# The Formal Definition of the Derivative, or Why Holes Matter

Lucky you! Two calculus posts in one day. Mainly because I don’t want some of these ideas to disappear in my hiatus from teaching it. This one deals with our favorite topic: the formal definition of the derivative.

$\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{(x+h)-(x)}$

I see that expression and my mind goes to the following places:

• Doing a bunch of tedious algebraic calculations for a particular function in order to find the derivative.
• I “see” in the expression the slope of two points close together.
• I envision the following image, showing a secant line turning into a tangent line

And I think for many teachers and most calculus students, they think something similar.

However I asked my (non-AP) calculus kids what the $h$ stood for. Out of two sections of kids, I think only one or two kids got it with minimal prompting. (Eventually I worked on getting the rest to understand, and I think I did a decent job.) I dare you to ask your kids and see what you get as a response.

What I suspect is that kids get told the meaning of $\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{(x+h)-(x)}$ and it gets drilled into their heads that they might not fully understand what algebraically is going on with it.

It was only a few years ago that I came to the conclusion that even I myself didn’t understand it. And when I finally thought it all through, I came to the conclusion that all of differential calculus is based on the question: how do you find the height of a hole? I started seeing holes as the lynchpin to a conceptual understanding of derivatives. I never got to fully exploit this idea in my classes, but I did start doing it. It felt good to dig deep.

The big thing I realized is that I rarely looked at the formal definition of the derivative as an equation. I almost always looked at it as an expression. But if it’s an equation…

$f'(x)=\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{(x+h)-(x)}$

… what is it an equation of? An equation with a limit as part of it?! Let’s ignore the limit for now.

Without the limit, we have an average rate of change function, between $(x,f(x))$ and $(x+h,f(x+h))$. And since we have removed the limit, we really have a function of two variables.

$AvgRateOfChange(x,h)=\frac{f(x+h)-f(x)}{(x+h)-(x)}$

We feed an $x$ and $h$ into the function, and we get an output of a slope! It’s the slope between $(x,f(x))$ and $(x+h,f(x+h))$!

Let’s get concrete. Check out this applet (click the image to have it open up):

On the left is the original function. We are going to calculate the “average rate of change function” with an x-input of 1.64 (the x-value the applet opens up with).We are now going to vary h and see what our average rate of change function looks like: $f(1.64,h)=\frac{f(1.64+h)-f(1.64)}{h}$. That’s what the yellow point is.

Before varying h, notice in the image when h is a little above 2, the yellow “Average Rate of Change” dot is negative. That’s because the slope of the secant line between the original point $(1.64, f(1.64))$ and a second point on the function that is a little over 2 units to the right is negative. (Look at the secant line on the graph on the left!)

Now let’s change h. Drag the point on the right graph that says “h value.” As you drag it, you’ll see the second point on the function move, and also the yellow point will change with the corresponding new slope. As you drag h, you’re populating points on the right hand graph. What’s being drawn on the right hand graph is the average rate of change graph for all these various distances h!

Here’s an image of what it looks like after you drag h for a bit.

Notice now when our h-value is almost -3 (so the second point is 3 horizontal units left of the original point of interest), we have a positive slope for the secant line… a positive average rate of change.

The left graph is an $x-f(x)$ graph (those are the axes). The right graph is a $h-AvgRateOfChange$ graph (those are the axes).

Okay okay, this is all well and dandy. But who cares?

I CARE!

We may have generated an average rate of change function, but we wanted a derivative function. That is when h approaches 0. We want to examine our average rate of change graph near where h is 0. Recall the horizontal axis is the h-axis on the right graph. So when h is close to 0, we’re looking at the the vertical axis… Let’s look…

Oh dear missing points! Why? Let’s drag the h value to exactly h=0.

The yellow average rate of change point disappeared. And it says the average rate of change is undefined! 0/0. We have a hole! Why?

