cheesemonkey wonders

cheesemonkey wonders

Monday, December 21, 2015

Burning Questions

The great psychologist and inner development teacher A.H. Almaas is one of my favorite authors of talks that make me think hard and clearly about what I do as a teacher and why I am doing it.

Because our semester just ended and I will have all-new classes in the new semester, I have been reflecting on how I'm dealing with homework. This is the first time I've ever felt good about my homework strategy — not only because it is working but also because it is aligned with something fundamental that Almaas writes about in his talk on "The Value of Struggling," in his book Diamond Heart Book One: Elements of the Real in Man:
  When you have an issue in your life, the point is not to get rid of it; the point is to grow with it. The point is not just to resolve the issue; the point is to grow through resolving it. So in many ways, you can see that maturity has to do with this growth, this broadening, this depth. (p. 128)
In my classes, this is the point of having homework and of doing homework; so the same should be true with the way in which we deal with points of struggle.

For this reason, the most important part about homework and homework review in my classes has become what I call burning questions. As Almaas says,
   In terms of working here, the question you bring to your teacher has to be a burning question. If you have a feeling one day and don't understand it, don't run to your teacher saying, "I was walking down the street, and this person said such-and-such to me and I felt scared. Why was I feeling scared?"  That is not a burning question. (pp. 127-8)
So for the first four minutes of class, while the intro theme music is playing and while the countdown timer counts down on the screen, students' job is to compare answers and methods in their table groups and to explain to each other anything they can to work through their routine questions and problems with the homework.

Their other goal is to identify any burning questions that they can not answer for themselves or each other. I tell them explicitly and repeatedly for the first two weeks of class that I will only take burning questions — in other words, "group questions" that they can confirm that they cannot answer for themselves.

This, I believe, is the most important classroom cultural thing I establish about homework review. In my classroom, homework review is not the place where you should collapse like a helpless baby and expect me to take your problems away. Homework review and questions to the teacher should be the place to bring your burning questions, which can help you to struggle better and to get that last little boost you need to work your way up to the next level of understanding:
   Respect your issues, grapple with them, struggle with them.  When an issue comes up, involve yourself in it, observe, pay attention, be present, understand it as best you can, using all the capacities you've got. Then, if the issue is hard for you to understand and you can't get through it and the fire is burning inside you, come and ask the question. It is that question which is the best question to ask a teacher. It is the right use of the teacher. When you ask that question, deal with it, and come to understand it, you will undergo a transformation that is not possible otherwise. Then you can take the realization and digest it, absorb it. But if you tell me to give you the enzyme and you haven't digested anything on your own, how are you going to absorb it? It's like trying to absorb big lumps that haven't been thoroughly chewed. No matter how much enzyme we put in, you'll probably only get a stomach ache.  (p. 129)
After the first two weeks of hammering this procedure and prioritization home, I have found that students really take more ownership of their own learning. They seem to better understand how I expect them to mature as learners in my classroom. And they come to appreciate — and ask — really deep and meaningful questions.

Sunday, November 22, 2015

On using privilege to combat racism: a love letter to #educolor, from an aspiring ally

It has been inspiring this past week to watch my young fellow Princetonians confront the legacy of institutional racism that is quite literally etched into stone at my well-intentioned, sometimes clueless, but deeply beloved-and-worth-improving Princeton (link).  Cornel West captured my feelings well with his support.  It is possible to love an institution and, because we love it, want it to grow and improve our society and our world.

It has also been quite moving to watch my Princeton classmate, University President Chris Eisgruber, as he wrestles with these issues in the public eye, working through layer upon layer of unconscious white privilege and commitment to anti-racist education. It has been impressive to see him come through it with open-mindedness, wholeheartedness, and a willingness to listen deeply, responding with integrity, and widening our commitment to inclusiveness at an institution that has not always supported inclusion. This is what I consider to be "Princeton in the Nation's Service."

So it was utterly disheartening to wake up this morning to a hate-filled screed on our Princeton Class of 1983 Facebook page from a different classmate of ours — a white woman who is a hedge fund manager on Wall Street. She was a leader at one of the hedge funds that nearly destroyed our country's economy. She and those she worked with have never been called to account for their crimes.

