# An Animal Problem

One of our math club leaders gave out this problem as the final problem of math club for the year. I had never seen it before, and after she handed it out, a number of math teachers were in a tizzy about finding the solution. So instead of planning for classes, we enjoyed working on this problem. But we got it! HUZZAH!

Here’s the problem:
In how many ways, counting ties, can eight horses cross the finishing line?

So we fully understand the problem, let me list all possibilities for three horses: Adam, Beatrice, and Candy. No, wait, those are better names for unicorns:

1st: A   2nd: B   3rd: C

1st: B   2nd: A   3rd: C

1st: A   2nd: C   3rd: B

1st: C   2nd: A   3rd: B

1st: B   2nd: C   3rd: A

1st: C   2nd: B   3rd: A

1st: AB (tie)   2nd: C

1st: AC (tie)   2nd: B

1st: BC (tie)    2nd: A

1st: A   2nd: BC (tie)

1st: B   2nd: AC (tie)

1st: C   2nd: BC (tie)

1st: ABC (tie)

That comes out to 13 different ways these horses unicorns can finish the race.

That’s the answer for 3 unicorns. What’s the answer for 8 unicorns?

(FYI: If you want to know if you’re on the right track… I have 541 for 5 unicorns…)

# Some Random Things I Have Liked

## The Concept of Signed Areas

In calculus, after first introducing the concept of signed areas, I came up with the “backwards problem” which really tested what kids understood. (This was before we did any integration using calculus… I always teach integration of definite integrals first with things they draw and calculate using geometry, and then things they do using the antiderivatives.)

I made this last year, so apologies if I posted it last year too.

[.d0cx]

Some nice discussions/ideas came up. Two in particular:

(1) One student said that for the first problem, any line that goes through (-1.5,-1) would have worked. I kicking myself for not following that claim up with a good investigation.

(2) For all problems, only a couple kids did the easy way out… most didn’t even think of it… Take the total signed area and divide it over the region being integrated… That gives you the height of a horizontal line that would work. (For example, for the third problem, the line $y=\frac{2\pi+4}{7}$ would have worked.) If I taught the average value of a function in my class, I wouldn’t need to do much work. Because they would have already discovered how to find the average value of a function. And what’s nice is that it was the “shortcut”/”lazy” way to answer these questions. So being lazy but clever has tons of perks!

## Motivating that an antiderivative actually gives you a signed area

I have shown this to my class for the past couple years. It makes sense to some of them, but I lose some of them along the way. I am thinking if I have them copy the “proof” down, and then explain in their own words (a) what the area function does and (b) what is going on in each step of the “proof,” it might work better. But at least I have an elegant way to explain why the antiderivative has anything to do with the area under a curve.

Note: After showing them the area function, I shade in the region between $x=3$ and $x=4.5$ and ask them what the area of that bit is. If they understand the area function, they answer $F(4.5)-F(3)$. If they don’t, they answer “uhhhhhh (drool).” What’s good about this is that I say, in a handwaving way, that is why when we evaluate a definite integral, we evaluate the antiderivative at the top limit of integration, and then subtract off the antiderivative at the bottom limit of integration. Because you’re taking the bigger piece and subtracting off the smaller piece. It’s handwaving, but good enough.

## Polynomial Functions

In Precalculus, I’m trying to (but being less consistent) have kids investigate key questions on a topic before we formal delve into it. To let them discover some of the basic ideas on their own, being sort of guided there. This is a packet that I used before we started talking formally about polynomials. It, honestly, isn’t amazing. But it does do a few nice things.

[.docx]

Here are the benefits:

• The first question gets kids to remember/discover end behavior changes fundamentally based on even or odd powers. It also shows them that there is a difference between $x^2$ and $x^4$… the higher the degree, the more the polynomial likes to hang around the x-axis…
• The second question just has them list everything, whether it is significant seeming or not. What’s nice is that by the time we’re done with the unit, they will have a really deep understanding of this polynomial. But having them list what they know to start out with is fun, because we can go back and say “aww, shucks, at the beggining you were such neophytes!”
• It teaches kids the idea of a sign analysis without explaining it to them. They sort of figure it out on their own. (Though we do come together as a class to talk through that idea, because that technique is so fundamental to so much.)
• They discover the mean value theorem on their own. (Note: You can’t talk through the mean value theorem problem without talking about continuity and the fact that polynomials are continuous everywhere.)

