cheesemonkey wonders

cheesemonkey wonders

Thursday, October 31, 2013

And *THIS* is why I love the MathTwitterBlogosphere, part 573 – Infinite Tangents interviews Lisa Henry, Part 1

There are some very good things about Halloween on a school day, but a mandatory classroom "party" in advisory last period of the day on a Thursday is not one of them.

When I had finished complaining to myself about the state of my classroom and cleaning up the last of what the 13-year-olds had left behind that really bothered me, I packed up my stuff and got into my car.

And I remembered that I had an Infinite Tangents podcast all cued up for myself — one I'd been saving for a moment when I needed it most. A moment like now.

So I got to spend the drive home with Ashli and Lisa Henry. Part 1. A glorious triumph of delayed gratification.

I feel lucky to have gotten to know Lisa first through Twitter and blogs and then in person at the first Twitter Math Camp in 2012. Lisa has a gift for teaching through community-building, and she has brought this gift to bear on Twitter Math Camp. I admire and appreciate her respectful and inclusive community-building, and it inspires me in my own classroom and in my life.

Now, like most teachers, I come from a family of storytellers, so it's probably no surprise that I love hearing other people tell the stories of events I participated in. I love the prismatic contrasts of perspective and memory – the way something that struck you as essential to an event gets bumped down or deflected sideways in another person's memory due to proximity or overtaking or whatever. So I love hearing Ashli and her guests telling stories of events I remember because that process invokes the same pleasure twice – the memory of the event itself and the joy in the retelling.

My mood lifted considerably as I gained distance from school and lost myself in the conversation and the memories they were weaving on my car stereo.

It was fun to hear about and remember the great Facebook "befriending" moment in 2011 or so—that moment over Christmas Break when a bunch of individuals who'd been nothing more than virtual colleagues on Twitter (but who were still basically strangers) decided to take the seemingly insane step of "friending" each other on Facebook.

It was a moment of enormous risk.

It's one thing to share teaching ideas or goof around on Twitter, but crossing that line between virtual and IRL felt profound. What if these people turned out to be crazy? unpleasant? dangerous? Or even worse — what if they turned out to have different political beliefs than I did?

The risk felt very real at the time, and sometimes it still does. I don't pretend to be something I am not. I am a liberal. I live in San Francisco. I am a practicing Buddhist and a Democrat. My representative in Congress is Nancy Pelosi. I believe in a lot of things I know that a lot of other people in other parts of the country do not.

But the one thing I know in my feet is that I am a teacher — and a learner.

And I knew that all of these other teachers all over the continent who had become my tentative friends and virtual colleagues on Twitter in exploring what it means to teach and learn math were every bit as committed to what that means as I am. So I guess I trusted it. I was willing to go with it, and to push myself beyond my comfort zone for the sake of connecting with a community of like-minded math teachers who want the same things for our kids and for our communities and for our countries — regardless of what we may believe at the grassroots personal level.

And with all of that as background, I have to admit — it was one of the best and most profound decisions I have ever made.

I was one of those crazy ten or fifteen people who was hellbent on attending Twitter Math Camp even if we had to hold it in a yurt outside a garbage dump. I knew that these were people I wanted to be connected to and spend time with and get to know, even if we seem like we'd be completely incompatible based on what you can see from examining our surfaces.

There was a (now-hilarious) period of several months when it seemed as though what my new friends most wanted us to do was to go on a cruise together and do Exeter or PCMI problem sets together. I remember that Julie looked into costs and group rates and I thought to myself, what in the name of everything sacred have I gotten myself into? I hate situations like cruises. I get seasick. I could imagine nothing worse than being trapped on the open ocean for days with people I don't know.

But there was something about the energy of the group that I innately trusted.

I kept my cruise-hating thoughts to myself, but I hung in there because I knew I did not want to miss out on what appeared to be happening. These were people I wanted to spend time with, and I supposed that if that meant I would HAVE to spend time on a cruise ship, I could probably get a prescription for some kind of anti-anxiety medication to have on hand in case I completely freaked out.