But first, a warning. Please note in advance that I strongly condemn this kind of hate speech. But I believe that hate speech needs to be called out because I believe it has no place in the power structure, much less in civil discourse. I also believe hate speech deserves no shielding or privacy. I wanted to capture these publicly-expressed pieces of hate speech before she could think better of it and delete them.

She wrote, "Churchill: 'You were given the choice between war and dishonor. You chose dishonor, and you will have war.'"

The she linked to the following truly reprehensible article (WARNING: this article contains contemptuous and morally disgusting attitudes that are blind to their own privilege. You may be as sickened as I was when you read it. I strongly condemn this hate speech): hate speech article link

Another classmate pushed back against this right away, writing,
as Eisgruber said “we should be aiming for a campus in which all students feel equally welcomed.” Commentary like the above and related blog posts are unwelcoming and also inaccurate (for instance saying all the students protesting at PU were black). There are many students, and people, of diverse colors and backgrounds, who support taking a hard look at campus life and assumptions. At PU there is a high value for tradition and a high value for making changes that make the educational experience the best that it can be.
But this woman kept on going with her racist rants. She lashed back.
I am quite prepared to believe that the BLM hysterics come in all colors. Their insistence on Maoist reeducation of their peers is a uniform pink.
And further:
Nothing says "welcome" like a mandatory Maoist reeducation program. Unless it's a building that you cannot access due to the color of your skin. Not to mention the black students who may not want to self-segregate -- if there is a component of pure evil to this profoundly racist and anti-educational movement, it is the pressure it will put on sane black kids to conform to the madness of the Maoists.
I could not believe my eyes, except that I spent four years with this woman and her entitled, privileged bullshit, so this was not the first of her objectionable ravings that I have been subjected to. Still, we are supposed to be older and wise. But apparently not everybody is actually committed to growing up.

The classmate who pushed back against her ravings wrote back:
That's your view and I am unlikely to influence it. However, I am quite prepared to think that the outcome will be orange and black, and not pink. And that there are apparently "hysterics" in various quarters. Just sayin.
This racist with a Princeton education could not stop herself, so she went on:
Except that if you're orange you can't get into the new "cultural" Affinity building.
And on:
Maybe Princeton should be renamed "Wilson University" to honor the new segregationism.
I couldn't take it any more, so I posted a reply objecting to her hate speech. But predictably, she screeched right back at me:
I hope that nobody would be surprised to find me pro-First Amendment and anti-Maoist. But you are welcome to a participation trophy anyway.
This is the voice of someone who benefited from a world-class education, as well as from our open, inclusive, and welcoming immigration policy. It causes me a deep and lasting sadness that these are the values she took away from these uniquely American opportunities and institutions I hold so dear. 

When people reveal their true values in public, it is important to document and witness their doing so. It is also important not to let this kind of evil go unanswered. The witnessing function is one of the most important roles of an ally. So I am doing my best to do so here, however imperfectly and stupidly I may be doing it. I continue to grow and learn from my #educolor colleagues on Twitter and on blogs. And I am training my students for their roles as the leaders of the rebel forces.

As the Buddha said, "Hatred never ceases through hatred but by love alone is healed. This is an ancient and eternal law."

And as Michelangelo wrote on a scrap of paper left behind in his studio in the wobbly handwriting of his old age, "Ancora imparo" — "I am still learning."

Thursday, November 19, 2015

A Modeling Manifesto — join the sense-making revolution

I have HAD IT with bogus PD.

As usual, there was a lot of warm and fluffy language about finding things that different students had shown us in their pieces of worksheet work that they are "smart at," but nobody mentioned the elephant in the room — namely, that students were not demonstrating any form of active engagement with the actual problem situations at hand.

Instead, all the pieces of student work that had been lovingly curated and assembled by respected professionals to showcase student smarts merely showed students blindly whipping out one procedural method or another and just guessing. Not guessing and checking, mind you. Just guessing and getting stuck. Or plugging values into the quadratic formula but forgetting where the negative sign goes. Or trying to factor or complete the square, but making minor but fatal computational errors.

Where, I kept trying to ask, is the sense-making???

Now, don't get me wrong — I know that kids are smart. Very smart. And these kids clearly understood that they were being measured on whether or not they slapped numbers into the quadratic formula or came up with enough lines' worth of symbols to create the appearance of the desired forms of problem-solving.

