure we have math content in that activity. We came up with *Rational Function Headbandz*, which was inspired by this post on *the agony and dx/dt*.

**The setup: **There are a bunch of cards (they could be index cards). On the front of them is an graph of a rational function. On the back is the equation of the rational function. The cards are attached to ribbons or headbands, so that when attached to the forehead only *other* people can see the graph on the front of the card — not the person wearing it. Sort of like this image below. You can re-imagine how to create these cards/headbands so they work for you.

**The Goal: **Since this was an introductory activity, participants picked one of two goals for themselves… (a) to figure out as many features as they could of their rational function and to sketch a graph from those features, or (b) to figure out the equation of their rational function.

**To Play: **I put all the cards/headbands on the table, and *covered up the graph with post-its* so the participants couldn’t see the graphs. I wrote on the post it if the graphs were graphs I considered sort of challenging, pretty darn challenging, or wow-you’re-going-for-it challenging! Then they attached their headbands to their head, and had someone else remove the post-it note.

Before starting they were told the following things about their rational functions:

- All the graphs are of rational functions.
- Some might be plain old polynomials. (Rational functions with the a 1 in the denominator!)
- If written in the most factored form, none of the terms has degree of more than two
- If written in the most factored form, most of the coefficients are really nice

Each person carried around with them a notebook, and they were allowed to ask *up to three questions* about the graph to each person (and a get to know you question to each person!). The rub? All questions had to be answered with a single word or a single number.

A valid question: “How many holes does my graph have?”

A valid question: “Is my rational function a line?”

A valid question: “Does my rational function cross or kiss the x-axis at x=3?”

An invalid question: “What is the coordinate of the hole?” (Because the answer will have two numbers as an answer — an x-coordinate and a y-coordinate.) You could instead ask “What is the x-coordinate of one of the holes of my graph?” and then follow up with “For the hole with x-coordinate BLAH, what is the y-coordinate?”

After three questions, they move on to a different person. Then another. Et cetera. From these questions they were supposed to gather information about their graph, and possibly about their equation.

You stop the game whenever you want. Everyone looks at their graphs and equations, and ooohs!, dohs!, and aaahs! result.

And then if you have time, you can debrief it with students by talking about what they thought was important information to gather in order to sketch or come up with the equation for the graph (holes? x-intercepts? y-intercepts? vertical asymptotes? horizontal asymptotes? slant asymptotes? end behavior?). And then if you had time you could have individual students present their graph, their thought process, and their solution.

**Our Graphs: **We really varied the nature of the graphs because we were working with precalculus teachers and we didn’t know their ability level with the material. And also I know I emphasize in my class working backwards from the graph to the equation, but that isn’t a standard thing taught. So I would highly recommend creating graphs of your own based on the level of work that you’re doing in your class.

**Trouble Spots: **One thing that was challenging for us when we played this was what someone does when they have figured out their own equation/graph. They came to us and we confirmed. But then what? We should have anticipated this because we had such varying levels of difficulty for graphs. I wonder if a good solution would be to then try to figure out the equation for the rational functions of others when they are being asked questions.

Another thing to keep in mind is that this will take a longer time than you think. We used this as a get-to-know-you activity, and so that extended everything even more. (In your class, your students probably won’t be using this as a get to know you activity.)

**Alternatives: **Just as I adapted this from a teacher using them for trig functions/graphs, these can easily be adapted for other topics. Some initial ideas:

Geometry vocabulary review: Students have a vocabulary word on their heads. They only can ask questions with one-word answers. (e.g. “Does it have to do with parallel lines?”)

Polynomial graphs (instead of rational function graphs), or even just parabolas [update: Mary did this!], or even just lines.

Students have derivative graphs on their heads, and they need to come up with a sketch of the original function (for this they should be allowed more than one-word answers).

[1] One thing I worked on in a group with four other people is how to get students to understand inverse trigonometric functions (a topic we collectively decided was challenging for students to wrap their heads around). I blogged about the result of our work here. I used it in class this past year, and although I didn’t use it completely as intended, it did really push home the meaning of what sine and cosine were graphically (the y- and x-coordinates on a unit circle corresponding to a given angle) and then what inverse sine and inverse cosine were graphically (the angles that are corresponding to a given y- or x-coordinate). Check it out!