And I just hung in there.

Eventually, the cruise ship idea fell apart, thank God, and the math camp idea came together. And nothing has been the same in my life ever since. And it's been good. Very good.

By the time I went through the toll plaza at the city end of the Golden Gate Bridge, I was not only not crabby any more, I was actually happy.

I felt connected to something much larger than my own daily grumbles, and that was enough to wash all the grouchiness away.

By the time I had parked the car, walked the dog, and poured myself a beer, I realized I needed to blog about my drive as a way of remembering what was good and sane and life-affirming about this experience I am having of being part of a worldwide community of math teachers who see teaching as something much larger than what is happening just in our classrooms.

So this is my "One Good Thing" for the day. Thank you, MathTwitterBlogoSphere, for being there on the other end of the Twitter line whenever I need to feel connected.


Wednesday, October 30, 2013

Mathalicious New-Tritional Info Lesson – CC Math 6, units, calorie burn rates, celebrities, and oh, yeah, BURGERS!

DAY 1 - Tuesday

We started Mathalicious' New-Tritional Info today in my Math 6 class. We are not a fast-paced group, so it is going to take us more time to get through the lesson. We spent Tuesday working on Act 1 of the basic lesson.

I would echo everything my friend the inimitable Julie Reulbach had to say about the Mathalicious pacing and student worksheets. As Julie says,
I have to give a shout out to Mathalicious lessons right now.  I’m impressed with the way the student sheets are structured.  The directions are very clear and accessible to students so they can get right to work without tons of questions or further explanation from me.  This allows me to walk around and observe so I can see where my students are and help the ones that are struggling.  The questions also progress in the lesson so that students use their previous work to make discoveries.  This is really tough to do when creating lessons.  Kudos guys!
A couple of students were quicker than others to catch on, so I gave them the off-the-cuff extension idea to research online the calorie "burn rate" values of other activities such as singing (since Selena Gomez was the first example person given in the table). Once we found the burn rate, I divided those by the given poundage of the person performing the activity, and voilà – we had the burn rate per pound per minute. It was rather low – like around 0.0114 calories per pound for 1 minute worth of singing.

Then I had those students go back through the table and figure out how many calories Justin would burn by singing, how many Abby would burn, and how many LeBron would burn. The kids enjoyed imagining LeBron singing and figuring out his calorie-burning for that combination.

You could add on any variety of activities for this extension, such as kazoo-playing, knitting, thinking, etc. It gave the faster students a job to do (finding and converting) and it gave the slower students an incentive to hurry their little butts up on what they needed to focus on (can you spell "sixth graders"???).

A strange thing happened in my second class (5th period). An argument broke out between two factions: one who believed with all their hearts that Larry Bird had come along and stolen LeBron's lunch, and another faction that believed with equal vehemence that it was, in fact, former President Bill Clinton who had skipped out with the lunch at the end of the commercial.

Everyone in this class (except me) was born in 2002, so they probably aren't the most reliable eyewitnesses in the world. But I thought it was interesting how many of them insisted that the lunch-filcher was Bill Clinton.

Anyway, food for thought.

DAY 2 – Block Day

We worked on side 2 of the New-Tritional Info worksheet today. Students were more confident today with the various units, burn rates, unit rate conversions, and corresponding activities. For my classes, this was a good pace.

Tomorrow is our other Block Day, so I'll do the same lesson with my other 6th grade class.

Can't wait to integrate Julie's Desmos ideas and extensions next week!

Sunday, October 27, 2013

Noticing and Wondering as a practice with my 6th graders

When I'm using MARS tasks with my 6th graders, I have found no structure to be more effective in aligning their attentions and energies to the task than the Math Forum's Noticing and Wondering structure.

We kind of go into a "noticing and wondering" mode, in which we are choosing to limit our our monkey mind attention to just plain noticing and we let go of any other kind of attention that comes up during that cycle.

Noticing, in particular, is a quiet and nurturing structure for kids to simply be present with what they notice. We are not privileging noticings or knocking down noticings, we are simply welcoming them as valued and arriving guests.

