(wow, been too busy to shave, have we Mr. Shah?)

For the first point, I don’t actually do much. I ask them to write, they write, I comment. And we discuss (more at the start of the year, but I always let this go and I forget to talk about it a lot). In Algebra II, they get one or two writing questions on every assessment. And each quarter they had problem sets where they had to write out their thought processes/solutions comprehensively and clearly. Even though I didn’t actually do anything systematic and formal in terms of teaching them to write (mainly I just had them write), I can say that I’ve seen a huge huge improvement in their explanatory skills from the beginning of the year. What I used to get just didn’t make sense, honestly. A random string of words that made sense in their heads, but not to anyone reading them. But now I get much more comprehensive explanations, which usually include words, diagrams, graphs, examples. They aren’t usually amazing, but they’re not ready to be amazing.

For the second point, I realized that the types of questions that we tend to ask (you know, those more routine questions that all textbooks ask) don’t always let me know if a student understands what they’re doing. It just lets me know they can do a procedure. So, for example, if I asked students to graph , I would bet my Algebra II kids would be able to. But if I showed them the question and the solution, and ask them to explain what the solution to that question means, I would expect that only half or two thirds of the class would get it right. (Hint: The solution is the set of all points (x,y) which make the inequality a true statement.) They can do the procedure, but they don’t know what the solution means? That’s what I’ve found. And you know what? Before asking students to write in the classroom, I had deceived myself into conflating students being able to answer with a full understanding of 2-D linear inequalities. [1]

Before having students write, I actually believed that if I asked that question (“What does this solution mean?”), almost all the students would be able to answer it. (“Like, duh, of course they can!”) But since asking students to explain themselves, explain mathematics, I’ve uncovered the nasty underbelly to what students truly understand. The horror! The horror! But now that I recognize this seedy underworld of misconceptions or no-conceptions, I’ve finally been able to get beyond the despair that I originally had. Because now I know I have a place to work from.

The counterside to this point is that when kids do understand something, they kill it.

This simple question I made for my calculus students early in the year, and this student response, says it all. I have no concern about this kid understanding relative maxs and mins. No traditional question would have let me see how well this student knew what was up.

For me the obvious corollary is that: we need to start rethinking what our assessments ought to look like. If we want kids to truly understand concepts deeply, why don’t we actually make assessments that require students to demonstrate deep understanding of concepts? I am coming to the realization that the more we keep giving the same-old-same-old-assessments, the more we are reinforcing the message (implicitly) that we don’t reallyreally care to know about their thinking. We are telling our kids (implicitly) that we are content if they show their algebraic steps. But as I’ve noted, my big realization is that students performing those algebraic steps don’t necessarily mean that the student knows what they’re doing, or what the big picture is.

I don’t know have an example of what I think a truly ideal assessment might look like, but I do know it isn’t anything like I gave when I started off teaching five years ago (has it really been five years? why am I not better at this?), and I do know that each year I am slowly inching towards something better. Right now, my assessments are fairly traditional, but with each year, they are getting less so.

Sorry if I’ve posted something like this before. I have a feeling I have. But it’s what’s been going through my head recently, and I wanted to get it out there before I lost it.

[1] Another good illustration might be having students solve . Sure, they can get . But does doing that really mean they understand that whole “if you divide by a negative in an inequality, you switch the direction of the inequality” rule that has been pounded in them since seventh grade? Nope. The traditional questions don’t tend to check if the kids know why they’re doing what they’re doing.

Knowledge is precious and vast, it keeps us curious and engaged in the world, and simple ideas can — when taken to their thoughtful conclusions — be extraordinarily powerful. And thought it may seem like we have forever to cull this knowledge, we don’t, so take advantage!

Without further ado, my letter to my seniors. I know, it always comes across as hokey. But when I get sentimental…

Yeah you, my super awesome teacher friend!

I am about to start composing a letter of recommendation for @cheesemonkey [blog, twitter], and I wanted your help. For me, her constant upbeat spirit and cheerleading of every one of us  in everything we do has been glorious. Heck, in my opinion, she’s a lynchpin to our online math math community. Full stop. The activities that she posts about are constantly on my list of things to steal. And just as importantly, the thoughtfulness that she writes about in all her interactions with students — whether it be in her zillion recommendation letters to her conscientious work to build up each student’s math confidence — is an inspiration.

I want to write a collective recommendation, one where the reader can see that @cheesemonkey has a broad impact on the math teaching world.

So let’s do a solid for @cheesemonkey. If she’s done something large or small, inspired you, helped you, given you something to use in her classes, keeps you engaged in teaching, pay back the favor. Throw your mini-recommendation in the submission box below (it will be emailed to me) and show her how much you care! A few sentences to a few paragraphs, just share. We’re a community that helps each other out all the time, and I need your help!