]]>

This was prompted due from a 30 minute mini-sesh that Justin Aion had around his 180 blogging adventure this year. For those not in the know, a 180 blog is something teachers started doing a couple years ago — posting once a day. (It is called a 180 blog because there are supposed to be — though I definitely don’t have — at least 180 school days in an academic year.)

The difference between regular blogs and 180 blogs are that 180 blogs tend to be a single snippet, every day. Sometimes it is just a photograph. Sometimes it’s just a paragraph. Sometimes it’s a brief reflection. And you know what? *You know what?*

I kept a 180 blog last year too. And I just realized I never mentioned it on this blog, nor did I ever give it a post-mortem or reflection. So tonight, the first evening of TMC14 inspired by a mini-session, that is what I am going to do.

My 180 blog all started because I have an incredible colleague and friend at my school who I know would get along with this community of math teachers online like gangbusters. I wanted to bring him into this world, but it stressed him out too much, and moreso, he didn’t have that much time. I took a stab at ensnaring him by showing him the idea of the 180 blog. It has a low barrier of entry. It involves only 5 minutes a day. And it has a basic structure to it that he could routinize: make a post each day. He agreed! We would both keep a joint 180 blog!

And thus: the very cleverly named ShahKinnell180 blog was born. (Click on the image to be taken there!) [1]

Back to Justin’s TMC talk. He spoke about how he wanted his 180 blog to be centered around *reflectiveness*. And I think that many people do use them for that. However I was 100% sure that reflectiveness *wasn’t* something I was looking for.

Besides getting my colleague/friend involved in this online math teacher world, I think my reasons for wanting to do this are as follows:

- I wanted a little archive of my teaching life. So the only rule I had in making it was that I would post a picture every day, and a few words. Nothing expansive, nothing overwhelming. I had in mind those people who take a photograph of themselves everyday for a year, and then splice them all together, resulting in this whole pastiche of the passage of time? I revel in the fact that I now have this little slice of my teaching life all beautifully laid out. Visual. Chronological. And what I kinda love the most: just like the blog is filled with snapshots of things that happened to me-as-teacher (usually from my classes, though not always), the blog itself is now a snapshot of who I am as a teacher.Although I haven’t done this yet (why not??? well I didn’t even think about writing about it here until after it was done for a whole year! so who knows where my head is at), I would love to send it to my parents. Heck, it’s a great way for non-teachers (wow, this could be awesome for teachers-to-be too!) to see a depiction of what people in our profession do, what we get our kids to do, what we think about, what experiences we have. It’s like a regular blog, but less reading — perfect for skimming and being non-threatening!
- I wanted something to keep me on the lookout for the good. My brain constantly tells me I am not good at what I do. And I am someone who can obsess over what’s not going right and just skip over the juicy deliciousness in front of me. (I was that kid in high school who would take a test, get stuck on one or two questions, and leave saying I knew I did horribly on it… not because I was being modest, but because I would focus totally on what I didn’t know, instead of seeing things in perspective.) All this brings me back to a few years ago when I was a contributor on the “One Good Thing” blog (my posts on that blog are here). If you don’t know about that blog, it is a collaborative blog where teachers just write something good — anything good — that happened. Big or small. The tagline to the blog is: “every day may not be good, but there is one good thing in every day.” That’s some powerful stuff. And you know what? Because I was posting on that blog, I had a shift in my mindset. Even in my worst days, especially in my worst days, I would force myself think back through the day for something good. And heck if I couldn’t find something. And then I started paying more attention to the good that was happening
*when*it was happening (I would think: “Heck yes! I need to blog this!”).I wanted my 180 blog to remind me that I do good things in the classroom. Even when I feel like I’m stagnant, when I’m not innovating, when my kids are lost and I’m at fault… I wanted my 180 blog to keep me on the lookout for things that I should feel proud of. Not every post is a “feel good” post on my 180 blog, but the point is: I was constantly on the lookout for something I would want to post about or an image I wanted to save from the day. - Finally, and probably least important to me, I wanted something to keep me accountable to being a good teacher. This probably sounds a bit weird… but as a regular blogger, I noticed I would get extra enthusiastic about something when I knew I was doing something or creating something and realized I could blog it. When my classroom wasn’t the only audience, and when what we did just disappeared in the temporal aether. Perhaps a 180 blog would help me do the same?