Unfortunately, that also meant that they clearly understood that, if they had taken the time instead to write out words, phrases, or whole sentences; to challenge the diagram by drawing a different diagram; or try to figure out what variables in the text go with what parts of the diagram, they were going to get marked down. A lot.

These thoughts left me wondering, where are these kids being challenged to engage in sense-making?

This has been on my mind a lot lately because over the past week, we've been working on systems of linear equations in Algebra 1. Textbook explanations simply suck. They are vast, multi-colored makeup experiments on dogs wearing bandannas, with slashes of red and blue arrows pointing everywhere and nowhere.

Textbook explanations do active harm to student reasoning and sense-making.

We needed to escape from the textbook.

I did a lot of thinking and reflecting and being willing to ask stupid questions over the course of a week and I came to a few important realizations. First of all, there are only three main categories of linear systems problems that show up to torment Algebra 1 students: (1) upstream/downstream problems, (2) mixture problems, and (3) number and digit problems.

So I decided we would take some time to work our way through the sense-making aspects of each of these types of problems. If nothing else, at least my students would have some skills for actively making sense of the crap we throw at them. But at best, they would have tools to connect their mathematics to their beautiful, intuitive common sense about the world around them.

I was rewarded after today's test with smiles and much more confident statements of belief by students about their own increasing success.

All of this has confirmed my newest hypothesis, which is as follows:

 if I'm not teaching sense-making, then I'm not actually teaching modeling.

The process of decoding and translation and re-encoding into symbolic form was so powerful for my students that I felt a need to document it for myself, so I do not forget or lose sight of this fact. I discovered that, if I spend time working through guided interpretation and translation of situations as a means of scaffolding the up-front part of the modeling process, it pays us all dividends in student engagement and clarity and success.

Here is how we started.


You are allowed to name anything!
You are allowed to use your own vocabulary!
It is often super-valuable to use a table to organize your info and build out your equations!


From working with the Exeter problems, I have learned that it is very helpful when you remind students of what they already know. It is particularly helpful to remind them that these are all based on d = r • t .

r is the trickiest part here. It consists of two elements: (1) the protagonist's basic speed, which I declared to be "b"— "b" for "basic speed." There is also the idea of (2) a current (air, water, wind, whatevs), which I labeled as "c" for "current."

The secret of these problems, my students explained to me today on our test, is figuring out what the relationship between b and c is. Is c helping the protagonist's progress... or is it hurting?

"Helping or hurting?" over and over again as we modeled different situations.

This required a lot of small-group and whole-class discussion and work on vocabulary. We made a word board (we don't have a whole wall):

             helping = downstream, tailwind, downhill, with the current

              hurting = upstream, headwind, uphill, against the current

If c is helping the protagonist's progress, then r = (b + c)
If c is hurting the protagonist's progress, then r = (b — c)

Once you figure that stuff out, you can make a table.

And at this point, students can usually build their equations and solve with minimal help. But they really need some scaffolding and guidance about how to take their common sense and mathematical understanding and bring this modeling problem to the next level, where they know what to do.


I was wracking my brain until my colleague Robert said, "Oh — I just draw it for them this way:"

Sometimes a simple solution is the best.


The hardest part of these is organizing the given information and identifying what you want or need to turn into a variable.

STEP 1: Name the numbers (I use capital letters A and B to start with)

STEP 2: Rewrite A & B using your knowledge of place value

A = _x__ _y__ = 10 (_x_) + 1 (_y_)   ; this means that  A = 10x + y
       tens   ones

B = __y_ __x_ = 10 (_y_) + 1 (_x_)   ; this means that  B = 10 y + x
       tens   ones

STEP 3: Revisit the problem and reinterpret the information in light of what you have organized.

x + y = 7

B = A + 27

STEP 4: Substitute what you figured out in Step 2 into Step 3

x + y = 7
10y + x = 10x + y + 27

STEP 5: Simplify

x + y = 7

9y — 9x = 27

Can you divide through by 9?

Final equations:  x + y = 7  AND  y — x = 3

STEP 6: Solve

STEP 7: What was the problem asking for — digits or numbers?