6th graders love having a structure, so they loved the structure of noticing. Then once we'd heard from everybody,we did a round of noticing. It's powerful when space is allowed to sit with this first round of work.

Look at all these amazing insights they had:

We took a little time to admire this collection. It's a great list!

They even did great a great job of thinking about the value of doing noticing and wondering at the start. The first item from the link is from @fnoschese, whose wisdom even my middle schoolers can grasp.

Doing this task together and then talking about it made students reflect in deep ways about what kinds of growth processes were going on for them.
I just wanted to share this one implementation for anybody who is interested in ways you can use this.

Sunday, October 20, 2013

Using Exeter problems as an intro to algebra tiles

We are an Apple 1-1 school, so I am always happy to figure out lessons my students can use their laptops for.

I also like to use manipulatives in Algebra 1. It's not easy to get all students to accept the need to use multiple representations (such as an area model), but they help enormously to extend kids' conceptual understanding of the distributive property — plus they make a return appearance a lot when we get to the Festival of Factoring in the late winter.

So the National Library of Virtual Manipulatives seemed like a natural fit. But what problems to use to introduce them?

Enter the Exeter Math 1 problem sets.

I have been using the Exeter problems with my advanced 8th grade students taking Algebra 1 almost every week during our Problem-Solving Workshops on block days. Each page is a self-contained "problem set" that builds from simplest principles and often loops back on itself later in the page. This gives students a chance to give themselves a pat on the back for having discovered and developed an intuition for activating their own prior knowledge. I then have them write up one of the problems they solved as a problem of the week to give them practice in blending symbolic and graphical representations with verbal representations (don't forget the verbal representations!). So much Common Core math in such a small span of time!

I will write more about using the Exeter problems as a resource for long-form problem-wrestling with my students, but here I just want to talk about the specifics of introducing algebra tiles.

One of the features of the Exeter problems that we do not get to take much advantage of is the way they build page over page. They will introduce part of a concept or skill in a problem on page 5, say, then introduce the next part of the concept in a problem on, say, page 8. This makes so much sense if you are teaching using one page per day and working through all of them. But for those of us who dip in and nick out once a week, that isn't really possible.

But... by the time we got to around page 21, it dawned on me that I could collect the five problems they use that introduce algebra tiles and put them on a single sheet of paper.

Then I could give that to students during Problem-Solving Workshop, along with a quick intro to the NLVM, to let them teach themselves how to use algebra tiles!

So that is what I did this week.  :)

A couple of programming notes if you want to try this yourself with your students:
1. Start at the very beginning with NLVM and PREPARE FOR TECH HICCUPS
NLVM can be extremely persnickety. This is probably due to some perverse desire to help us cultivate CC standards of mathematical practice #1. Encourage yourself and others to persevere.

Your network may have restrictions on how students can use Java-enabled apps on school equipment. We had some hiccups getting NLVM to run on everybody's system at first. Firefox seems to be the most reliable browser for NLVM. Also, you need to have the most up-to-date version of Java on the student's system.

On our network and systems, students can only update Java by logging out and in again or by restarting their computers. No matter how many times I explained this, some kids still didn't quite figure it out. So much for being "digital natives." Plan to go around to each kid the first time to help them get their systems up and running.

Our system throws up a modal "Security Warning" dialog that forces you to check "I accept the risk" and "Run" before NLVM will load in the browser window. Again, a minor pain in the butt, but you do need to make sure that every kid gets through the security gauntlet to use the system.

Refresh the browser window if need be and be patient which Java and the applet cooperate in loading.
2. Get everybody to the *FIRST* page of the Algebra Tiles site on NLVM
For reasons that pass my understanding, NLVM dumps you into the sixth page of the algebra tiles site (the activity panel on the right, which loads as "Multiplying Binomials - 1").

You need to have students click the leftward-ho button at the top of this right-hand panel SIX TIMES to get back to the first page, which is called "Distributive Law - 1."

