Proof Portfolios

Over the last five years of teaching proofs in Geometry, I have learned two things: (1) the most effective student understanding comes from writing about their proof process, not from the proving itself, and (2) the most effective feedback process for students is a peer-to-peer reciprocal feedback process.

So this year, when I had to be out of school for a few days, I designed a Proof Portfolio project for them to do in my absence.

Each day had four small, reasonable proofs students had to do — and they could collaborate on these. But then... they had to write a number of short-answer reflections to analysis questions based on their own proofs in the day's set.

In addition, they had to find a peer to trade with and to give a rubric-based peer review and reflection.

In my class, they did this for several consecutive days. I made it worth a quiz/project grade.

When I returned, there was a great deal of wailing and moaning and gnashing of teeth about How Hard This Project Was and How Hard They All Worked.

It was clear that this project was a rite of passage for my classes.

But as I'm reading their work, I am blown away by how much they seem to have learned!

Their mastery of proof is not perfect. But it is authentic and it is growing. And to me, that is the most important point at this stage.

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I made up four days' worth of activities. Each day is two double-sided pages (proofs & reflections).

Here is a link to the G-drive folder with the four PDFs:

https://drive.google.com/open?id=1Mcb-AueXujpiWI2wD1FkuXAGTtWFjKAY

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More photos:

The Festival of Reassessment (SBG)

December is when I am truly grateful for strong routines. They mean I can split the class up and still count on everything moving forward. I am running behind in my pacing, so I needed to set up a mass SBG reassessment opportunity for slope skills yesterday. I set up all the reassessors along the window side of the classroom and all of the non-reassessors on the hallway side of the room. The non-reassessors worked on linear systems skills and problem-based learning while the reassessors worked on demonstrating mastery of slope skills.

Whenever a reassessing student finished their work, they brought it up to be rechecked. I went through it right then and there while they watched. “Uh huh... uh huh... uh huh...” until “Oh noooooo! Why is this negative?!?!? It ruins everything! Go back and fix it!!!” And then the next student moved forward in the queue.

We went through this process of working and my checking and sending kids back for about 40 minutes. “Nooooo!!!” I would circle something and pretend to freak out, sending them back to fix stuff. It became a carnival. “AAACK!" I would say. "Go back and fix this!” Seeing this happen in real-time changed kids’ relationships with their misconceptions. It reminded me of piano practice as a child. Something would splatter in the midst of a phrase or figure and I would have to go back to the beginning, willing the figure to come out right through my fingers on the keyboard.

We kept going until all 12 students had triumphed over slope and point-slope form. What I loved about this process was how it transformed the social focus of the process of understanding to us (the whole class) against the skills. Kids were going wild and cheering when somebody finally triumphed over the "slope and point-slope” process, jumping up and down and hugging their newly non-reassessing classmates and friends.

The goal became one of getting everybody in the classroom over the finish line. From individual mastery to collective. Students stopped focusing on who had status in the classroom and who did not. They stopped thinking about their place in the social hierarchy and instead lost themselves in the flow of doing this f***ing slope and linear equation problem.

And then as each new classmate finally triumphed over the skill, the whole class felt truly victorious.

Everybody got the individualized attention they needed as they drove their skills forward, and we also bonded as a collective community, advancing our work as a group.

In these divisive times, I think this might be an important process to cultivate.

Chords and Secants and Tangents, oh my!

Once you get to circles, chords, tangents, secants, arcs, and angles in Geometry, it's all just a nightmare of things to memorize — which means things to forget or confuse.

Rather than have students memorize this mishmash of formulas, I decided to have them focus on recognizing situations and relationships. And a good way to do that is with a four-door foldable.

Make it and use it.

Humans are tool-using animals. So let's do this thing!

Doors #1, 2, 3, and 4 each have only a diagram on their front door with color-coding because, as my brilliant colleague Alex Wilson always says, "Color tells the story."

Inside the door, we wrote as little as we possibly could. At the top we wrote our own description of the situation depicted on the door of the foldable. What's the situation? Intersecting chords. What is significant? Angle measures and arc lengths. What's the relationship? Color-code the formula.

Move on to Door #2.