(When h=0 exactly, our average rate of change function is: $\frac{f(x+0)-f(x)}{(x+0)-x}$ which is 0/0. YIKES!

But the height of the hole is precisely the value of the derivative. Because remember the derivative is what happens as h gets super duper infinitely close to 0.

We can drag h to be close to 0. Here h is 0.02.

But that is not infinitely close. So this is a good approximation. But it isn’t perfect.

And this is why I have concluded that all of differential calculus actually reduces to the problem of finding the height of a hole.

Here are three different average rate of change applets that you might find fun to play with:

one (this is the one above)     two     three

In short (now that you’ve made it this far):

• Look at the formal definition of the derivative as an equation, not an expression. It yields a function.
• What kind of function does it indicate? An average rate of change function. And in fact, thinking deeply, it actually forces you to create a function with two inputs: an x-value and an h-value.
• Now to make it a derivative, and not an average rate of change, you need to bring h close to 0.
• As you do this, you will see you create a new function, but with a hole at h=0.
• It is the height of this hole that is the derivative.

PS. A random thought… This could be useful in a multivariable calculus course. Let’s look at the average rate of change function for $f(x)=x^2$:

$AverageRateofChange(x,h)=\frac{(x+h)^2-x^2}{h}$

Let’s convert this to a more traditional form:

$z=\frac{(x+y)^2-x^2}{y}$

Now we have a function of two variables. We want to find what happens as h (I mean y) gets closer and closer to 0 for a given x-value. So to do this, we can just visually look at what happens to the function near y=0. Even though the function will be undefined at all points where y=0, visually the intersection of the plane y=0 and the average rate of function should carve out the derivative function.

If this doesn’t make sense, I did some quick graphs on WinPlot…

This is for $f(x)=x^2$. And I graphed the plane where y=0. We should get the intersection to look like the line $f'(x)=2x$.

Yup. Cool.

I did it for $f(x)=\sin(x)$ also… The intersection should look like $f'(x)=\cos(x)$.

# u-substitution, visually

I created some calculus Geogebra applet thingies last summer that I wanted to use last year. Alas, time ran short and we never got to use them. However since I’m no longer teaching calculus (at least not next year), I figured I’d throw them up in case anyone else out there finds them useful.

They deal with u-substitution. I’ve always had a problem with teaching it. Here’s how it goes… You have some integral in terms of $x$. You convert all the $x$s and $dx$s into $u$s and $du$s. And viola! It works out. It’s very powerful. And it’s procedural. And kids have throughout the years learned this “substitution”-y thing works [1]. So kids tend to like it.

But here’s the thing. For my kids, it’s just a random method to evaluate an integral. They don’t conceptually understand what is going on… what this changing of variables is doing.

When I thought deeply about this, I realized what truly is happening is that we are transforming space… From the $x-f(x)$ plane to a much convoluted $u-f(u)$ plane. But it is through our particular choice of $u$ that makes the change in space beautiful, because it turns something that looks particularly nasty and converts it into something that looks rather nice. Ish.

Here is a screenshot from one of my geogebra applets illustrating this (you can click on the screenshot to be taken to the applet):

We start with a pretty ugly function that we’re integrating. But by using this substitution to morph space, we end up with a much nicer function. I mean, throw both of these up and ask your kids — which one of these would they rather find the integral of. They’ll say the one on the right! The u-substitution one. Although not perfect [2], it’s pretty kewl.

The applets are here:

And the applets are dynamic! You can change the lower and upper bounds on the $x-f(x)$ graphs and the lower and upper bounds automatically change on the $u-f(u)$ graph! But because math is awesome, the areas are preserved!