This is stupid but necessary because on the first two pages of this site, you can do things you need at the beginning that will quickly drop away as students gain fluency.

For example, the Distributive Law pages are the only ones where you can easily represent both multiplication over addition AND an area addition model in the same window.

3. Familiarize yourself with the syntax of the NLVM Algebra Tiles pages
You'll need to tinker with this a bit, to get comfortable with the syntax of the applet, but there are two essential features of the Algebra Tiles distributive property pages:

4. Click to CREATE tiles in the workspace; drag to MOVE tiles
You can create instances of any of the area blocks that are possible by CLICKING them in the menu bar along the bottom of the workspace. When you click the "x" button, for example, NOTICE that NLVM creates a single instance of a 1-by-x rectangle in the workspace. You can create as many "instantiations" of any of these blocks as you need for any expression you want to represent.

NOTICE that you can drag these critters around in the workspace and add them up, like LEGOs. Or you can drag them into the x-axis tray or the y-axis tray to represent lengths and widths of various area blocks of multiplication.

ALSO NOTICE that you can mouse over the corner of a block in the workspace to rotate it into the position you need.

5. How to show multiplication over addition (i.e.,  how to show  x (y+2) :
The x- and y-axis are basically x- and y-axis "trays" that students can drag tiles into. Drag a 1-by-x tile into the y-axis tray and it creates an x coefficient. Drag a 1-b-y tile and two unit blocks into the x-axis tray and they become the quantity in parenthesis over which your x coefficient will drape itself in multiplication.

NOTICE that as blocks snap into place in the second axis tray you fill, a red area outline appears in the main workspace between the x-axis tray and the y-axis tray.

6. How students can confirm for themselves that area addition and multiplication over addition produce equivalent area values (i.e.,  how to show that  x (y+2) = xy + 2x :
In this window, students can create blocks to fill in this red outline and verify for themselves that the area they get using the distributive property is equivalent to the area they can get using the area addition postulate approach.

Have students click to create blocks and then drag them around to fill the red outlined area perfectly.

My bastardized worksheet of the five Exeter Math 1 problems that introduce algebra tiles and an area model can be found here on the Math Teacher's Wiki.

NLVM Distributive Property pages are here and 6 pages to the left.


Monday, October 7, 2013

Teaching Mathalicious' "Harmony of Numbers" lesson on ratios, part 1 (grade 6, CCSSM 6.RP anchor lesson)

I started teaching Mathalicious' Harmony of Numbers lesson in my 6th grade classes today, and I wanted to capture some of my thoughts before I pass out for the night.

The Good — Engagement & Inclusion
First of all, let's talk engagement. This made a fabulous anchor lesson for introducing ratios. The lesson opens with a highly unusual video of a musical number that every middle school student in America knows — One Direction's "What Makes You Beautiful."

You'll just have to watch the video for yourself to see how the surprise of this song gets revealed.

What I wish I could capture — but I can only describe — was the excitement in the room as my 6th graders realized what song was being played. It took about eight measures for the realization to kick in. Imagine a room full of South Park characters all clapping their hands to their cheeks and turning around with delight to see whether or not I really understood the religious experience I was sharing with them.

Every kid in the room was mesmerized. Even my most challenging, least engaged, most bored "I hate math" kids were riveted to the idea that music might be connected to math. It passed the Dan Pink Drive test because suddenly even the reluctant learners were choosing to be curious about something in math class. My assessment: A+

We started with a deliberately inclusive activity to kick things off — one whole-class round of Noticing and Wondering (h/t to the Math Forum). Sorry for the blurry photography of my white board notes. They noticed all kinds of really interesting things and everybody participated:

From noticing and wondering, we began to circle in on length of piano strings and pitch of notes. This was a very natural and easy transition, perhaps since so many of the students (and I) are also musicians of different sorts. Five guys, one piano, dozens of different sounds, what's not to like?