## The Backwards Polynomial Puzzle

As you probably know, I really like backwards questions. I did this one after we did  So I was proud that without too much help, many of my kids were really digging into finding the equations, knowing what they know about polynomials. A few years ago, I would have done this by teaching a procedure, albeit one motivated by kids. Now I’m letting them do all the heavy lifting, and I’m just nudging here and there. I know this is nothing special, but this course is new to me, so I’m just a baby at figuring out how to teach this stuff.

[.docx]

# Green’s Theorem and Polygons

Two nights ago, I assigned my multivariable calculus class a problem from our textbook (Anton, Section 15.4, Problem 38). Even though I’ve stopped using Anton for my non-AP Calculus class, I have found that Anton does a good job treating the multivariable calculus material. I think the problems are quite nice.

Anyway, the problem was in the section on Green’s theorem, and stated:

(a) Let $C$ be the line segment from a point $(a,b)$ to a point $(c,d).$ Show that:

$\int_C -y\text{ }dx+x\text{ }dy=ad-bc$

(b) Use the result in part (a) to show that the area $A$ of a triangle with successive vertices $(x_1,y_1),\text{ }(x_2,y_2),$ and $(x_3,y_3)$ going counterclockwise is:

$A=\frac{1}{2}[(x_1y_2-x_2y_1)+(x_2y_3-x_3y_2)+(x_3y_1-x_1y_3)]$

(c) Find a formula for the area of a polygon with successive vertices $(x_1,y_1),\text{ }(x_2,y_2),...,(x_n,y_n)$ going counter-clockwise.

Today we started talking about our solutions. We all were fine with part (a). But part (b) was the exciting part, because of the variation in approaches. We had five different ways we were able to get the area of the triangle.

• There was the expected way, which one student got using part (a). This was the way the book intended the students to solve the problem — and I checked using the solution manual to confirm this. What was awesome was that even though we as a class understood the algebra behind this answer, a student still asked for a conceptualgeometric understanding of what the heck that line integral really meant. I knew the answer, but I left it as an exercise for the class to think about. So we’re not done with this problem.

• There was a way where a student made a drawing of an arbitrary triangle and then used three line integrals of the form $\int_C y\text{ }dx$ to solve it. In essence, this student was taking the area of a large trapezoid (calculated by using a line integral) and subtracting out the area of two smaller trapezoids (again calculated by using line integrals). Another student astutely pointed out that even though we had an arbitrary triangle, the way we set up the integral was based on the way we drew the triangle — and to be general, we’d have to draw all possibilities. You don’t need to understand precisely what this means — because I know I”m not being clear. The point is, we had a short discussion about what would need to be done to actually have a rigorous proof.

• There was a way where a student translated the triangle so that the three vertices weren’t $(x_1,y_1),\text{ }(x_2,y_2),\text{ }(x_3,y_3)$ anymore… but instead $(0,0),\text{ }(x_2-x_1,y_2-y_1),\text{ }(x_3-x_1,y_3-y_1)$. Then he used something we proved earlier, that the area of a triangle defined by the origin and two points would involve a simple determinant (divided by 2). And when he did this, he got the right answer.

• Another two students drew the triangle, put it in a rectangle, and then calculated the area of the triangle by breaking up the rectangle into pieces and subtracting out all parts of the rectangle that weren’t in the triangle. A simple geometric method.

• My solution involved noticing that $\frac{1}{2}(ad-bc)$ is the area of a triangle with vertices $(0,0),\text{ }(a,b),\text{ }(c,d)$. And so I constructed a solution where a triangle is the sum of the areas of two larger triangles, but then with subtracting out another triangle.

The point of this isn’t to share with you the solutions themselves, or how to solve the problem. The point is to say: I really liked this problem because it generated so many different approaches. We ended up spending pretty much the whole period discussing it and it’s varied forms (when I had only planned 10 or 15 minutes for it). I liked how these kids made a connection between a previous problem we had solved (#28) and used that to undergird their conceptual understanding. I loved how these approaches gave rise to some awesome questions — including “what the heck is the physical interpretation of that line integral in part (a)?” In fact, at the end of class, we were drawing on paper, tearing areas apart, trying to make sense of that line integral. All because a student suggested that’s what we do. (Again, I have made sense of it… but I wanted the kids to go through the sense making process themselves… their weekend work is to understanding the meaning of this line integral.)