Some things I maybe would have done with the applets in my class:

• Let kids play with the applets and get familiar with them.
• For the first applet (starting simple), have kids count the boxes and estimate the area on one graph, and then do it on the other (careful though! the gridlines are different on the two graphs!). Whoa, they are always the same!
• For the first applet (again, starting simple), ask them to drag the upper limit to the left of the lower limit. Explain what happens and why.
• The second applet is my favorite! Put the lower limit at x=0. Drag the upper limit to the right. Explain what is happening graphically — and that tie that graphically understanding to the particular u-substitution chosen.
• In the second applet, can students find three different sets of bounds which give a signed area of 0?
• In the fourth applet, have students put the lower and upper bounds on x=6 and x=7. Have them calculate the average height of that function in that interval (the area is given!). Do they have visual confirmation of this average height for this interval?Now Looking at the u-graph, the bounds are now u=8 and u=10. Have them estimate the average height of that function in that interval (again, the area is given)! (The average height “halves” in order to compensate for the wider interval. It has to since the areas must be the same) Have students do this again for any lower and upper bounds for this graph. It will always work!
• In the fifth applet, have students put the lower bound at x=0, and have them drag the upper bound to the right. What can they conclude about the areas of each of the pink regions on the $x-f(x)$ graph? (Alternatively, you can ask: you can see from the $u-f(u)$ graph that the signed area on the original graph will never get bigger than 1, no matter what bounds you choose. Try it! It is impossible! Armed with that information, can you conclude about the pink regions in original graph?)

I’m confident I had more ideas about how to use these when I made them [3]. But it was over a year ago and I haven’t really thought of them since. But anyway, I hope they are of some use to you. Even if you just show them to your kids cursorily to illustrate what graphically is going on when you are doing u-substitution.

***

[1] Though I bet if you asked a class why they can use “substitution” when solving a system of equations, what the reasoning is behind this method, they might draw a bit of a blank… But that’s neither here nor there…

[2] What would actually be perfect would be a copy of individual Riemann Sum rectangles from the $x-f(x)$ graph “leaving” the first graph, then in front of the viewer stretching/shrinking their height and width for the appropriate $u-f(u)$ graph, and then floating over to the $u-f(u)$ graph and placing itself at the appropriate place on the $u$ axis. And then a second rectangle does that. And a third. And a fourth. You get the picture. But even though the height and width morph, the area of the original rectangle and the area of the new rectangle will be the same (or to be technical, very very close to the same, since we’re just doing approximations). In this sort of applet, you’d see the actual morphing. That’s what is hidden in my applets above. But that’s actually where the magic happens!

[3] I recall now I was going to make kids do some stuff by hand. For example: before they use the applets, kids would be given lower and upper x-bounds, and asked to calculate lower and upper u bounds. And then use the applets to confirm. Similarly, given lower and upper u-bounds, calculate lower and upper x-bounds. Use the applets to confirm.

# Some Geogebra Fun

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

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

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

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

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

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

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

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

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

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

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

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

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

***

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

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

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

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

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

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

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

# Teacher Growth, the MTBoS, and TMC14

As usual, going to Twitter Math Camp has caused me to be all reflective and stuff. Barf. About myself and about this online community. And trust me, this self-introspection will be over soon and I’ll be back on my regularly scheduled program of procrastinating doing my prep work for school. But I should probably get it all out first, in one giant word-vomit. Ready? SPEW!

## I’m a Hobbit

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

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.

## Inadequacy and Change

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.

## Personal Inadequacy as Part-of-the-Deal

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.

# Rational Function Headbandz

TL;DR: An interactive activity having kids ask each other questions to guess the rational function graph they have on their foreheads.

***

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.

[.pdf, .docx]

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!

# Musings on the 180 blog

I’m at Twitter Math Camp 2014. Normally my inclination at a conference is to take a moment to recap the day from start to finish, as an archive to what I learned. Little things, big things, trying to capture every little morsel. Instead, I think I’ll just write about one thing I’ve been thinking about today, based on sessions and conversations.

## 180 blogs: Mine from this year

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.

## 180 blogs: An idea for the future

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!

# Playing with Math

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

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

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

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

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

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

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

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

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

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

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

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

Sue speaks about the origins of this book here:

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