It is a bit time sensitive, so if you could do it soon (translation: in the next day or two), I would be ever grateful.

Thank you, all!

Always,
Sam

This past weekend and this week, I’ve been consumed with writing narrative comments on all my students. In the past two years of teaching, I have been trying to be more thoughtful about what I’m writing. To put all the cards on the table, I don’t think that comments themselves really effect change in students. However, I do think there is a powerful thing that comments can do: it is a way to tell students I see you and I care about you and I am thinking about you and your learning. Not literally, but a comment can send that message implicitly.

So even though I have serious doubts about the efficacy about what I write in helping students to change their practices, I hold firm to the belief that the implicit message is worth it. So I write, and hope that for a few kids, it matters.

It’s almost 9pm. I’m at a coffeeshop now, and I just finished my last (my 49th) comment of the year. 58 pages later, I am breathing a sigh of relief that I’m done.

I’m totally drained.

I’m so tired of writing that I don’t have it in me to talk about how my comments have evolved in the past two years, or how standards based grading has made writing comments so much easier. Or list the places I know I could still improve on. And maybe I will at some later point.

For now, I just wanted to write a post now sharing the good news with everyone:

I am done!

(If  you want to see the type of comments I wrote in my first three years of teaching, I’ve archived that here.)

I know I don’t use this blog to talk about my non-school life, but that’s only because it’s only about 1% of my life.

So at the start of this spring break, I did something I’ve been dreaming about for years. You see, when I was in college I had a bout of insomnia so I started to listening to Supreme Court oral arguments to focus my mind on something boooooring so I could fall asleep. Little did I know I would become a Supreme Court junkie. And so I went with a friend (who teaches history and constitutional law at my school) to Washington DC where I had a glorious time. The night before the oral argument, I invited @rdkpickle to dinner and didn’t get psychopathkilledtodeath. You’ll all be pleased to know that she’s just as personable in person as she is online.

The following day I got to Supreme Court

early enough that we got tickets to hear the arguments. It was similar to what I expected in terms of the argument, and also nothing like I expected in terms of the room. It wasn’t as grandiose as I imagined — I imagined the justices to be higher up, the room to be wider, and the seating for the visitors to be nicer (we were like sardines put on very cramped wooden chairs). The two cases we heard were Astrue v. Capato and Southern Union Company v. United States, both fascinating. (And for those of you who are dying to know, yes, I took off my hat in the courtroom.)

In DC, I also got to meet up with two dear old friends who I hadn’t seen in ages, and just in time, because they are moving to Korea for two years, soon. And one high school friend who I consider one of my besties even though we never see each other or keep in touch. He’s that kinda guy.

In addition to my trip to DC, I had my sister in NYC for a day, where we ate delicious food, traipsed around a lot, walked the high line, read a bit in Bryant Park, went shopping at the Strand (I didn’t buy anything!), and then met my parents and family friends for dinner. It was a full and lovely day.

Then I scampered to San Francisco for a whirlwind trip. I got to see a ton of high school and college friends, do a bunch of shopping, eat delicious food, watch the Hunger Games, and throw a party! That’s right — one of my best friends from high school just moved back and I convinced her throw a house party — and I invited all my friends.

Additionally, and this is going to make all of you jealous, I got to hang out and have dinner with the following math twitter people at Bar Tartine: @woutgeo, @btwnthenumbers, @cheesemonkeysf, @ddmeyer, and @suevanhattum. I only wish we had started earlier. It was totes amazing (@cheesemonkeysf wrote about it). And again, I didn’t get psychokillerkilled. Although when I talked smack about ed researchers, I thought the towering Dan Meyer was going to kill me with his laser stare! But he is too much of a Good Guy Greg for that.

And then I got back, and have basically been doing nothing but watching bad TV and thinking (but not doing anything) about all the work I have to do but haven’t done. I even finished the two seasons of Party Down (amazing, btdubs), and the season of Summer Heights High (also amazing, btdubs). Go me!

So even though I felt like that I could have done, all those roads not taken and all that, I think I’ll always feel that way. It’s just the way I am. And I have to learn to appreciate all that I have done, instead of focus on all that I could have done. In fact, that’s probably a lesson for me in teaching. There you go — I have a sickness. Everything is about teaching. 

With that, I’m out.

PS. I would love to have shown more photos, but I feel weird using photos of people who might care if their photo is out in the world. Dan, he’s probably okay with it. He has a TED talk and all that.

Anyway, I wanted to share with you two things.