We did the same thing with the chord, secant, and tangent segment theorems.

These two foldables are the only tools students can use (plus a calculator) for all of the investigations we have done with this section. They are learning to use it like a field guide — a field guide to angles and arcs, chords, secants, tangents. I hear conversation snippets like, "No, that can't be right because this is an intersecting secant and tangent situation."

My favorite is the "two tangents from an external point" set-up, which we have dubbed the "party hat situation" in which n = m (all tangents from an external point being congruent).

A big part of the success of this activity seems to come from teaching students how to make tools and to advocate for their own learning. Help them make their tools, set them loose, and get out of their way.

That's a pretty good strategy for teaching anything, I think.

Here's an example: how I use Talking Points both before and *for* mathematical conversation

OK, here's an example of how I used Talking Points first to get students primed for listening and considering other viewpoints, and then to get them to listen to and consider other viewpoints that can cause them to change their minds.

As our first activity following our first test of the semester, we did these Talking Points to start class.

These talking points were not especially successful, but they opened the door for the similar triangles discussion that followed.

We debriefed a bit, then I handed out this lovely, subtle activity from Park Math (Book 3, #20), and I asked them to change (a) to become a Talking Point, as in, "Triangle PRQ is similar to triangle STU." They were, as always, charged with doing three rounds and justifying their opinions.

Ten minutes of conversation ensued.

Next, I wrote three headings on the whiteboard (Agree, Disagree, Unsure) and asked each table in turn to tell me which conclusion they had come to and why. One by one, I wrote the table numbers under the categories where they located themselves (Agree, Disagree, Unsure).

And I held my tongue as table after table disregarded the order of vertices to tell me that, Duh, of course, they are similar triangles. I held my tongue because I trusted the process and had a felt sense that in a room full of 37 people, surely SOMEBODY would express a different, correct opinion.

And lo, it came to pass.

Table 6 bravely offered their belief that the triangles named were not similar because the order of vertices in each was not corresponding.

And one by one, the little lightbulb moments popped around the room.

I kept the discussion going until we were through with all 9 tables. Then, and only then, did I give tables another round in which they could change their opinion about what was actually going on in the diagram. 

Afterwards, we discussed what had happened. What did happen, I asked them. And they responded that something they heard made them realize they wanted to change their minds.

So that was my perfectly imperfect day of Talking Points. On the one hand, kids understood (some for the first time) that listening to somebody else could have value for them. On the other hand, many spent most of the exercise not listening to each other and simply waiting for their own turn to talk.

This doesn't mean that it was a failure. It just means it was a first step. 

I believe that if you want students to take ownership of their own learning (and listening... and opinions), then you have to allow space for them to do it in their own perfectly imperfect way. I have found that when I trust the process, I get the best results.

I am posting this to help you understand that every round of Talking Points I do is not a cornucopia of unicorns and rainbows.

WEEK 1 - PROJECT 'MAD MEN' -- Classroom Rules PSA Skits

Leonard Bernstein once said, "To achieve great things, two things are needed — a plan and not quite enough time." I decided to put that principle to the test this first week at my new school by assigning a project on Day 1 that thew together strangers with an absurd but achievable goal: given a particular classroom rule or guideline, create a Public Service Announcement  (i.e., a 30-second "TV commercial" in the form of a skit) whose purpose was to motivate viewers to follow the rules/guidelines for the good of the group.

I created a set-up, instructions, and a rubric for the group project. And my students did not disappoint.

The idea was to get students to think about the consequences of their actions and choices, but their ideas for implementation exceeded even my wildest dreams. Most skits followed a "slice of life" strategy, but the ones that really blew us all away were the ones that parodied existing campaigns.

Two brilliant PSAs started from already-iconic Geico insurance commercials, but the one that left me with tears running down my cheeks was a take-off on Sarah McLachlan's ASPCA spots. The song, "In the arms of the angels..." began playing, and student "Sarah" appears onstage making the exact same kind of appeal she makes in those ads. They had the tone, cadence, and music exactly right, and they clearly understood the emotionally manipulative rhetorical strategy — the seemingly endless list of forms of ignorance designed to eventually provoke self-recognition in almost everyone. Their "mathematical justification" was as follows: the narrator enters and says, "In the past year alone, texting in class tragically cost 5 of Doctor X's students their lives. Remember, think twice before texting in class — there may be fatal consequences for your grade, and for you!