I don’t know the real point of posting this — except that I wanted to archive this unexpectedly rich problem. Because it’s not that it is algebraically intensive (though some approaches did get algebraically intensive). Rather, it’s because it is conceptually deep.

# Ellipses

What are the ways we can generate ellipses?

We’ve been working with ellipses. I have talked about some of these this year. Others I haven’t. But I like this list for future reference.

• The polar equation $r=\frac{1}{1-k\cos\theta}$ gives rise to ellipses if $0<|k|<1$
• An ellipse arises out of squashing or stretching a unit circle horizontally or vertically (or both)
which means that algebraically…the rectangular equation is $(\frac{x}{\square})^2+(\frac{y}{\triangle})^2=1$
• An ellipse arises out of looking at a circle straight on (so it looks like a circle) and then tilting that circle.
• Ellipses can be created by taking a cone (or cylinder) and slicing it at a variety of anglesThis is equivalent to shining a flashlight at a wall at an angle:
• The set of points from two points (called foci) which have a set sum of distances from these two pointsand for a cool video illustrating this (alongside the reflective property of ellipses):
• Drop a planet in space near a massive object, and give it an initial push (velocity)
[
not drawn to scale, obvi.]

# CUPCAKES! ALGEBRA II! BEST ACTIVITY EVAR!!!

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.

# Introducing Conic Sections

## One Ring Equation to Rule Them All

On Monday, in Precalculus, I am starting conic sections.

I’ve made the decision to introduce conics through polar equations. This is totally backwards to the way that most people do it. Our textbook even sticks the polar version of conics at the end of the chapter of conics. However, I think it will be more powerful to do it this way.

You see, we just finished a unit on polar a hot minute ago, and I want to capitalize on that so students can draw connections between polar and conics. Additionally, we did a project on families of curves.

And in case you didn’t know this, the polar equation $r=\frac{1}{1-k\cos(\theta)}$[1] give rise to all the conic sections by varying the parameter $k$. In other words, you can see all conics as a family of curves with a varying parameter.

## Noticing and Wondering

For you, I decided to take a few seconds to plot them on geogebra:

Instead of teaching them anything, on the first day I’m going to have them work in pairs. The plan:

• Have students get in pairs
• Have students use desmos.com to explore the family of curves using a slider to change $k$
• Have each pair notice things and wonder things about what they see as they change the parameter — and record their observations on a google doc (this sample doc is set so you can view it)
• After each pair is done noticing and wondering about $r=\frac{1}{1-k\cos(\theta)}$, I’m going to have kids spend a few minutes noticing and wondering about $r=\frac{1}{1-k\sin(\theta)}$

I’m handing out this worksheet [.docx] to get us to do these things.

I don’t know how long this will take. Maybe 20 minutes, maybe 50 minutes? It really depends on how into the noticing and wondering. I’m a little uncertain where to go next with this — how to share the noticing and wondering… Each pair is going to have a group letter (A, B, C, … H, I, J). So I might have them spend a few minutes looking at the documents of those that preceed and follow their group (e.g. C will look at group B and D’s group’s observation).

Then I’m not sure how to wrap this all up. I think what I may do is leave it there, and not do a whole-class share. But after that, I’ll collate their noticings and wonderings, and as we introduce new things, I’ll tie them back to statements from these documents. For example, if a group notices “when $k$ is a huge number, the graph looks like two intersecting lines,” when we get to hyperbolas, I’ll start the day reading that statement, and after delving into them, we’ll return back to that statement to see how it relates to the algebraic work we did. Or if a group wonders, “When $k>>1$, I wonder if there’s a relationship between the value of $k$ and the angle between the two intersecting lines,” I could build that into the questions in the worksheets I’m writing for this unit.

Or maybe I’ll do nothing with them. Just the mere act of exploring, and coming with the conclusion that one family of polar curves can general four distinct general shapes (circle, ellipse, parabola, hyperbola) is good enough for me. Just paying attention to what’s happening to the graph, and learning to ask questions about what’s happening, that’s a skill in itself I should be content with cultivating.

