cheesemonkey wonders

cheesemonkey wonders

Wednesday, May 17, 2017

Take Time to Save Time – Hall of Fame reference sheets

Inevitably, teachers get known for their mottos. Sam's mottos are justifiably world-famous. Personally, I love "Don't be a hero." Mine are known mostly around my school, but it is interesting to see how they trickle down into students' unconscious minds.

Color telling the story
Mottos pay off. My favorite is one I stole from my former colleague Alex Wilson: "Color tells the story." I don't understand how anybody can do math at a deep conceptual level without colored pencils. Color really does tell the story, especially in Geometry (see popular worked example at right).


One of my best math class mottos comes from published patterns for knitting. It is, "Take time to save time." In knitting, this means to make sure that the tension of your actual knitted work — your hands, your needles, your yarn — match the tension or gauge described in the knitting pattern. There are no shortcuts here. My knitting gauge tends to be extremely big or loose compared to most pattern-makers. I often have to use much smaller needles than specified in order to achieve a good match with the specified knitting gauge.

In my classroom, "Take time to save time" means, synthesize your learning into a reference sheet. For all tests but the final, I allow students to have and make a half-page reference sheet.  The first rule is, you can have anything you want except a photocopy of my work on your reference half-sheet. The second rule is, if you have more than a half sheet of 8.5 x 11 inch paper, then I get to tear it in half and choose which half you get. This rule gets tested even when I emphasize it. Every year somebody tests this rule. "But Dr. S! I only wrote a half-page worth of stuff on the paper!" It doesn't matter. I usually rip the whole thing lengthwise so they only get the right-hand half of the paper.

It makes its point.

In knitting, this point gets made by the scale and size of your finished object. If you insist on not checking your gauge, at some point, you will end up with a finger-puppet-sized sweater or a scarf the size of Lake Tahoe.

Clearly this student is going to ace the final.
In our classes, this point gets made by your performance on our common final exam. Students who have been practicing making clear, concise, summaries and examples of their work and key points tend to turn in consistently strong performances. So on the final, I allow a full-page reference sheet (both sides). I emphatically want students to consolidate their understanding and create their own examples. That is where the learning happens.

So I was thrilled today when I asked to see examples of in-progress reference sheets. Many of them made my Hall Of Fame request to scan for posterity. This Algebra 1 student has totally nailed her understanding of mixture problems. This is the best example I've seen of a student consolidating her understanding of these modeling challenges.


Thursday, December 1, 2016

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.

Tuesday, November 22, 2016

Blogs feed my teaching soul, but Twitter helps me feel less alone

As I've been working on my National Board certification, I have come to appreciate something that the #MTBoS gives to me every single hour of every single day.

My tweeps' teaching blogs feed my teaching soul. Seriously, any time I feel stuck, I can just fire off a search in the MTBoS search engine and BOOM, I get drenched in a downpour of borrowed genius.

But in the moment, when I am freaking out or feeling lost, Twitter makes me feel ever so much less alone.

So thank you, my tweeps, and keep going. I am thankful for you Every. Single. Day.

Sunday, October 23, 2016

Graphing Stories Meets Estimation 180: A Love Story

On Friday we started our Functions unit, and as always, my go-to is to start with Dan Meyer's Graphing Stories.

After they pick up a Graphing Stories worksheet, I give them a brief set-up, explain that the guy in the video is my friend Dr. Meyer (though now I have to explain that this was a long time ago, in a galaxy far, far away, when he was a young Jedi-in-training), and promise them that I will rewind the videos as many times as they want, to whatever point they want, for as long as they want, so we can figure out as closely as possible what their graphs ought to look like.
screen shot of Dan leaning over the railing of the footbridge
Portrait of the artist as a young Jedi warrior
Only this time, we encountered a spontaneous twist.

The first video went off without a hitch. Using QuickTime, I have an action-only version of the first 15-second video (Dan walks over a bridge in a Santa Cruz park), so that it doesn't accidentally reveal anything I'm not ready to reveal yet.

I explained that my friend Dr. Meyer is unusually tall (I gave his actual height, which can be found on his web site) and this gave students the idea to "measure" him onscreen so they could try to better estimate the rise of the bridge.

This gave me an idea.