A = 25
B = 52

Sunday, September 20, 2015

DANCE DANCE TRANSVERSAL - logistics and playlists in a very crowded room (Geometry)

I originally received the very useful Dance Dance Geometry game (in PowerPoint) about six years ago from the very generous David Sladkey of Naperville High School in Naperville, Illinois. It is an amazing way to get students to practice identifying the essential angle pairs in a parallel-lines-plus-transversal situation.

A few years later, @algebrainiac (Jessica Marie) and Julie Reulbach put their own spin on it (recasting it as "Dance Dance Transversal") and gave me new ideas for how to use it.

But then... last year I arrived at my current school, with the world's tiniest classroom and 36 kids in every class. No space for everybody to move around. So no more Dance Dance Transversal for me.  *sad face*

But now... I'M BACK, BABY!

Now that Matt Vaudrey has turned me into a monster with musical cues. I've got musical cues down pat. The other day I was so pressed for time I forgot to play the theme music for my Geometry class (the opening from the old Hawaii 5-0 show), and the next day, my students said, Hey, where the heck is our theme music?!?!?

I love a self-regulating classroom.

So now I've figured out a way to do DDT even in my tiny room. Everybody gets a half-sheet-sized "game board" with two parallel lines cut by a transversal. Students will do DDT with two fingers while seated. Chair-dancing is encouraged.

I'm not sure how long it will take my students to master each level, so I've created a couple of alternate playlists in iTunes for Levels 2 and 3.

Also, at the end, we are going to have a dance-off!

Playlists for each level are as follows:

LEVEL 1 - The Honeyhive (I doubt we'll stay at Level 1, if we do it at all) 
LEVEL 2 - Herb Alpert, Mexican Shuffle
LEVEL 2 ALT - Raymond Scott, Powerhouse (middle section only, on a loop)
LEVEL 2 ALT 2 - Herb Alpert, Spanish Flea 
LEVEL 3 - Yakety Sax
LEVEL 3 ALT - The James Bond Theme (Original Version) 
LEVEL 4 - Mission Impossible
UPDATED 09/21/15:

And it was glorious. Here are two tiny video clips (one of the screen and one of the "dance"):

Saturday, September 19, 2015

Proportional Reasoning Capture Recapture with Goldfish Activity (Algebra 1)

I recently ran Julie's version of Capture Recapture with Goldfish activity with my Algebra 1 classes today. It was a huge hit!

This is a middle school topic, but proportional reasoning is so important it needs to be repeated every year in basic high school courses. Driving past the topics is not supposed to be the point! In HPL terms, I thought of this as an "activate prior knowledge" task that we tried to fit firmly into place with an engaging transfer task.

Julie's recommendation about showing the video is spot on: you absolutely MUST show the video. I broke it down as follows:

  • the first 1:30 to reveal the problem and the general idea (but without revealing the math or the solution)
We then discussed what was going on (activating prior knowledge about proportional reasoning) and together, we remembered how to do these problems.

Only after we had done our own work to rediscover the mathematics did I show the next 33 seconds of the video:

  • the next 33 seconds to reveal the mathematics needed to estimate
Together we wrote down all the figures and elements of the problem as Johnny Ball had revealed them so far. Then everybody in our class did their own work to figure out how many ping pong balls they estimated would be in his fish tank.

Once we had done that, then I did the final big reveal:

  • the next 13 seconds to reveal the exact number of balls in the tank

Once they had seen all of this, it was time for a transfer task.

I had students work in groups, recording their four trials on their data worksheets, taking an average of their estimates for each trial, and finally counting out their exact number of goldfish and writing a summary statement about how close (or far off) they had been. When they were done, they sent a representative to the whiteboard to add their group's data and best estimates to our table of whole-class results.

When we had everybody's data and estimates added to the table on the whiteboard, we discussed how accurate this method seemed and came up with other situations in which this method would be useful.

The hardest thing in Algebra 1 in my opinion is getting students to stop seeing skills and concepts as being discrete topics that you can mentally "put away" after the chapter test and to start seeing skills and concepts as new tools you want to "keep close at hand" in your mental tool belt.

So connecting proportional reasoning to tangible (and often edible) results in the real world is at least as important as the content of the lesson itself. This goes beyond merely "spiraling" back; it requires integrating into mathematical thinking and problem-solving from each moment forward.