I don’t have any grand pronouncements from the experiment. I definitely didn’t learn anything about teaching from keeping the 180 blog. I am almost certain I will not return to the 180 blog for teaching ideas, or to see how a particular lesson went. I definitely did not become a better teacher because I kept the blog. (At least not in any tangible way.)

But here’s the thing: looking at this experiment on the whole, I am **beyond thrilled I started my 180 blog and kept up with it. **Why? Because when I have moments (be it days, weeks, or even month-long-stretches) when I feel like I’m not doing a good job, I simply can pull up the blog and browse through it and recognize:

*I don’t do the same things every day. I am thoughtful about stuff a bunch of the time. I have pretty great kids who do some pretty great and possibly hilarious things that are worth recording/remembering. *

Which them reminds me: I’m lucky that I get to do what I do. I enjoy thinking about what I get to think about. I really do enjoy working with kids (which definitely needs reminding because… well kids are rarely easy). And that: if this is my job, if this is what I get to do and get paid for it, then things are pretty great.

So as I noted, I was blogging mainly to archive. And archive I did. I have no desire to archive again next year. However I had been thinking at the TMC14 session I was at: is there anything that could get me to do another 180 blog?

And I dawned on the answer. I could create a 180 blog around *one* specific thing I was working on as a teacher. And this 180 blog would force me to stay accountable.

Examples:

- I’m not an expert at deep questioning in the math classroom. So I would be forced to blog about one question I asked, if I had time write about some of the context in which the question was asked, and what happened when I asked it in the math classroom. I would then briefly evaluate whether the questioning was good and/or if there was a better way to have asked the question.
- I am trying to make groupwork the central way kids in my classes learn. So I could write one blogpost each day about how I facilitated some part of groupwork — either in the planning of the class, during the class, or after the class.
- I am trying to be more conscientious about formative assessments. So I vow to have one formative assessment each day in one of my classes (not even all of them! just one!). It doesn’t have to be even a big one… even a 10 second “thumbs up if you get this, thumbs to the side if you’re slightly confused, and thumbs down if you’re totally lost” counts.
- I struggle with wait time. So each day, I vow to record with a timer how many seconds I wait after one question (only one question!), and I post the question and the wait time on the blog.
- I know I’m terrible at “closing” class. I have kids work until the end, we rarely take the time to summarize what we did, the big questions we tackled, the big questions we have lingering. Very often it is: “Eeep, sorry, we’re out of time. Check the course conference for your nightly work. Missyouloveyou!” Okay, maybe not the missyouloveyou part, but you know what I’m talking about. So blogging about the close of one class each day.

I’m not saying I’m going to do anything of these. If I do, it will definitely only be *one* of them. But the idea is that it is targeted about something I want to improve upon, and doing it will hold me accountable.

[1] As a follow up, my colleague who did the 180 blog with me blogged many — but not all — days. But heck if he’s not been so inspired that he’s starting Geogebrart, his own blog about making art with geogebra which has been knocking my socks off this summer. Once you peruse his entries on our 180 blog and you peruse his new Geogebrart blog, you probably understand why I feel lucky beyond belief to get to work with this guy!

]]>

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!

]]>

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.

]]>

I have them come up with a prospectus and I individually talk with kids about their proposed project and timeline for completion. Then when they get started and start envisioning a final product, they are asked to write a description of the final product out clearly, and come up with a rubric for grading that product. They are also asked to make a 20-25 minute presentation to their classmates, their parents (if they choose to invite them), math teachers, and administrators. This year, they wanted to give their presentations during senior thesis week, which means that lots of their friends could come to their talks.

And they have been! In the past week, students have given their talks and I have been way impressed by them. Honestly they’ve been more independent than in years’s past, so I was unsure of whether they were putting together a solid final project or not. They did.