First, how I introduced the idea of optimization to my kids. Instead of going for the algebra/calculus approach, I wanted them to toy with the idea of maxima and minima, so I had them spend 35-40 minutes working on this in class:

[doc]

I thought it was pretty cool to see my kids engaged. I rarely do things like this, but I did it (I was being videotaped during this lesson… and I had never done it before… and I had the idea to create it the night before…). It was fun! And although I cut the debrief the next day short (ugh, why?), I enjoyed seeing kids engaged in problem solving through various strategies. And there was a healthy level of competition. (The winners for the 1st and 2nd tasks got a package of jelly beans, but they were so gross I threw them out! One student gave them to his rabbit who likes jelly beans, and even the rabbit didn’t like them!) But when it came down to it, it drove home the idea that optimization was something that trial and error is good for, sometimes we do it intuitively, sometimes our intuition is terrible and sometimes it is good, and sometimes we get an answer but we don’t know how to prove there isn’t a better answer (e.g. in problem #3). Some kids liked that this felt more “real world” than this world of algebra and graphing that we’ve been meandering in.

Second, I have allotted a few days for students to work on this project during class (it’s the week before Spring Break and kids are overburdened, so I didn’t want to have them do something which involved a lot of at-home time). They’ve been working on it this week, and I’ve heard some good conversations thus far. (They’re doing this in pairs, and I have one group of three.) The fundamental question is: with a given surface area, what are the dimensions of a cylinder with maximal volume?

Now I don’t quite know how their posters will turn out yet, or whether students will have truly gotten a lot of “mathematical” knowledge out of it. But each day, I’ve had a couple kids say things that indicate that this isn’t a terrible project. (I don’t do projects, so that’s why I’m very conscientious about it.) A few said something equivalent to “Wow, the companies could be giving me x% more creamed corn!” or how they like doing artsy-crafty things. At the very least, I can pretty much be assured that students — if I ask them if there is any question that calculus can answer at the grocery store — will be able to say yes.

Next year I will probably add the reverse component (for a given volume of liquid you want to contain, how can we package it in a cylinder to minimize cost… what about a rectangular prism… what about a cube… what about a sphere… etc.?).

[1] The one thing I found in this book my friend gave me (on science and calculus) was an experiment where you shoot a laser at some height at some angle into an aquarium, so that it hits a penny at the bottom (remember the laser beam will “change” angles as it hits the water) to minimize the time it takes for the photon to travel from the laser to the penny. I almost did it, but deciding to do it was too last minue.

I’ve posted much of my quadratics materials before, but I thought I’d share some new/updated ones. I’m a bit exhausted, so forgive the shortness of my descriptions.

1. My Vertex Form worksheet was motivated by my frustration with students just memorizing that has a vertex of because you “switch the sign of the -2 and keep the 3.” Barf. (FYI: we haven’t done function transformations yet.) So I created this sheet to “guide” students to a deeper understanding of vertex form.

[.doc]

2. My Angry Birds activity was inspired by Sean Sweeney, but modified. I had taught students how to graph (by hand) quadratics of the form and . Students also had been exposed to the vertex form of these basic quadratics. But they hadn’t been exposed to quadratics where the coefficient in front of the term wasn’t “nice.” So all I did was give them four geogebra files, and had them play around. By the end of the activity, students recognized how critical the “a” coefficient was to the shape of the parabola, they started conjecturing that if you had the “a” value and the vertex and whether the parabola opens up/down that you could graph any parabola, and one pair of kids were able to convert a crazy angrybirds quadratic (with a really nasty “a”‘ value) to vertex form.

[.doc] [files]

If I’m teaching Algebra II next year, I want to ask if I can get rid of quadratic inequalities or some of the other more technical things we do, and make an entire unit/investigation on using geogebra and algebra and angrybirds to investigate quadratics.

3. My discriminant worksheet is below. It worked okay, but students still didn’t quite understand the difference between and , which was the goal of the sheet. So it needs some refinement.

[

[doc]

4. Finally, below are my attempts to get students to better understand quadratic inequalities. I started with a general sheet on “visualizing function inequalities,” and then I made a guided sheet to bring more detail to things. I found out that students didn’t quite understand the meaning of the schematic diagram we drew, nor did they understand why to solve we have to draw a 2D graph. Well, to be more specific, students could do the process but didn’t fully grasp why we graph . I changed up this worksheet this year, but maybe I should go back to last year’s worksheet.

[doc]

[doc]

C’est tout. With that, I’m exhausted and going to bed.

I have been mulling over if I should do a project in calculus in the fourth quarter. And I had a thought. I have been really trying to focus on the fundamental underlying ideas in calculus, and shooing away the algebraic gobblygunk. Why? Because my kids aren’t taking AP Calculus. Most won’t be taking math in college. So I want my kids to leave calculus saying: “Yes, I understood the ideas. Calculus is about ideas.”

I wonder if a good final project, which would force them to grapple with the Big Ideas, might be having students create a collective time capsule, which will be stored in some deep underground facility, and will be the only remnants of “Calculus” that may exist after some horribly apocalyptic disaster.