Sunday, September 13, 2015

"How People Learn" and how people learn

How People Learn (HPL) is back in the blogs again, and for me, that is always a good thing. There is so much value, depth, and humanity in this slim, free book by the National Academies Press that any time anybody wants to talk about it at all, I say let's mark that as a win in the 'Wins' column.

There seems to be some misunderstanding, though, about exactly what HPL proposes an effective learning cycle ought to look like. Since in HPL, there is a place for everything, here is my 30,000-foot understanding and implementation of the four-stage process it advocatesI don't claim to be the definitive voice in any of this. I'm just taking this opportunity to document and share my practices in using their model because I believe that understanding this framework can go a long way toward helping teachers make good instructional decisions that can help their students to learn and thrive.

Specifically, HPL advocates:
STAGE 1 - a hands-on introductory task designed to uncover & organize prior knowledge. In this stage, collaborative activity provides an occasion for exploratory talk so that students can uncover and begin to organize their existing knowledge;
STAGE 2 - initial provision of a new expert model, with scaffolding & metacognitive practices woven together. The goal here is to help students bring their new ideas and knowledge into clearer focus so that they can reach the next level. Here again,  collaborative activity can provide a setting in which to externalize mental processes and to negotiate understanding, although often, this can be a good place to offer some direct instruction;
STAGE 3 - what HPL refers to as "'deliberate practice' with metacognitive self-monitoring." Here the idea is to use cooperative learning structures to create a place of practice in which learners can work within a clearly defined structure in which they can advance through the 3 stages of fluency (effortful -> relatively effortless -> automatic)
STAGE 4 - working through a transfer task (or tasks) to apply and extend their new knowledge in new and non-routine contexts. 
As with all good models, there is a lot of fluidity and variation in each stage, depending on how the teacher "reads" the learners in her classroom.  Here are some of my notes on each of the stages and how I have learned to look at each stage realistically and pragmatically:

A good discovery activity can be a powerful catalyst for learning  in Stage 1. But unfortunately, sometimes there just really isn't a great discovery activity that leads students captivatingly but inexorably to a blinding insight that will transform their learning forever.

Sometimes the best you've got is a mediocre discovery activity from a textbook that kinda sorta leads students in the general direction — but not without a lot of heavy-handed guidance. Or perhaps there is some other deficiency in what is available to you.

Like Gattegno, I believe that all learners have an energy "budget," and that means I have to make savvy and strategic decisions about how I'm going to ask my students to apply theirs. A boring or mediocre discovery activity requires just as much energy as a great one, but without the payoff of leaving students energized.

So sometimes I've learned I have to ask myself, is a discovery activity the best choice I can make here at Stage 1? Or do I have some other kind of introductory task I could use — such as a simulation, a story, a funny or interesting deleted scene, or some other kind of analogy — that will get my class into the learning episode faster and free up more of their energies to developing the necessary fluency that a rich and interesting transfer task may require?

To me, the most important thing that can happen in Stage 1 of a learning episode is that students come sharply to appreciate the Burning Question of this segment. Whenever possible, I really like for my students to arrive at a Burning Question through a collaborative discovery activity that they own because when they own it, they buy into it.

But realistically, this is simply not always possible with every single topic in the curriculum. So I have a range of strategies for Stage 1 that can get my students to a Burning Question even though there may be a gap in my pedagogical arsenal.

If the purpose of Stage 1 is to motivate students to ask a Burning Question, then the purpose of Stage 2 is to provisionally "pay off" the Burning Question — and to whet their appetite for knowing more. I say that my purpose here is to provisionally pay off the Burning Question because I believe a huge part of growing up as a learner is developing your own internal capacity for identifying questions and finding ways to pay them off and extend them.

So for me, this is where I "earn" the right to give my student a little bit of lecture, although when I work with them, I always call it "doing some notes" or "organizing our ideas" or "investigating ways in which others before us have thought about this problem." I say this not because I'm trying not to admit that I am lecturing (I am lecturing here) but I am also modeling note-taking and annotating practices that they will need when they arrive at a class where there is no other learning mode than lecture. No matter who you are and no matter where you study, at some point, somebody is going to lecture at you. If you are lucky (like I was at Princeton), those who lecture at you will consider it a high art form and will put great thought and care into their storytelling and argumentation modes.

