Author: samjshah

Students communicating mathematics has opened my eyes to mathematical ugliness (and what that means to me)

This year, as I have been in the past few years, I’ve been attempting to incorporate more writing in my math classes [note: Shelli found a post from 2009 I wrote on this endeavor]. It’s been extraordinarily enlightening, because what this has done is show me two things: (1) kids don’t know how to explain their reasoning in clear ways, and (2) I’m usually extraordinarily wrong when I think my kids understand something, and the extent to which I am wrong makes me cringe.

(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 y < 2x+3, 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 y < 2x+3 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 -3x<6. Sure, they can get x>-2. 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.

Senior Letter 2012

Each year at the end of the school year, I say goodbye to my seniors. And each year, I’ve written a letter to the seniors with some imparting thoughts as they go off in the world. And each year, the message in the letter stays fairly constant, even though the way I say my message might slightly change. It always goes something like this:

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…

Let’s do a solid for @cheesemonkey

Dear you,

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.

← Back

Thank you for your response. ✨

Thank you, all!

Always,
Sam

Comment Time Is Over!

This is a post of celebration.

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.)

Spring Break 2012

As this Spring Break comes to a close (it’s Friday, school starts on Monday) I am a little wistful — thinking about all that I could have done, and all that’s still on my plate to do. But I do that to myself. I don’t take time to appreciate all that I do and stop looking for what’s next. So in this post, I’m going to recount some awesome things about this Spring Break.

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.

Optimization: An Introductory Activity & Project

I switched things around with optimization in calculus this year, and I realized if I had the time, I would spend a month on it. [1] I wonder if this shouldn’t be a crux of the class. Not the stupid “maximization and minimization” problems but finding some real good ones — in economics, physics, chemistry, ordinary situations. There have got to be tons of non-crappy ones!

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?

[.doc]

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.

Recent Quadratics Stuffs from Algebra II

I am just finishing up my quadratics unit in Algebra II. We spend a lot of time on quadratics, doing everything from factoring, to completing the square, to the quadratic formula, to all sorts of graphing, the discriminant, 1D and 2D quadratic inequalities, quadratic linear systems, systems of inequalities, etc. Tons. And we didn’t even get to do the project I enjoy involving pendulums and quadratic regressions. Le sigh.

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 y=(x-2)^2+3 has a vertex of (2,3) 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 y=x^2+bx+c and y=-x^2+bx+c. 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 x^2 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 y=ax^2+bx+c and 0=ax^2+bx+c, 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 0<x^2-4x+3 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 y=x^2-4x+3. 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.