Without further ado:

M.C.

**Title: Mathematical Change We Can Believe In**

Description: This presentation shows how one region can be manipulated to form something more interesting, a process called Transformation of Axes. The 2D and 3D analogues, use of rectangular and rounded shapes, and proofs of the properties of transformations abound in this exciting journey through the wonders of the world of multiple (MANY) variables.

B.W.

**Title: Pursuit Curves: The Ultimate Game of Tag**

Description: Pursuit curves are the paths formed when one point chases another point. In this program, we will be looking at the mathematical explanations of pursuit curves, and then using a computer program I have built to model a few.

J.B.

**Title: What’s Our Vector, Victor**

Description: This will be an investigation into the history, origins, and evolution of vectors, their analysis, and notation.

I.E.

**Title: Economists working with Models: Understanding the Utility Function**

Description: Firstly, we will gain a foundational understanding of economics as a discipline. Secondly we will discuss the utility function and the questions which it raises.

C.D.

**Title: From Chemistry to Calculus: a study of gas laws**

Description: For my project I have constructed a “textbook” that analyzes the idael and real gas law through the lens of multivariable calculus. In my “textbook” I compare and constrast these two laws by means of graphical and derivative analysis.

E.F.

**Title: Knot Theory**

Description: Knots are everywhere around us, from how we tie our shoes to how the proteins in our body wind themselves up. My presentation will give an overview of their place not only in the “real world,” but also the world of classroom math and calculus.

]]>

More than anything, I have enjoyed watching the editors become independent leaders, organizing something involving so many people and moving parts, and presenting their creation to administrators, math teachers, science teachers, computer science teachers, and other students. I feel like I’m coming to understand the niche I play in my school: I find ways to make math exist outside of the formal curriculum for kids who want to get more involved. *Intersections* is one of those spaces — both for editors and for those students who submitted.

If you want to check out this year’s issue, please click on the cover photo (designed by a student) below and it will take you to the website.

(You can also click here.)

More than anything, if you have the time, just click around and see what cool things you discover!

Although it’s a lot of work, if you have any thoughts about starting something like this at your school, I highly recommend it.

]]>

So even though I currently hate it, here is this year’s senior letter.

It came packaged with their “who I am” sheet that they wrote about themselves on the first day of class, and two cards I had printed.

]]>

And luckily, I have the opportunity to try something new. **Next year, I will be giving up Calculus to teach an Advanced Geometry course for the first time. In fact, it’s the first time I’ll have ever taught geometry at all.**

When I first began teaching, I was scared of geometry. Partly because as a student in high school, I found geometry to be uninteresting. It certainly didn’t have the elegance of algebra, at least the way I was taught it. Partly because I realized in that course — more than any other course — you as a teacher really have to focus on hard things. If you want kids to be able to do a proof of any kind (two-column or not), **you are really teaching intuition building and connection making.** Which is tough, and daunting for any new teacher, and this is why I recoiled at the thought.

Right now, I am not anywhere about how to teach this course. And in fact, I’m only teaching one section and the other teacher is teaching three sections. But he’s very open to really revitalizing the course. So now we’re in exciting territory. Before I go bananas on scouring everything out there, I thought I’d crowdsource.

For any of you geometry teachers out there, if you have time to answer one or two (or all!) of these questions in the comments, I’d be ever so grateful!

1) What are your favorite geometry teaching resources — both online and offline? I’m talking books, websites, applets, manipulatives, whaever?

2) What are your favorite math teacher blogs that focus on geometry?

3) Is there a lesson you absolutely could not imagine teaching Geometry without?

4) Do you teach the course with a connective thread? Like: We are studying *space* and the properties inherent in space as we *build space*? Or: We are studying exactitude –and in particular, how we define mathematical entities so they yield uniquely understandable creatures? Or: We are studying “measurement” (in the vein of Paul Lockhart’s book).

5) I’m concerned that our kids lose a lot of their Algebra I skills when they take geometry. The other teacher and I have talked about putting coordinate geometry front and center from the beginning to help with this. Do y’all do anything else that helps keep their algebraic skills sharp, and maybe even push them forward?