Realistically, though, a lot of the lecture we encounter in our lives is not thrilling. But you need a certain degree of note-keeping and annotating skills that will enable you to survive those instructors and their inanimate lecturing practices so you can take what you need from their teaching and move on in your life.

So I use Stage 2 to also teach my students these note-taking/note-keeping/annotation survival skills as well as some metacognitive practices that will help them to get the greatest possible "bang" for their note-taking "buck."

As HPL clearly says, Stage 2 is about the "initial provision of an expert model." This is the place where we are sharing what students cannot find or develop on their own — or at least, what they cannot find or develop very efficiently given the time constraints of teaching and learning.

So please don't tell me there's no place for a transmission model in the HPL learning cycle. It's there, we all do it, and we all need to do it from time to time. Enough said. Let's move on.

With some new knowledge or ideas in hand, and having borrowed a more expert model from me as a tradeoff for accelerating the learning cycle, students need time to practice thinking these new thoughts, using the new model, and discovering what happens when they take it out for a spin. Deliberate practice with metacognitive self-monitoring is not the same thing as drill-and-kill. It's a form of experiential learning, like what a young child develops as they are integrating new vocabulary words. I've heard that a toddler needs to hear a new word used appropriately in context between 10 and 20 times before s/he can try it out for herself or himself. Mathematical ideas are no different. Students need to try and stumble, try and wobble, try and fall over, dust themselves off and try again until something takes hold in their unconscious. Nobody really knows what this secret crossover point is for every learner in every subject and every topic. So we provide a range of experiences for our students to help them find this crossover point for themselves.

Once students achieve some degree of "relatively effortless" fluency, they can dive into a transfer task.To me, an inspiring transfer task is more important than all of the mediocre discovery tasks in the world combined. An inspiring transfer task takes a learner seriously as a professional, and offers him or her an engaging, in-context opportunity to apply their new learning with all its glorious, messy, gravity-driven moving parts. One lightbulb moment from a transfer task — say, as Barbie is launched over a balcony railing, held aloft only by a series of looped rubber bands in answer to the question "How do we balance 'thrilling' and 'dangerous' to give her the greatest possible bungee jump that does not split her little plastic skull open?" — can last a lifetime. Being tasked to figure out experimentally and quantitatively whether or not Double Stuf Oreos do indeed contain double the "Stuf" as regular Oreos... or whether they are another marketing fraud being perpetrated on the Oreo-eating public can easily push a student over the fence into losing themselves in doing mathematics.

And frankly, to me, that is the whole point.

Wednesday, September 2, 2015

"Find What You Love; Do More of It"— #MTBoS Edition, a report from the field

While my friend @TrianglemanCSD Christopher Danielson is holding down the fort at the Minnesota State Fair and the Math On A Stick exhibit, I'm in my third full week of school avec students, and I realized I needed to peel off another layer of my persona and take his advice from his keynote address at #TMC15:
Find what you love; do more of it.
I love storytelling and stories. I come from a family of storytellers, where dinnertime was always a time of sharing stories.

I also love history — ancient history — and I know more about it than I usually give myself credit for.

So I did more of both of those things today. In my Algebra 1 class, no less.

It felt like a huge risk. But I decided to feel the fear and do it anyway.

So I told them what I have long known about the origins of algebra and equations. In its earliest uses, "algebra" means "balance." An equation is a metaphor for what everyone in the ancient world knew and understood with their own inherent sense-making and mathematical reasoning: just as a vendor at a market weighs out what a customer wants to buy — weighs it out with standards-based measures that are sanctioned by governments and universally accepted — so too is an equation a representation of these scales... and our goal is to balance out abstract or concrete quantities using that familiar structure from daily life.

I asked students what they knew about weighing and measuring and they told me honestly what their experiences have been at the farmer's markets all over our city.

We investigated what we knew about what happens when you place a known amount of weights on one side of a balance, and we imagined — and in some cases, acted out — what happens as you pour a continuous quantity of something onto the other side of the balance.

We talked about what it means to bring a scale into balance and we applied what we knew — the best we had — to the ideas at hand.