It was pure and inspired genius.

I also loved the spot-on impressions of my teacher persona. One student gave a pitch-perfect parody of my "Function Basics" talk that made me both cringe and laugh my ass off simultaneously.

The best thing about this assignment was that it really pushed the voice of authority downward, into the student community itself. Whatever they made of the experience, they owned it.

I am going to try and remember this for later in the semester, when we've become too routinized.

This is definitely going to be an ongoing part of my repertoire of Day 1 activities. I got through what I neded to,  then gave them the rest of the abbreviated period to collaborate. The time pressure was a thing of art.

It was perfectly imperfect — exactly the way first days ought to be.

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UPDATE:

Here is the link to a generified Word document that you can customize for your own class:

PROJECT MAD MEN- classroom rules PSA generic.doc

And here are the three sample 30-second PSAs I showed my classes to give them ideas:

'You Lost Your Life!' – game show hosted by the Crash Test Dummies (Since Vince & Larry, the Crash Test Dummies, were introduced to the American public in 1985, safety belt usage has increased from 14% to 79%, saving an estimated 85,000 lives, and $3.2 billion in costs to society)
'
What could you buy with the money you save?' - throwing things over a cliff (You could purchase TVs, bicycles, and computers with the money most families spend on wasted electricity)
'Five Seconds' – at highway speeds, the average text takes your eyes off the road for 5 seconds (Five seconds is the average time your eyes are off the road while texting. When traveling at 55mph, that's enough time to cover the length of a football field)

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!

"Wow, what a great question!" or Repeat After Me: A Unit Test is Still a Learning Opportunity"

I have met teachers who refuse to answer questions of any kind at all during tests, and I admit that this puzzles me because from where I sit, those are some of the highest-leverage teaching and learning opportunities I will ever have.

I just hate to waste them.

For they are the moments when I have my students' complete and undivided attention.

And that means they are my best hope for encouraging and guiding students in the process of productive struggle.

During a test, I will happily entertain anybody's question about anything. Really. Ask me anything. I will gladly offer encouragement and encourage their courage because for many students, THAT is the moment at which they are most deeply engaged and present in the process of struggling with their learning.

But I am apparently the most frustrating person in the world because the best answer I will ever give students is to say, "THAT is a GREAT QUESTION!"

I smile and nod and encourage them and urge them to keep going. And at first, they really think they hate me for it.

During today's Math 8 test, kids kept asking questions and I kept answering, "That is a GREAT question! What a terrific insight!" and leaving to move on to the next questioner. "But is this RIGHT?" They would ask, sounding wounded. And I would say, "You are asking a FANTASTIC question! Keep going!" and move right along.

Finally somebody thought to ask me in a tiny and supplicating voice, "Dr. S, is THIS a good question to ask?" And I peered over and looked at their paper and exclaimed, "Yes! That is a super-fantastic question!"

I was certainly the most annoying person in the room, but they are starting to catch on to this whole productive struggle business. Eventually it became a humorous trope. "Oh, yeah — don't bother asking. I'm sure that's a really GREAT question."

To which I would chime in, "Yes — it really IS a super-great question!"

Don't get me wrong — this is NOT an easy thing to do. It takes strength and practice and intestinal fortitude. It will never be featured as "great classroom action." But it is the most precious and valuable thing I know how to offer my students.

I can say this because I have also been on the receiving end of this kind of teaching. It is a teaching about the value of struggle. It is an incredibly precious gift, but nobody can ever explain it to you. You actually have to experience it in order to understand what a profound act of respect it is for the primacy and centrality of your own personal experience to your own personal learning.

I spent plenty of years complaining bitterly about meditation teachers who practiced this kind of bounded containment. But I sat with it. I stayed with it. I learned first to accept it, and then to embrace it. After a lot of struggle, it humbled me. It changed me in ways big and small. It opened my heart and empowered me to discover the value of struggling within my own life's journey, as well as in subjects like mathematics. That kind of preciousness and wholeheartedness is all too rare in America, but I hope that all human beings will at least taste it at some point in their lives.