6) Anything else? Problem solving? Sangakus? Geogebra use? Things you throw out because you feel strongly it’s only taught because it’s always been taught?* *Incorporation of Euclid’s *Elements* or math history? Graphic-design-y projects? Math art?

**UPDATE:** WOW, everyone, thank you so much for your resources and advice and for taking the time to type out so much great stuff. Now I’m genuinely THRILLED and CHOMPING AT THE BIT to get started re-learning geometry (and then teaching it). I am going to sort through things this summer!

]]>

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.)

]]>

- Last quarter students scoured the web and did 5 different mini-explorations which exposed them to all the neat math that exists outside of our standard curriculum. This quarter students will be doing
*up to two*more in-depth explorations. - Because I don’t want this to be seen as busy work, doing “Explore Mathematics!: Part II” is going to be
*completely optional.*I was glad to read that almost every kid who did the five mini-explorations last quarter didn’t end up finding it busy work, but I suspect doing it a second time would feel tedious. - To have some sort of incentive for those who do it, I am going to make each of the two explorations worth 12 points. These explorations will count as a mini-assessment (normal assessments are around 50 points). This is useful for kids because our fourth quarter only has 18 days of instructional time (seriously) — so there are only two major assessments and one minor assessment scheduled. Doing these explorations can act as a way to get another mini-assessment grade in there, that will be low-stress, high-reward. [1]
- I’m not framing it around the grade boost it will likely provide, but around the fact that it’s an opportunity to do some awesome math explorations, for anyone who wishes to do so.
- It is still pretty open-ended, but I’m now looking for students to write something to
*get others*to see what they find interesting/intriguing/awesome about something.

Here’s the document I just emailed my kids:

Here it is in .docx form in case you want to modify it.

[1] Yes, I do SBG with my calculus kids. Yes, I know how ridiculous this sounds, me playing the “point game.” I almost wanted to make it so that there was no external reward, but our kids are so busy with so many things that I know even a little incentive will go a long way. I’ve been at my school long enough, and know our kids well enough, to know this is doomed to failure without a little external reward.

]]>

It was a cover letter that gave a link to a really simple website, and on that website was an educational philosophy, a few sample tests, and some student work. Although it was pretty basic, what I liked was that on that simple site I got a *much* better sense of who this candidate was. I loved the idea. And I decided then and there that I would create my own teaching portfolio online that would capture who I was as a teacher.

This past summer, I did it.

To be clear: this isn’t a *reflective* teacher portfolio. It’s a *descriptive* teacher portfolio. It is something that I put together — a mishmash of snippets — that together hopefully gives a solid sense of who I am, what I do, what I believe in. I think calling it a **visual teaching resume **or a** wunderkammer** best describes it. (Click on the image to go to the site.)

There are a few missing things that I would like to add to this site at some oint:

- I would like to add everyday samples of student work. Not projects. Just everyday stuffs.
- I would like to add a section about the two week history of science course I designed and implemented with another teacher this year. (See Days 80-87 on my 180 blog for more.)
- I would like to add a section about the “Explore Math” project (more info here and here) I did in Precalculus this year.
- I would like to finish the student quotation page. I actually have quotations typed for a number of previous years, but I do not have more recent years ready.

It was pretty simple to make (I used the free website creator weebly) and I hope if I ever were to go on the job market, it would catch the eyes of whoever had the giant stack of cover letters and resumes in front of them. I wasn’t really going to make a post about my visual resume, or share it with anyone, because I thought: *who would care?*

But heck: maybe someone out there is going on the job market and thinks the idea is worth replicating? So I decided to post.

]]>

**The question in the survey:**

The “Explore Math” project is something I’ve never done before. I explained my reasoning behind it — which is I wanted to encourage you to see that there is so much more than our curriculum covers, and let you just have fun looking at math stuff outside of our curriculum… and get some easy credit for it (almost everyone is getting full credit for the first batch of things I’ve seen). However, as a teacher, I know something like this could easily be seen as busy work, and that was my big concern — that it would feel like a chore rather than something you actually want to do.

This is me laying my cards on the table. If I came to you in the student center and told you this and asked you for your thoughts, what would you say?