We talked about national and international standards bodies and about how even a perfect model degrades over time, which is why it needs to be monitored and occasionally replenished.

And so, by the time we got to the addition, subtraction, multiplication, and division properties of equality, we were already deep into our own connections with the metaphor and with the human history of algebra and with our own active and vivid imaginations.

By the end of class, nobody even complained about having to do the 2-2 homework. I am hoping that's because it was a little more deeply connected to their humanness than it ever had been before.

Saturday, August 29, 2015

New Geometry Unit 0 — Intro to Logic

Since, as @samjshah reminds me, we blog partly as an reflective archive for ourselves, I wanted to capture some of what I did in the new introductory logic unit I created for Geometry this year. I also wanted to capture some of who and what inspired me to do the things that worked out best!

I had no idea how prescient it would be to have an introductory unit that both adds tremendous rigor and depth while simultaneously being somewhat optional. These first two weeks of school have been somewhat chaotic as we discover who has been placed and scheduled correctly, who needs to be placed into a different section due to unavoidable schedule changes, corrections, or updates, how many new sections of certain courses we need, and so on. There were days when I felt like I didn't have the same students in any class two days in a row — even though some days this was more of a feeling than an actuality. So it was wonderful to have an adaptable "throughline" these two weeks — a river into which students might step, flow, or reenter without too much extra craziness. The unit was hard in the good way that students at my school really love — at the Zone of Proximal Development (ZPD), way up on a shelf just high enough above them that they have to stretch to reach it.

DAY 1 — Attacks and Counterattacks — What makes a mathematical definition?
We started with @samjshah and Brendan's Attacks and Counterattacks, which is now the recommended default starting unit for all Geometry students in our district. It was a great icebreaking activity, prompting students to activates their prior knowledge about what constitutes a definition. Plus it involved defining a narwhal, so how could that end badly, right? Table groups passed their definitions around the room, then used table-sized whiteboards to come up with a counterexample that broke the defining group's definitions. This involved both collaboration and presentation skills, as well as a good memory for definitional trivia. Did you know that the horn of the narwhal is ACTUALLY a tooth?

DAY 2 — Statements, Compound Statements, and Truth Tables
I also used this unit to set up classroom norms. Each day, when students come in, the "Welcome" slide is projected onto the board with introductory instructions and the Home Enjoyment (HE) assignment to copy down into your notebook.

We don't have bells at our school, so as an auditory cue, I am taking a page from the amazing @MrVaudrey and embedding a one-minute "welcome to class" music button that tells students this is a short, specific, and finite portion of our show and they know what they need to do. Thank you, Mr. Vaudrey! The theme music for Geometry (my 1st block class at 7:35 a.m.) is the music from Hawaii 5-0 (in honor of my principal and reminding me to ask myself, What Would @wahedahbug Do? with her brilliant mathematical classroom intro routines).

The instructions sometimes tell students to grab a handout from the handouts hanger but they always tell students to get out their HE and compare answers. I have realized that if I need to include homework review during a time of greatest primacy and recency, then I am going to make it count (thank you, @druinok and @pamjwilson!).

Next up was a dramatic table read of a deleted scene from the first Harry Potter movie. It's just terrible what ends up on the cutting room floor but, you know — Hollywood. "Harry Potter and the Logical Statement" was a SMASH hit. Students taught themselves the basics of statements, negations, equivalence, and truth tables and it beat the living crap out of giving them a boring lecture. After they were done, we summarized and organized what we had learned and I set up expectations for Home Enjoyment.

Day 3 —Advanced Equivalence & Rules of Replacement; Intro to Conditional Statements
The beginning of the truth table Olympics. Getting students to use what they know and extend it and — hooray — organize their work logically on the page was a huge win. Plus the kids liked learning something grown-up and hard. We did more practice set up homework review routines and expectations. Students are getting better about coming into class and getting started right away. Overview of the Professionalism score and expectation-setting.

Day 4 —Conditional Statements, Part II
Students discovered the Law of Non-Contradiction (Law of the Excluded Middle: A ⋁ ~A, but not both), and introduction to converse, inverse, and contrapositive using the basic conditional statement, "If I am Batman, then I am a superhero" (I'm looking at you, @mgolding). Whole lot of truth table whiteboarding going on.