The Not Actually 'Bad,' But Somehow Slightly Less Good
One thing I noticed right away was that, while the scale of the drawings on the worksheet worked out very neatly, it was kinda small for 6th graders to work with. The range of fine motor skills in any classroom of 6th graders is incredibly wide. At one end of the spectrum, you have students who can draw the most elaborate dragons or mermaids, complete with highly refined textures and details of the scales on either creature. At the other end of the spectrum, you have the students I've come to think of as the "mashers," "stompers," and "pluckers." These are the kids who haven't yet connected with the fine motor skills and tend to mash, crush, or stomp on things accidentally. Some will pluck out the erasers from the pencils in frustration ("Damn you, pointy pencil tip!!!").

This made me want to rethink the tools and scales of the modeling. It might be good to have an actual manipulative with bigger units (still to scale). Cutting things out is a good way for students this age to experience the idea of units and compatible units. Simply measuring and mentally parceling out segments is a little tough for this age group. Ironically, within a year or so, this difficulty seems to disappear. I'm sure there are a lot of great suggestions for ways to make this process of connecting the measurements to the ratios through a more physically accessible manipulative or model. But then again, I'm just one teacher, so what do I really know? My assessment: B

The Not Ugly, But Still Challenging Truth
The most difficult thing about this lesson is that 6th graders go S L O W L Y. Really slowly. My students' fastest pace was still about three times longer than the initial plan.

I am fortunate that this pacing is OK for me and my students. They need to wallow in this stuff, so I will simply take more time to let them marinate. We'll try to invent some new manipulatives for this, and I'll blog about them in a follow-up.

But the reality is that this lesson is going to take us three full periods to get through. They will be three awesome, deeply engaging learning episodes filled with deep connections as well as begging to have me play the video again (Seriously? Three times is not enough for you people???).

Even though this is a much bigger time requirement, I still give this aspect of the lesson an A+. Getting reluctant learners to be curious about something they're very well defended against is a big victory.

I'm excited to see what happens tomorrow! Thanks, Mathalicious!

Sunday, October 6, 2013

Reading aloud in math class — it's a developmental thing

One of the things you would see if you were to observe my mathematics teaching is that we spend a lot of time reading aloud, decoding, and rereading all kinds of texts.

A LOT of time.

We read word problems and problem set-ups aloud, we read instructions aloud, and we read texts by mathematicians and scientists aloud. We read texts and scripts that I have written out loud and we read texts that students have written out loud. We read boring stuff and serious stuff and whatever silly stuff I can sneak into an investigation set-up.

And what this has taught me is that all students need to do a lot more reading aloud in math class.

Research shows that kids whose parents read to them early become fluent and confident lifelong readers. I used to think that only high-poverty kids don't get much opportunity to be read to, but now I can confidently tell you that kids in wealthy communities need this too. Even kids with many advantages whose parents did and/or do read to them early and diligently are often still weak readers by the eighth, ninth, or tenth grades.

And what I have learned is that even well-trained students need and benefit from regular and intensive practice in reading aloud in their math class.

They need more than just primitive "reading comprehension" skills. They need practice in what gets called "textual response and analysis" in the ELA standards, in spite of the fact that these are actually just fundamental literacy skills for anyone who hopes to gain access to the kinds of career opportunities that provide socio-economic mobility.

Reading aloud, like counting circles, is a bedrock practice for students. It is often a great leveler and differentiator in the math classroom. When I announce that I need six readers for the overview we are going to dissect to begin a class, I routinely get three times as many volunteers as that. Even my most discouraged math learners will volunteer to read aloud. Reading is a window into another world. And it's an activity in which every learner can be actively included. It's an equal opportunity invitation into the concert of intention.

And as I often tell my students, much of adult life consists of responding to badly written instructions for ill-defined tasks. Doing your taxes, interpreting and responding to job posting, following your boss' instructions, interpreting conflicting instructions in e-mail sent by customers or clients with competing or downright troubling motivations.

So this is my plea to everybody to consider adding reading aloud as one more practice in your quest to introduce low-barrier-to-entry, high-ceiling problem solving in your classroom.