**Every Student Response In Entirety:**

I really liked the Explore Math project and I definitely would say it was an overall success. I loved how many options we were given for what we could do, and the fact that you gave us the options was great because otherwise it can feel like you are just trying to desperately research and find a topic to write about. My Explore Math topics I thought were extremely interesting, and it was cool to even connect some to the stuff we were learning in class. It was a lot of writing, which is something foreign for math classes, and also made it kind of difficult to grasp exactly how to format what we were writing (five page essays for each topic?). One other thing that was a little stress-inducing was the deadline and I know it was for a problem for most people that it often happens that when there are multiple assignments due on one day, students leave them all and do them in bulk. Because of this, having the deadline of the first three due in February was definitely helpful. Overall, I really loved the assignment.

I really liked this project! I found a lot of things about math that I would have never known about if we weren’t assigned this project. I learned new formulas, new (very addictive games), great youtube channels and informative popular articles. I found an entirely new community online that I did not know existed.

At first I expected it to feel like a bit of a chore but when I actually sat down and did it, it was pretty fun. I think it was great that there were multiple ways you were allowed to “explore math.” I also thought it was amazing I could play around with the project a little bit to find areas of math that are aligned with my personal interests. Being able to think about how math affects our society, in a math class, was an amazing interdisciplinary activity. I think it’s good that not every option was a math puzzle — that would have felt constrictive.

I would say as long as the students are innovative, interested and patient people the project sounds wonderful. The student, if very interest in math, should be encouraged to further their mathematical understanding, and find means in which math is even more interesting to them as it was prior. Emphasizing the point that one (the student) does not need to seek the more difficult problem or most tedious theorem is also very helpful, as the student will be encouraged to explore areas of math in which really interests them.

I would say that I absolutely love the explore math project. I have always been a person who enjoyed math that connected with the world. Being in a classroom memorizing formulas was never my interest and I was psyched when you announced the project. I think that this project can be very helpful in putting math on the global scale for students who only see it as a class in a school. This opens their eyes to new heights math can taken and how much math actually helps outside of the classroom.

I agree it felt like busy work some. I find it weird that something that’s supposed to be us having fun exploring math had a grade and time constraint attached to it. That’s one thing I didn’t like.

All I have to say is that this was not busy work; in fact it was productive and learning work. I found this to be incredibly intensive and interesting, and it broadened my horizons of the understandings of applied mathematics and sciences, and introduced me to things that I had previously trembled [at] before, like string theory, for instance. I thought this was a great project and a simple and easy way to get us thinking in a mathematical mindset, and I am definitely reaping the benefits from it, because I have come away with much more knowledge about certain aspects of math that I had previously not known. I really wouldn’t know what to change because I liked these individual explorations so much and they intrigued me so much. Thank you for giving a projected that I was thoroughly interested in, seriously!

For someone who is very interested in math in and out of the classroom, I am generally engaged with math concepts that are not a part of our curriculum. Thus, this was a good experience for me in that I was able to get credit for simply enjoying and exploring math; it also perhaps pushed me a little bit to go further than I normally would in exploring mathematical concepts online. However, for students who don’t love math outside of the classroom, I could definitely see how this might have seemed like busy-work. If you don’t genuinely enjoy math, then writing a lot about it and research about it is going to be cumbersome, but if you do, it’s enjoyable.

I really liked doing the explore math assignment. I liked that you were giving us an outlet for us to not just do the math that needs to be done in order to complete the class. This assignment allowed me, personally, to dive deeper into how math can be applied to the world and that math is actually occurring all the time. Also, I remember not really understand[ing] infinite series and then I did an explore math with infinite series that really helped me because it was a visual representation that really clicked with me.

I think that initially I thought the project might just be busy work and I didn’t really understand what we were expected to be doing. Once I read over the assignment and saw the scope of the projects we were allowed to do, I was much more interested and saw the project completely differently. I think that it is important to highlight, when giving the assignment, how broad a range of options you have when doing this, and that there are so many math projects that relate to everyday life that could be interesting if you just *think* about it, rather than relying on the assignment sheet completely to guide you.