(I should point out that I have rarely used the notice/wonder thing… so this isn’t fluent for my kids.)

## Where I go from Here

From this activity, though, we are definitely going to talk about how we see four qualitatively distinct shapes, and we’ll name them:

circle, ellipse, parabola, hyperbola

And since we aren’t relying too much on the textbook, I am going to want them to make a schematic chart to organize what they’ve uncovered through observation.

And then we go on to the icky algebra, identifying various polar equations as different types of conics, and then eventually converting polar to rectangular form. (But in our polar unit, they already were asked to convert equations like $r=\frac{1}{1-3\cos(\theta)}$ to rectangular coordinates, so this will be a bit of review.)

[.docx]

This is all subject to change, obviously.

And then… then… when students have qualitatively understood conics as all emerging through one equation… when students see that the conics all gently slide into and out of each other as a single parameter changes… when students see two things they already know (parabolas and circles) and see two things they don’t know (ovals, weird pairs of curves that look almost like crossing lines)… then we’ve motivated this luscious mathematical journey we’re going to embark on. [2]

Then we can get to the rectangular form for conics and see how they come to look so similar, and why the differences arises… Why certain things open up and certain things open sideways… all the traditional stuff… But motivated by this untraditional beginning.

UPDATE: It’s Sunday, before I try this on Monday. I decided I want kids to understand why hyperbolas have asymptotic behavior from the polar form (and why ellipses don’t!). So I made this sheet [scribd online, .docx] which I think will get them to discover some algebraic connections behind some of the visual things they will have uncovered from their noticing and wondering.

UPDATE 2: My kids did their noticing and wondering. Because they were comfortable with Desmos from our polar unit, it went really smoothly. It took them about 25 or 30 minutes before they had exhausted all their observations. I walked around and pressed a few on some of the things they were doing/saying without giving anything away… Like if they said when $k=1$ that the graph is a parabola, I had them graph $k=0.99$ to confirm that it wasn’t… and then they had to zoom out to see it truly was an ellipse. Or if they said that very high values of $k$ give rise to intersecting lines, I would ask them to record in their noticings the point of intersection was (so they’d zoom in and see there was no point of intersection!). All my kids’s observations are recorded here in their google docs: Group A, B, C, D, E, F, G, H

[1] And, technically, $r=\frac{1}{1-k\sin(\theta)}$

[2] I’m in the middle of Paul Lockhart’s Measurement and what’s amazing is I was reading it on the subway to school, and today of all days, I started his introduction to conics. He introduces it in a stunning way, through projections, and showed me one of the most elegant proofs I’ve seen dealing with ellipses and why the sum of the distances from the foci to the ellipse is a constant. I wish I could move away from our traditional curriculum to work as qualitatively and beautifully as he has done.

# Dan Meyer says JUMP and I shout HOW HIGH?

On a recent blog post, Dan Meyer professed his love for me. He did it in his own way, through his sweet dulcet tones, declaring me a reality TV host and a Vegas lounge act [1]. LOVE!

He was lauding a worksheet… well, a single part of a worksheet… I had created. You see, I’m teaching Precalculus for the first time this year, and so I have the pleasure of having these thoughts on a daily basis:

What the heck are we teaching this for? IS THERE A REASON WE HAVE KIDS LEARN [fill in the blank]? WHAT’S THE BIG IDEA UNDERNEATH ALL OF THIS STUFF?

[Btdubs, I love teaching a new class because these are the best questions EVAR to keep me interested and to keep my brain whirring!]

And I went through those questions when teaching trig identities. And so I concluded the idea of identities is that two expressions that look different are truly equal… and they all derive from a simple set of ratios from a triangle in a unit circle. Equivalent expressions. When things are the same, when things are different…

So my thought was to make graphing central to trig identities. For the first couple days, every time kids were asked to show an identity was true, they were asked to first actually graph both sides of the equal sign to show they truly are equivalent. (And half the time, they weren’t!)

[If you want, my .doc for the worksheet above is here... and the next worksheet with problems to work on is here in .doc form too.]