The second video in the series is of Dan descending some exterior condo stairs, but after the first viewing, an argument broke out in the discussion as they tried to find some hook they could use to improve their estimations. "A car is, like 6 feet high!" "No, the stairs are about 5 feet in total!" Blah blah blah.

Enter Estimation 180 thinking.

My classroom is right next to the door to the stairs up to the small faculty parking lot. I pointed to two kids. "OK, you and you — take a yard stick, carefully cross the driveway, and go measure some average car heights in the teachers' parking lot."

The arguments continued so I pointed out another kid, one who was extremely interested in the stair height but who has never before piped up in class. "Pick a helper, grab a yard stick, and go out to the stairs and find an average for the stair height."

While our data gatherers performed their missions, the estimation arguments continued inside the classroom. "The car is only about 4 feet high!" "No, it's not!" "What about the lamp post—can we use that?"

Five minutes later, our intrepid explorers returned and we harvested our newfound information.

Then we continued working on the Graphings Stories task.

The funny thing was that our estimates led to wildly wrong answers, but that wasn't the most important thing. Instead, what was powerful was the level of engagement in discovery that electrified the room.

Instead of considering the need to quantify to be yet another tedious task that stood in the way of getting "the right answer," students started to lose themselves in the flow of the process of mathematizing their world.

And isn't that the whole point?

So thank you to Dan (@ddmeyer) and Andrew (@mr_stadel) for giving me the tools to slow my kids down and help them to find the wonder in everyday situations.

PS — I still haven't revealed the "answer" to Graphing Story #2 yet. But I'm curious to see what unfolds in class when we come back tomorrow. ;)

Wednesday, October 19, 2016

Scaffolding Proof to Cultivate Intellectual Need in Geometry

This year I'm teaching proof much more the way I have taught writing in previous years in English programs, and I have to say that the scaffolding and assessment techniques I learned as a part of a very high-performing ELA/Writing program are helping me (and benefiting my Geo students) a lot this year.

I should qualify that my school places an extremely high value on proof skills in our math sequence. Geometry is only the first place where our students are required to use the techniques of formal proof in our math courses. So I feel a strong duty to help my regular mortal Geometry students to leverage their strengths wherever possible in my classes. Since a huge number of my students are outstanding writers, it has made a world of difference to use techniques that they "get" about learning and growing as writers and apply them to learning and growing as mathematicians.

We are still in the very early stages of doing proofs, but the very first thing I have upped is the frequency of proof.  We now do at least one proof a day in my Geometry classes; however, because of the increased frequency, we are doing them in ways that are scaffolded to promote fluency.

Here's the thing: the hardest thing about teaching proof in Geometry, in my opinion, is to constantly make sure that it is the STUDENTS who are doing the proving of essential theorems.

Most of the textbooks I have seen tend to scaffold proof by giving students the sequence of "Statements" and asking them to provide the "Reasons."

While this seems necessary to me at times and for many students (especially during the early stages), it also seems dramatically insufficient because it removes the burden of sequencing and identifying logical dependencies and interdependencies between and among "Statements."

So my new daily scaffolding technique for October takes a page from Malcolm Swan and Guershon Harel (by way of Dan Meyer).

I give them the diagram, the Givens, the Prove statement, and a batch of unsorted, tiny Statement cards to cut out.

Every day they have to discuss and sequence the statements, and then justify each statement as a step in their proof.

This has led to some amazing discussions of argumentation and logical dependencies.

An example of what I give them (copied 2-UP to be chopped into two handouts, one per student) can be found here:

-Sample proof to be sequenced & justified

Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts! The conversations about intellectual need have been spectacular.

I am grateful to Dan Meyer for being so darned persistent and for pounding away on the notion of  developing intellectual need in his work!

Saturday, October 1, 2016

Algebra 1 inequalities unit - notes on a conceptual and problem-based approach

My Algebra 1 inequalities unit is now the unit with which I am most satisfied, including a solid conceptual launch and framework, some serious problem-based learning, a deep treatment of quantitative reasoning and logic, and an excellent amount of discourse-rich practice activities designed to achieve both procedural and conceptual fluency, so I want to document here how it works. If there is useful stuff in it for you, then great! But I'm not going to be posting a lot of user-ready materials here to print off and give to students, so I want to be totally up front with you.