Personally, I have enjoyed what I have done so far. Just recently, I voiced my concerns about the state of math in America and was able to comprehensive research about the bitcoin that I would not have done on my own. That being said, some of this has seemed like busy work and stuff “I just have to do for credit.” Since it seems like you genuinely want us to enjoy the project, it might be made better by making it extra credit. That way, we could be able to explore as much as we want without worrying about our grade.

I had a really awesome time doing my Explore Math assignments, but the one thing you could do to make it less busy work is make it 3 different assignments, rather than 5, and make them a little more in depth, and more interesting in that regard. I think that if the students only had to do 3, they could expand more on what they were interested in.

I really like the idea, but for me personally, it turned into busy work. Not because I find it boring but because I have so much other work that it gets pushed back towards the end of my load. I would like to spend more time on them, so possibly have it on top of the nightly work for math, designate a night specifically to explore math.

This is practically the farthest thing from busywork we can do! Repetitive problems often seem like busywork. Practice is always good, but once you have something down, it can be quite annoying to practice it over and over again. Sometimes i feel that way about homework, but with this project we’re choosing any math-y thing that interests us! We have a lot of freedom, and hopefully it piques an interest in math outside of the curriculum. This project is great, personally, I wish I had taken more time with it. As long as you don’t procrastinate too badly with it, I don’t see how this project could be a chore, unless you claim to hate math.

I LOVED this project, and I wish we got to do more things like this throughout the year. (I know we can do things like this whenever we want, but it’s really nice to get some recognition and the chance to formally share your math ideas with others.) As a side note, this project was also interesting to be doing while looking at colleges for the first time. I know that sounds like a really strange thing to say, but getting to enjoy math in new contexts, such as music theory, has given me new ideas of things I would like to pursue and take classes [on] while I am at college because we don’t always get to learn about things like this on a daily basis in high school.

I do admit that I wasn’t very enthusiastic at the start of the project, but as soon as I started I completely changed my mind. Most of the work that I did was stuff I had never done before and might never do again. I was genuinely interested in what I was doing, and it was great to be able to choose what I focused on instead of being told what to look at.

I understand why you assigned this project, and I think it is very important to see the relevance math has in the world. This breathes life into the abstract “why are we learning this” type that doesn’t appear to have anything to do with life outside the classroom. However the problems with this assignment are that I didn’t know what I was searching for. When I found the Sloane’s Gap video and paper I felt like I struck gold after seemingly endless mining. However the mining part is very un-exciting. Not un-exciting enough to undo the excitement of finding the cool stuff, but it’s not very encouraging either. I wouldn’t want this assignment to turn into a chose 5 of these pre-determined projects because that wouldn’t make anyone feel like anyone feel like they’re venturing outside the classroom. I’m not really sure what I would do to change this assignment, but I think it really is a good idea that with some refinement could become a really dynamic way to get into math. I think keeping it low pressure and “easy credit” is the way to go because stress + ambiguity about an assignment is a terrible combination that would only end in resentment from your students, and students not enjoying their work.

Honestly, I had quite a bit of fun with the “Explore Math” project as I saw many cool analogies of real-world applications of math. For example, one of my five “research topics” was the probability and randomly guessing on every SAT multiple choice question. I learned that the probability is horrifyingly low — I already knew this, but not to such an extent. Furthermore, I saw some very cool analogies in this SAT topic; for instance, if a computer were to take the SAT 1 million times a day, for five billion years, the chance of any of the SATs resulting in a perfect score on just the math section would be about 0.0001%. Crazy, I know!

]]>

At the end, he said something powerful. The first thing one needs to do to when leading a purposeful life is to say what it is that you want to do. Articulate it aloud. And that is scary. Making it public so you can hear yourself say it, but also so someone else can hear you say it. So it becomes *real* instead of this thing that bounces around in your head but never gets out. And so at the end, he told everyone to be quiet, and he was going to say something he wanted to do, and then afterwards there should be silence… and when anyone else wanted to say something they wanted to do — something they would declare out loud — they should stand up and say it, and then remain standing. This was an open invitation to the students in these honors societies, but also to the parents and teachers there as well.

The speaker said: “I want to change the world.”