I’m not sure what would go in the capsule, but I like the idea that all students would be asked to contribute a few items. Maybe we’d break the course into chunks, and each student would be responsible for writing an accessible explanation of each chunk — and we bind these together into a book? And each student would create set of drawings/graphs/photograph/images that (for them) represent the Big Ideas of Calculus, and they have to explain each one of them… What’s the idea, and why is it so important?

In addition to these required items, students could have their choice of what else to contribute… Things like:

1) A video of the student explaining the weirdnesses/paradoxes/strange ideas of (or relating to) calculus
2) A short research paper on the history of calculus
3) A letter to the future explaining why calculus is an important swath of knowledge that shouldn’t be forgotten (including uses / applications of calculus)
4) A challenging calculus problem, and it’s solution
5) A “concept map” for calculus
6) Audio recordings of students reading quotations about calculus that resonated with them, and then students explaining why it resonted with them.
7) Designing a cover to the collective calculus book we bound together, and on the back cover, an explanation of how the cover exemplifies the course

Or other things?

I don’t know. It felt like a cool idea when it jumped in my head a few minutes ago, but now that I’m writing it, I can’t quite picture it … yet. Any ideas of how to take this idea and turn it into something good? Throw it in the comments!

In that vein, of super thoughtful intentional stuffs, I wanted to share two crazy good “do nows” from last week. Not because they’re deep, but because they were so thought-out.

For one calculus class, I needed my kids to remember how to solve (that equation was going to pop up later in the lesson and they were going to have to know how to solve it). I also know my kids are terrified of logs, but they actually do know how to solve them.

I threw the slide below up, I gave them 2 minutes, and by the end, all my kids knew how to solve it. I didn’t say a word to them. Most didn’t say a word to anyone else.

How I got them to remember how to solve that in 120 seconds, without any talking, when they are terrified of logarithms and haven’t seen them in a looong while?

I can’t quite articulate it, but I’m more proud of this single slide than a lot of other things I’ve made as a teacher. (Which is pretty much everything.)  Not deep, I know. It’s not teaching logs or getting at the underlying concept, I know. But for what I intended to do, recall prior knowledge, this was utter perfection. The flow from each problem to the next… it’s subtle. To me, anyway, it was a thing of perfection and beauty.

The second slide is below, and I threw it up before we started talking about absolute maximums/minimum in calculus.

As you can imagine, we had some good conversations. We talked about (again) whether 0.9999999… is equal to 1 or not (it is). We talked about a property of the real numbers that between any two numbers you can always find another number (dense!). I even mentioned the idea of nonstandard analysis and hyperreal numbers.

So I know it isn’t anything “special” but I was proud of these and wanted to share.

Usually, I introduce this in a really unengaging lecture-format. But he inspired me to … copy him. And so I did, extending some of his work, and I have had an amazing few days in calculus. So I thought I’d share it with you.

The Main Point of this Post: By creating the need for a word to talk about inflection points on graphs, we actually saw the math arise naturally. And through interrogating inflection points, we were able to articulate a general understanding of concavity. In other words… the activity we did motivated the need for more general mathematical concepts.

First, definitely read Bowman’s post. All I did was formalize it, and extend it in a few ways, by making a worksheet. I put my kids in pairs and I had them work on it (.docx):

What naturally will happen when students generate their graphs is they will get a logistic function. (Which has a beautiful inflection point! But they don’t know the word… they just see the graph.)

So here we are. The students have a graph, and they’ve been asked to explain their graph for (a) the layperson and (b) the mathematician. Most get some of it done with their partners, and then they take it home to finish individually.

The next day, at the start of class, I assign students to work in groups of 3 (with different people than their partners the previous day). They are asked to take a giant whiteboard and:

(Now I want to give credit where credit is due. I have really been struggling with using the giant whiteboards well, and having students present their work effectively and efficiently. My dear friend Susanna, when I told her about this activity, suggested the groups, the underlining of the mathy words, etc.)

This worked splendedly.

(click to enlarge)

And they had such great observations. Some groups picked up on that change where the function was increasing in one way to increasing a different way. Others talked about how the rate of change (of infected over time) was greatest. Others talked about how the function was “exponential” for the first thing, seemingly linear for the middle third, and “something else” for the last third.

Those gave rise to good short discussions, and we came up with the language for inflection points (which I call INFECTION POINTS!!! GET IT!?!) and concave up/down.

After they had a sense what those words meant, I had students work in partners on the following (.docx):

The point was to get students comfortable with the ideas before we delve into the heavy mathematical lifting. It was powerful. Especially the last page, which got students thinking about patterns, exceptions, and ways to generalize. Our big conclusions:

And with that, I’m too exhausted to type more. But that’s the general sense of what went on in an attempt to teach how to use calculus to analyze the shape of a function.