So even though it was in some ways a crappy day and a frustrating day and an exasperating day, it was also a kind of gift day too. I want to remember that.

Substitution with stars

This one is for Max, who asked about it on Twitter, and for Ashli, who interviewed me for her Infinite Tangents podcasts.

As Ashli and I were talking about some of the struggles we see as young adolescents make the transition from concrete thinking to abstraction, I mentioned substitution.

For many learners, there comes a point in their journey when abstraction shows up as a very polite ladder to be scaled. But for others (and I count myself among this number), abstraction showed up as the edge of a cliff looking out over a giant canyon chasm. A chasm without a bridge.

This chasm appears whenever students need to apply the substitution property of equality — namely, the principle that if one algebraic expression is equivalent to another, then that equivalence will be durable enough to withstand the seismic shift that might occur if one were asked to make it in order to solve a system of equations.

Here is how I have tinkered with the concept and procedures.

Most kids understand the idea that a dollar is worth one hundred cents and that one hundred cents is equivalent to the value of one dollar. I would characterize this as a robust conceptual understanding of the ideas of substitution and of equivalence.

One dime is equivalent to ten cents. Seventy-five pennies are equivalent to three quarters. You get the idea.

We play a game. "I have in my hand a dollar bill. Here are the rules. When George's face is up, it's worth one dollar. When George is face down, it's worth one hundred cents. Now, here's my question."

I pause.

"Do you care which side is facing up when I hand it to you?"

No one has yet told me they care.

"OK. So now, let's say that I take this little green paper star I have here on the document camera. Everybody take a little paper star in whatever color you like."

Autonomy and choice are important. I have a student pass around a bowl of brightly colored little paper stars I made using a Martha Stewart shape punch I got at Michael's.

Everybody chooses a star and wonders what kind of crazy thing I am going to have them do next.

We consider a system of equations which I have them write down in their INB (on a right-hand-side page):

We use some noticing and wondering on this little gem, and eventually we identify that y is, in fact, equivalent to 11x – 16.

On one side of our little paper star, we write "y" while on the other side, we write "11x-16":

I think this becomes a tangible metaphor for the process we are considering. The important thing seems to be, we are all taking a step out over the edge of the cliff together.

We flip our little stars over on our desks several times. This seems to give everybody a chance to get comfortable with things. One side up displays "y." The other side up displays "11x–16." Over and over and over. The more students handle their tools, the more comfortable they get with the concepts and ideas they represent.

Then we rewrite equation #1 on our INB page a little bigger and with a properly labeled blank where the "y" lived just a few short moments ago:

"Hey, look!" somebody usually says. "It looks like a Mad Lib!"

Exactly. It looks like a Mad Lib. Gauss probably starts spinning in his grave.

"Can we play Mad Libs?" "I love Mad Libs!" "We did Mad Libs in fifth grade!" "We have a lot of Mad Libs at my house!" "I'll bring in my Mad Libs books!" "No, mine!"

It usually takes a few minutes to calm the people down. This is middle school.

I now ask students to place their star y-side-up in the blank staring back at us.

Then it's time to ask everybody to buckle up. "Are you ready?"

When everybody can assure me that they are ready, we flip the star. Flip it! For good measure, we tape it down with Scotch tape. Very satisfying.

A little distributive property action, a little combining of like terms, and our usual fancy footwork to finish solving for x.

Some students stick with substitution stars for every single problem they encounter for a week. Maybe two. I let them use the stars for as long as they want. I consider them a form of algebraic training wheels, like all good manipulatives. But eventually, everybody gets comfortable making the shift to abstraction and the Ziploc bag of little stars goes back into my rolling backpack for another year.

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I'd like to thank the Academy and Martha Stewart for my fabulous star puncher, without which, this idea would never have arisen.

I wore out my first star puncher, so I've added a link above for my new paper punch that works much better for making substitution stars. Only eight bucks at Amazon. What's not to like? :)

Sometimes I teach, and sometimes I just try to get out of the way...