Silence.

A little more silence where everyone looked around and felt uncomfortable.

Then a student — one courageous student — got up and said something. And remained standing.

And then another. And another.

The head of the upper school said something. Then more students. Then a parent. Then me. Then another math teacher. Then more students.

At the end, every student made a declaration, and a few adults too. *It is scary*. But it also showed me how much courage our kids have. Their declarations ranged from showing others that girls can do math and science *to* spreading love *to* making people laugh *to* promoting peace *to* inventing something *to* becoming a biochemist *to* making a mark on the world. Big things and small things, lofty things and concrete things, but all things that share with the room a sense of self and a sense of purpose.

I loved watching this.

I also loved and hated how hard it was for me to come up with my thing. My purpose in life. I said:

*I want to make it so that kids see math as an artistic and creative **endeavor*.

And I meant it. Because you know what has been bouncing around in *my* head that I have been having trouble articulating? I am now pretty good at coming up with deep and conceptual approaches to mathematical ideas. And I’m okay at promoting mathematical communication. And I’m transitioning to having kids do groupwork all the time, to learn from each other — so I am not the sole mathematical authority in the room.

But all of that said: **I don’t think I teach math in a way to shows how it is an art form, how deeply creativity and mathematics are intertwined.** And I know that this is one of my charges as a teacher moving forward. It’s going to be an uphill challenge, and one that will likely take me many years to wrap my head around. The hurdles are significant. Having a set non-problem-solving-based curriculum which doesn’t allow time for much mathematical “play,” nor for the inclusion of rich problems with multiple entry points, is the largest hurdle. But there must be ways — activities or units here and there — that can illuminate the artistry and creativity of doing and discovering mathematics. And I want to be involved in finding ways for this to happen. Yes, this happens at math circles. Yes, this happens at math clubs. Yes, this happens at summer math programs. That’s where the love and excitement and understanding of the *beauty* of mathematics unfolds for many students. But I want to find a way for this to happen in a normal classroom, with normal students, with the normal constraints. That (one of) my purposes.

]]>

Tonight I wanted to see if I could re-derive it like I did before — and lo and behold I did. I’m curious if any of you have done it the way I did it, or if there are other ways you’ve learned to approach this problem. (There is a student who I had last year who created this amazing 3-d version of this using the edges of a cube and some string. I love the idea of asking — for this 3-d figure — what *surface* is generated by the intersections of these strings.)

We start out by having these lines which form a family of curves. But of course we’re not graphing *all *the lines. If we were, we’d get something more dense like this.

The main idea of what I’m going to do to find that curve… I’m going to pick two of those lines which are *infinitely close to each other *and find their point of intersection. That point of intersection will lie on the curve. (That’s the big insight in this solution.) But I’m not going to pick two *specific* lines — but instead keep things as general as possible. Thus when I find that point of intersection for those two lines, it will give me *all* the points of intersection for *all* the lines.

Watch.

First we pick two arbitrary lines.

We’ll have one line move down on the y-axis units (and thus over on the x-axis units). And the second line will be moved down on the y-axis just a tiny bit more (down an additional units). Yes, we are going to have that tiny bit, that , eventually go to zero.

The two lines we have are:

A little bit of algebra is needed to find the point of intersection. Setting the y-values equal:

And then doing some basic algebra:

Now solving for we get:

So the point of intersection is:

Here’s the kicker… Remember we wanted the two lines to be *infinitely *close together, right? So that means that we want to go to zero. Thus, our point of intersection of these *infinitely* close lines will be:

or .

Beautiful! And recall that we picked the lines arbitrarily. By varying and plotting , we can get any two lines on our doodle.

But I want an equation.

Simple. We know that . Thus .*

Since , we have

Let’s graph it to check.

Huzzah!!! And we’re done!

I wonder if I can do something similar with this cardioid:

I think I must (for funsies) do some investigation of “envelopes” this summer. I mean, Tina at *Drawing on Math* even introduces conics with these envelopes!

**An extension for you. Do something with this 3d string-art.**

*Of course you might be wondering why I don’t say . Since is between 0 and 1, we know that must be positive.

]]>