Inequalities are still one of the most procedurally taught of all Algebra 1 topics (see Holt Algebra 1, CPM, Exeter, or anything else you can find, if you have any doubts about this), and yet, it has seemed to me for a long time that they offer the opportunity for students to ground their new understandings in one of their most accessible, intuitive existing mathematical understandings — namely, their sense of when a quantity is "more than" or "less than" some other quantity.

As Nunes and Bryant clearly show, children have a very deep understanding of comparing quantities from a very young age — including both discrete and continuous quantities. For this reason, it has long seemed essential to me to hook Algebra 1 students' work with understanding with their own authentic and meaningful prior knowledge.

So in my unit,we start by activating prior knowledge with Talking Points and problem-based learning from the Exeter Math 1 sequence. In the initial part of the unit, we investigate whether quantities are going to be more than or less than other quantities. We go from comparing known, computable quantities (such as one sum or difference to another sum or difference) to more abstract quantities (such as, If x is greater than -1, then x + 2 is greater than 1), always pressing them to rely on their own understanding or constructed understanding.

The conversations that were sparked during the initial phase — both this year and last year (with Patrick Callahan in my room) — were fascinating to all.

My rule for this unit is this: whenever I give students a "rule" (or whenever we develop a "rule" together),  it has to be a guideline that will help them to ground their new ideas in their old understanding of something. This has led me to come up with much more sensible and conceptually-based guidelines that students both remember and rely on correctly. For example, one of our "rules" is that when you have some kind of inequality, it makes sense to put it into "Number Line Order" — in other words, always organizing statements or representations about smaller quantities to the left of statements or representations about larger quantities. This has had the benefit of getting students to realize that the real number line is a tool that they can use for their own benefit. It is not just another arbitrary math teacher whim.

Another "rule" we co-developed as a community was the idea of the subjectivity of x  namely, that in mathematical sentences and statements, we should organize our symbolic representations so that x is the subject of our sentences. Another way that students expressed this idea was that x is the hero or heroine of our story, so x should be the subject of our sentences.

This too has had a profound effect on students' understanding of inequalities. Most textbooks emphasize equivalence of the statements 2 > x and x < 2, but from the perspective of student understanding, privileging both statements equally is just stupid. What we are hoping for students is that they will see that as they are trying to represent values of x, that makes x the hero of our story. So our mathematical statements should be organized to reflect the subjective status of x — not the "correct" or "incorrect" equivalence of the statements. This gave students the idea that they have permission to change statements around to suit their own  learning, and that they do not need to distort themselves to please whatever mathematical authority decided to place given statements in a form that is backwards for them.

We have also developed a "rule" for distinguishing absolute value expressions from other groupings in equations and inequalities. The idea we have developed is that absolute value expressions need to be "isolated" and "unpacked" into all possible cases of ordinary equations or inequalities (i.e., NON-absolute value statements) before you can apply any solving or simplifying techniques. Our shorthand for this method is: 1 - ISOLATE, 2 - UNPACK, 3 - SOLVE. Then once you have solved, you should put everything into Number Line Order so you can INTERPRET and GRAPH your findings.

This idea of unpacking absolute values first into all possible cases is a strong way of getting students to reason quantitatively and abstractly rather than blindly applying rules. Since they have some guidelines that are grounded in their own logical understanding and personal experience, my students have been much more thoughtful and intentional when they approach absolute value inequalities and equations. It has also led to a deepening of students' understanding of when the distributive property can and should be applied to a grouping and when it should not. This has greatly deepened the discussions of the functioning of groupings among students to the better.

Because we have build a solid conceptual foundation for our work with absolute values, practice activities have become opportunities for investigation and application of our conceptual understanding, rather than blind shooting at a list of targets from 1 to 35 odd.  At every step of the way, we have been using Speed Dating as a discourse-rich activity in which students can apply their understanding to a wide range of different problems of varying levels of complexity. It also demands that they share their understanding and speak about it with their classmates, sometimes giving and sometimes receiving assistance.