To be honest, I still have some thoughts about trig identities that I need to sort out. I am still not totally satisfied with my “big idea.” I still have the “so what” banging around in my brain when thinking about equivalent expressions. I have come to the conclusion that the notion of “proving trig identities are true” is not really a good way to talk about proof. There’s also the really interesting discussion which I only slightly touched upon in class: “Are $1$ and $\frac{x}{x}$ equivalent expressions?” I have something pulling me in that direction too, saying that must be part of the “big idea” but haven’t quite been able to incorporate.

If I were asked right now,  gun to my head to answer, I think I suppose I’d argue that “big idea” that a teacher can get out of trig identities are teaching trial and error, the development of mathematical intuition (and the articulation of that intuition), and the idea of failure and trying over (productive frustration). Because I think if these trig identities are approached like strange mathematical puzzles, they can teach some very concrete problem solving strategies. (To be clear, I did not approach them like strange mathematical puzzles this year.) Now the question is: how do you design a unit that gets at these mathematical outcomes? And how do you assess if a student has achieved those? (Or is truly being able to verify the identity the fundamental thing we want to assess?) [2]

[1] Except I got my teaching contact for next year, and I’ll be making more than the tops of those professions combined. YEAH TEACHING! #rollinginthedough

[2] Different ideas I remembered from a conversation on Twitter… Teachers have contests where they see how many different ways a student/group/class can verify an identity. And another idea was having students make charts where they have an initial expression, and they draw arrows with all the possible possibilities of where to go next, and so forth, until you have a spider web… What’s nice about that is that even if students don’t get to the answer, they have morphed the original expression into a number of equivalent and weird expressions, and maybe something can be done with that? I also wonder if having kids make their own challenges (for me, for each other) would be fun? Like they come up with a challenge, and I cull the best of the best, and I give that to the kids as a take home thing? Finally, I know someone out there mentioned doing trig identities all geometrically, with the unit circle, triangles, and labeling things… I mean, how elegant is the proof that $\sin^2(\theta)+\cos^2(\theta)=1$? So elegant! So coming up with equivalent expressions using the unit circle would be amazing for me. Anyone out there have this already done?

# Being an SBG teacher in a non-SBG world

I’ve posted about my adventures in Standards Based Grading… more so when I first started doing it, less so this year.

I overall have been happy with it, but three weeks ago I had a moment of frustration when I almost decided to abandon SBG completely in the fourth quarter (which will be upon us soon). The impetus: in my two SBG classes (non-AP Calculus), over half the students did not do the nightly work.

For some context, I require nightly work, but it is not graded. (This is not everyone’s philosophy, but I believe in nightly work.) I used to go around and check to see that it was done, but it never factored into their grades. For the most part, kids did it, so I stopped walking around. Things were going well.

For some more context: All the students in the class are seniors. I teach at a really awesome independent school in Brooklyn, and in general, kids are motivated. The kids in my class mostly don’t hate math (there are some exceptions), but they mostly don’t loooove math either (there, again, are some exceptions). It’s a really good group, and I felt things were going solidly for the first two quarters.

Then came the fateful day when I came to class and I saw the apathy and excuses for not doing the nightly work. It was horrible. In one class, I kinda lost it, and in the other, I tried to reign myself a bit. I mean, I didn’t start throwing chairs and stuff, but I definitely raised my voice, expressed my serious dissatisfaction, and (I admit it) I threatened to start grading nightly work. It wasn’t a moment I was proud of, because usually I’m less impulsive and don’t let things get to me. But I was M-A-D.

I talked it over with the other calculus teacher, who shared that he had been noticing a serious apathy and lack of work ethic.

One student emailed me apologizing for his lack of work, and I don’t know why, but that really meant a lot for me. It helped me table my frustration.

I needed spring break to cool down and see if I really needed to revisit SBG or not. That one incident really floored me. There are a few things I needed to think about:

1) There is a good possibility that this happens every year around this time. It’s 3rd quarter, a couple weeks before spring break. I have such a terrible memory and block out all the bad things… so this may be an annual occurrence. The inauguration of senioritis.

2) Although I talked about SBG a lot at the beginning of the year, I did not revisit it/talk about it as regularly as I have in the past. So I don’t know how much my kids have really understood about what we’re doing. I mean, they understood at the beginning. But I think it has morphed into something else in their minds.