We are in the midst of our giant 8th grade culminating assessment extravaganza — a multi-part project that includes a research paper, a creative/expressive project, a presentation with slides, and several other components I'm spacing out on at the moment.

I have to admit something here: I used to be an unbeliever when it comes to projects.

I used to think they lacked rigor and intellectual heft.

But I was wrong.

Two years of this process has made me a believer in the power of project-based learning.

Sometimes the creative projects are merely terrific, but every year, there are a few that are incredible. This year, this has already happened twice... and only two projects have been turned in so far (they are due on Monday, 22-Apr-13).

Sometimes it is the quiet, timid kid who really blows my mind. Sometimes it is a kid who is kind of rowdy who reveals another, hidden side. But I never fail to be humbled at the potential inside each of these people, and I am honored to teach them.

So this is a reminder to myself that sometimes my job is simply to get out of their way.

And this is why I teach...

It was another crappy Friday in an arithmetic series of crappy Fridays that were running together and threatening to define the limit of my patience for fall trimester as x approaches a mid-sized number that is nowhere near infinity. So I have no idea what possessed me to wake up even earlier than usual to pull together an extra day's practice activity for my right-after-lunch class of rumpled and discouraged algebra students — the ones who believe to their core that California's Algebra 1 requirement is God's own punishment for unremembered karmic crimes they must have committed in previous lifetimes.

But I did it.

The topic was solving and graphing compound inequalities — a skill set that must be mastered in order to have any hope of making sense of and mastering the next topic traditional algebra curricula force-feed to our students: the dreaded topic of absolute value inequalities.

There's really nothing I can say to convince a roomful of skeptical eighth graders that compound inequalities will prove not only useful in business planning (which, after all, is simply algebra writ large across the canvas of the economy) but also amusing and possibly even interesting little puzzles to delight the mind.

To this group of students, they're simply another hoop to be jumped through.

So something in me understood that I needed to reframe the task for them, and to do so using Dan Pink's ideas about intrinsic motivation from his book Drive.

Nothing unlocks the eighth grade mind like an authentic offer of autonomy. As I explained recently to a room of educators at a mindfulness meditation training, middle school students suffer emotionally as much as adults, but they have comparatively little autonomy. A little well-targeted compassion about this can carry you for miles with them, though I usually forget this in the heat of working with them.

For this reason, I like to save practice structures such as Kate Nowak's Solve—Crumple—Toss for a moment when they are desperately needed. I have learned to withhold my Tiny Tykes basketball hoop for moments like this, when students need a little burst of wonder in the math classroom. And so even though I was tired and very crabby about the ever-increasing darkness over these mornings, I pushed myself to pull together a graduated, differentiated set of "solve and graph" practice problems to get this group of students over the hump of their own resistance and into the flow experience of practicing computation and analysis.

And oh, was it worth it, in the end.

The boys who are my most discouraged and resistant learners came alive when they understood that a little athletic silliness was to be their reward for persevering through something they considered too boring to give in to. They suddenly came alive with cries of, "Dr. X— watch this shot!" from halfway across the room. One boy who can rarely be convinced to do the minimum amount of classwork completed every problem I provided, then started tutoring other students in how to graph the solution sets and perform a proper crumpled-paper jump shot.

The girls in the class got into it too, but they seemed more excited about the possibility of using my self-inking date stamp to stamp their score sheets. So I gladly handed over the date stamp to whoever wanted to stamp their own successfully solved and graphed inequalities.

I was far more interested in reviewing their mathematics with them. One of the things I love best about practice structures like this one is that they give me an excuse to engage one on one with discouraged students under a time crunch pressure that adds a different dimension to their motivation. Suddenly they not only want to understand what they have done, but they want to understand it quickly, dammit, so they can move on to another problem, another solution, another graph, another bonus point.

Ultimately, Solve—Crumple—Toss becomes an occasion for conceptual breakthroughs in understanding.

I can't tell you why this happens. I can only tell you that it does happen — often. It makes me feel lighter, more buoyant about teaching them algebra. And it makes them feel happier too.

I wanted to write this down so I could capture it and remember this for a few weeks from now, when it stays darker even longer in the mornings and when I feel crappier and crabbier and more forgetful.