In this way, practice activities have become a new form of conceptual investigation themselves, and are not mere mental exercises designed to enhance procedural efficiency. As I am discovering through my work on National Board Certification, our profession has a major problem with privileging "discovery" activities as the only ones that occasion and showcase conceptual understanding. As How People Learn clearly lays out, it is often the case that students do NOT have a 'Eureka' moment during the initial 'discovery' segment but rather, like the young learners they are, they have a lightbulb moment once they have experienced a problem in multiple different contexts. This leads me to one of my pet peeves about why teacher certification and National Board Certification overvalue only the initial discovery activity as a venue for showcasing student insight, but that is a complaint for a different blog post.

Another benefit of treating practice as an opportunity for applied investigation is that it avoids the contrived, dog bandana pseudo-context problems that strive to provide a real-world context, but only end up twisting themselves into senseless knots. The contexts should not be harder for students to make sense of than the absolute value problems themselves! The most accessible real-world absolute value problem we have worked with is the question, When is water NOT a liquid? This makes sense to my 9th graders. Convoluted problems about the conditions under which a value will land within a range of possible values are not helping here, people. I am finding that this is a case where treating absolute value problems as a system of equations or inequalities to be unpacked and investigated as number sense problems has been far more worthwhile — and far more rewarding for students. As Deborah Ball has said, Sometimes mathematics is its own best context.


Wednesday, September 7, 2016

Katamari and Speed Demons and Not-Knowing

I want to keep track of how I am learning from my speed demons and katamari in Algebra 1.

I want to be clear about two things. Not all speed demons are alike. And also not all katamari are alike.

Slowly I am finding my way to the most discouraged and shut-down learners. I group and regroup, based on formal and informal assessments. Also based on intuition. The great thing about using PBL and the adapted Exeter problem sets is that together they are giving me a lot better information and opportunities to identify, target, and support my students. Most of what we do in class is problem-solving in groups — interpreting, reasoning, and making sense using group whiteboards and whatever other tools we need.

I notice the need to provide much more intensive support to most discouraged katamari than I had expected. These are the most shut-down of my learners. These are the students who are being quiet to avoid being noticed. In math class, they live in a defensive psychological crouch. I know this posture well. But the only way I can understand it is to join the process and probe.  I ask questions: when you are AT such and such, how far away are you?

I am really shocked at how shut down and disassociated  these students are from their intuition and their natural intelligence. I wonder how long they have been cultivating this posture of hiding.

I draw a crude map on my paper and point to different positions. When you're here *pointing to the diagram*, are you further or closer to X? *moves pencil closer to destination* What about now? what about now?

By probing, I discovered something amazing — three tables of students didn't really understand what it meant to be AT the destination.

I tried different tactics until finally I asked, when you're IN your kitchen at home, how far away are you from your refrigerator? How many miles away are you? It took a while to convince them that they actually knew they were ZERO miles away.

Once they figured out what the meaning of being AT someplace or crossing the destination was/is, they could begin to build a table. One step after the other. Let's work backwards -- when you're one hour away, how many miles away are you?

This is a lack of basic trust in their own innate natural functioning. A lack of trust in their own intelligence and problem-solving skill. It takes skill to get this going. They have to be prompted/encouraged to start from where they're at — not to hide the fact that they are lost.

These students are good hiders. They know how to hide in plain sight. They are skilled spies. They know how to evade detection. I am more tenacious than most. I do not accept evasion. I probe, I question, I support, and if need be, I help. I help because they are too shut down emotionally to take the risk of humiliation. They are exhausted from not-knowing and from hiding their not-knowing. But not-knowing is the start of knowing. You can't begin to know until you know that you do NOT know.

Not-knowing is an empty space in which knowing can arise.

Katamari don't realize this yet, but they are closer to finding out. Speed demons have no clue about it yet, and they are so far away from discovering it there's not even any point in raising the issue yet. Katamari are so much closer to the truth. I don't mention any of this to any of them. I just keep probing and asking questions and asking what if and how much and how do you know.

At the end of class, one of my speed demons told me she would really like to collaborate (unlike the other speed demons at her table, who are just zooming along and only occasionally conversing). She asked if she could move to Table X, where some very discouraged Ss are. These are the students I was probing the most with. I was thrilled. She was eavesdropping on us when I was working with them. These are the questions she herself likes to ask and think about.

I don't have any conclusions to offer, just these noticings and wonderings.

We are still in the early community-building days, when discovery is young.