The thing is, though, I started to worry that SBG wasn’t serving the purposes I adopted it for:

1) Independence and Responsibility

2) Students learning about their learning process

3) Clarity about what students know and what they don’t know

4) Making mistakes, but learning from them

I was having, and still sort of am, having a true crisis of faith. Because if SBG didn’t address these things, what’s the point? And clearly much of this is on me. Because careful implementation is crucial and that is my responsibility. I’ve seen it be wildly successful with students since starting, students who wouldn’t have a chance in hell in a traditional class to learn and be awesome at calculus. And for those kids, the kids SBG really works for, it’s enough to keep me invested and wanting to continue.

But the force I’ve been working against since I started doing it is:

I’m SBG in a sea of non-SBG.

In a school where kids are focused on traditional grades. And so when push comes to shove, even kids who understand and are on board with my nuanced understanding on SBG have to make decisions: it’s 1am and I am exhausted and have Spanish homework and calculus to do.

Calculus goes.

It just does. And yeah, once in a blue moon, I’m okay with that. It’s life. They’re learning to make choices. But it doesn’t happen once in a blue moon, because I’m fighting a system I have no chance of winning in. Because what I’ve found is that kids who make this choice don’t learn from this choice. And so some kids (not all, mind you) aren’t really putting in the effort.

So kids come to class without having done the work. Class goes slower because we aren’t all on the same page. Students take assessments and don’t do awesomely the first time. That’s okay, occasionally, because that’s the point… But ideally, reassessments should be more like a safety net, not crutches. That’s what we’re supposed to get them to learn, that’s what we want them to see. They can do it, if they figure out their learning process. And so by the third quarter (where I’m at), this should be where we’re at.

And here’s the thing: for the kids who have it as a crutch, I am not that successful at getting them to change their habits so reassessments become less frequent, and more like a safety net. They aren’t learning about their learning, and making changes. It’s just ignore, reassess, ignore, reassess, reassess, ignore… And I know some of that is on me. But I can’t get over how hard it is for me to do SBG effectively when calculus will always come last, when push comes to shove. And I hate that.

I’m not really asking for commiseration. I just wanted to post my thoughts here, because that’s why I have a blog. Duh.

If you do want to chime in, I’d love to hear:

1) What concrete things do you do to keep the philosophy, spirit, understanding of SBG alive… so that it doesn’t become a mechanized system by the third quarter?

2) If you are in a school that isn’t SBG, have you found any ways to combat the notion “SBG class can come last”?

3) If you are teaching SBG in any school, what mechanisms/procedures do you have to help kids individually understand how they learn, and how SBG can help them learn how to learn better? Do any of you have individual conferences with your kids or anything? Do you have them reflect about what they’re learning (or not) through SBG regularly, and do you respond to those reflections?

I realize that I have the systems of SBG smoothly set up, but I need to work on the other stuff… keep my eye on the reasons I’m doing it and making sure they come into sharp relief, instead of fading into the background as they tend to do as the keeping-afloat-teaching-day-to-day takes hold.

PS. I don’t want this to imply that kids don’t do their work with SBG, if you’re reading this out and thinking of starting it. It’s not like it’s a epidemic. But there are enough of them who are inconsistent enough with their work that it has become problematic. And then that day, that horrible day, has gotten me to think about some big issues. To be clear, I ended last year by making the statement: “My conclusion: although not perfect, this was a wildly successful year for Standards Based Grading in calculus” (read here for every student response to the survey I gave).

# Some New Things On The Interwebs & HOLY COW WHAT IS HAPPENING!

## THREE INTERNET THINGS YOU SHOULD DEF KNOW ABOUT

Here are three quick things I wanted to mention are out there on the interwebs which have me twitterpated!

1. The Productive Struggle blog. A blog which anyone can submit to. The way I see it: we have a tendency to post about what works, but not about our process when something just bombs. This blog is a great repository to share our failures and learn from them (and each other). Consider submitting  or cross-posting. Here’s a nice short post which spoke to me.

2. The Infinite Tangents podcast. ZOMG! Here’s the thing: we are enough of a community now that we have our own podcast! Ashli Black (aka @mythagon, blog) has been taping podcasts which focus around math teaching. The inaugural podcast was an interview with second year math teacher Daniel Schneider (@mathymcmatherso, blog). It’s pretty  totally fantastic. Of course I hear the excitement and experimentation that he is doing in his classroom, it makes me think how tepid I was in my second year. In fact, he makes me feel tepid right now. Which is good, because this podcast reminded me to be more thoughtful about my practice.

It also is really fun to listen to on the subway. It sure beats listening to that crackly faux hiphop coming out of that person’s headphones sitting next to you.

3. DailyDesmos blog. Here. This. This is another collective effort of a number of people in the mathteacherblogotwittersphere (full disclosure: I begged, and I’m now, a regular contributor to the site). As a little background, desmos.com is the most superior online graphing utility which is designed for teachers, and is so amazing, that I didn’t even teach my kids in precalculus to graph polar on the graphing calculators. (No, they aren’t paying me to say this. But they should! Hint!)

Each day two different graphs are posted (a basic one and an advanced one):

And then you use desmos (or any other graphing utility) to try to find the equation that matches the graph. It sort of reminds me of greenglobs (remember that awesome game!?) when I was a wee lad. But this is so much better. I’ve pulled a lot of muscles doing these challenges, and I love the feeling when I make a breakthrough. My favorite, so far, is here. And of the two I’ve contributed, my favorite is here. I have a really beautiful graph coming out next Thursday (3/28) so keep your eyes peeled!

***

## THIS IS A SYMPTOM OF THINGS HAPPENING. GOOD THINGS HAPPENING.

One thing that is now crystal clear to me is that we’re shifting into a new phase. (“We’re” meaning our little math teacher online community.) Initially, we had blogs, and these blogs are where conversations happened (in the comments). Then we added twitter, and soon blogs were the asynchronous way for us to communicate and the “real” conversations started happening on twitter. (Blogs became this archive or repository, and less for discussion. Of course this isn’t true for all blog posts.)

Now in the past year or year and a half, there has been an explosion of activity. and this explosion seems to center around (a) collaboration and generating things which are (b) not really centered about us and our individual classrooms. We’re thinking bigger than ourselves.

I’m talking the letters to the first year teachers, I’m talking the Global Math Department, I’m talking the visualpatterns website, I’m talking the month long new blogger initiation, I’m talking the freaking inspirational One Good Thing group blog, I’m talking Math Munch, I’m talking the collaborative blog Math Mistakes, I’m talking MathRecap to share good math PD/talks with each other. And of course, now we have the Productive Struggle blog, Daily Desmos, and the Infinite Tangents podcast. [1]

We’re still keeping our blogs, and archiving our teaching and sharing ideas, and talking on twitter. But now we’re also moving into creating these other things which are crowdsourced and for people other than just those in our little communit…

It’s been a freakin’ pleasure to see all this stuff emerge out of the fertile soil that we already had. We’re starting to create something new and different… and… and… I can’t wait to see what happens. [2]

[1] There are more out there too. I’m trying to archive them here, but they just keep on coming!

[2] I have a session proposed (with two other people) at Twitter Math Camp 2013 about all this stuff that has been banging around in my brain… this seismic shift that we’re witnessing.

# Families of Curves #3

I have now printed out my Families of Curves projects at school, and hung them up. I still have to look through the actual booklets that students turned in and give feedback, but the actual way that these look — once hung — is pretty awesome.

I used just some tags I had lying around (I love buying random useless stuff from Staples and hoarding it at my desk at school) and dissection pins. I photocopied their artwork on cardstock.

Being honest, I hung them because I wanted the kids to think “Hey, Mr. Shah liked these enough to take the time to do this.” Implicitly. I wanted the kids to know I was proud of their creations (and to let them know that they should be proud of them too.) No kid in my class has said “Hey, that’s awesome.” So I don’t know if I accomplished that goal. But I have heard a zillion other people say how much they have liked seeing them there. A number of other teachers have randomly come up to me unsolicited to tell me how cool they think they are. And the head of the Upper School gave them a shout out in the Upper School meeting. And just recently, yesterday at the subway, I ran into a student who graduated a couple years ago. And she was at our school because her brother goes here, and she said she was looking at them thinking “how cool! Mr. Shah!”

Another great moment with these was having two of my kids go to a neighboring school which holds a math art seminar, and watch these kids talk with other students about their work. It was clear how invested these two kids were. Watching them articulate their process just made my heart melt.

[Here is Families of Curves #1, and Families of Curves #2]