Day 5 —Four Basic Laws of Inference
Modus ponens, modus tollens, simplification, disjunctive syllogism, and intro to proof. By this point are really getting into the technical language, to my great surprise. We start whiteboarding proofs and advanced truth tables.

Day 6 — Biconditionals & Definitions
We return full circle to where we started, with definitions, but now we define biconditional (iff) statements and how to prove them. Students start to groove on the idea that you have to prove both if A, then B and also if B, then A to prove a biconditional. There are lightbulb moments about the importance of counterexamples. Routines are starting to gel. Students are still transferring into the class and between/among sections/instructors, but other students help to indoctrinate them into our emerging culture.

Day 7 — Intensive Practice Day — Truth Table Practice aka Whiteboarding Madness
I give each table a sheet of truth tables to build and I use this activity as a Participation Quiz to further solidify our norms. Many groups start passing the marker around to ensure equitable participation. Everybody does splendidly on the Participation Quiz.

Day 8— Assessment
Home Enjoyment Packet #1 is due to be turned in. Students take the Unit 0 Logic test. It is a bear, but my students are "scared but prepared." Some students need more time so they come in at 7:15 a.m., during 3rd, 4th, 5th, or 6th block to finish. Their stillness and concentration is impressive. Algebra 1 students are doing their own crazy stuff all around them, but they persist and persevere. I am over the moon about them. They are proud of themselves for the advanced logic they have learned.

Day 9— Chapter 1 Launch: The Elements of Geometry — the poetry of primitives
We explore undefined terms, learn a little math history, and do some Think-Pair-Share. Next year I want to have another table reading/deleted scene activity for all this stuff instead of a boring lecture. But that is OK. Our routines are solid and we are moving forward.

Onward to constructions on Monday.

Now I have to score 73 logic tests (and 73 HE packets, but only for completion), but it was totally worth it. My Geometry classes are off to a great start, and we have a solid foundation to build from, even if my sections have 36 kids each.

Saturday, August 22, 2015

TMC15 reflection: The Story Of TMC or, How We Didn't Get Lucky

We had our first week with students this week and boy, am I tired.

But all week long, I feel like I've been carried along on the current of good energy I have forged over the years with my TMC (Twitter Math Camp) math teacher tribe.

I was thinking about how other people keep telling me, Oh wow, you're so lucky you've got that.

And I finally realized I've been wanting to say, "No — we didn't get lucky."

A little over four years ago, a bunch of us who had met on Twitter and blogs decided we wanted to get together in real life. In December of 2011, over winter break, there was what is now known as The Great Facebook Friending of 2011. One night during a rampage of funny, crazy, meaningful tweeting among math teacher tweeps, we made the decision to "Facebook-friend" each other.

At the time, that felt like a HUGE risk — letting other people into our real, personal lives.

I was worried that the next morning I would wake up and discover that it had all been an enormous mistake and I would need to go into internet witness protection to get away from these crazies. I was worried that I was going to have such a hangover.

But no. I discovered that these really WERE the people I wanted to be connected with. And other people did too.

So even though Julie Reulbach still wanted us to go on a cruise together, we all decided it would be safer — and saner — to meet on land somewhere. The Mary Institute and Country Day School in St. Louis was gracious enough to offer us a free space to have our math teacher jamboree, and we all traveled from remote parts of North America on our own dimes to get there.

And as I like to remind people who say how lucky we were this year to have TMC at Harvey Mudd College in Southern California — remember that at the first TMC in St. Louis, I was the only person from California who showed up.

We didn't get lucky. We took small, incremental risks with our teaching and with our professional development until we felt safe enough and ready enough to form something larger.

And as the great psychoanalyst and cantadora Clarissa Pinkola Estes has said, when you step forward and truly embrace your whole life with your whole life, other like-minded people will "mysteriously show up, announcing that this is exactly what they have been looking for all along."

In other words, there are rewards for courage.

So my biggest TMC15 reflection is a reminder to myself that we did NOT in any way just "get lucky" with TMC. We stepped forward and showed up in our professional development lives — over and over and over. We stepped over the negative chatter of people all around us saying "there's no such thing as good PD" and we pushed past people who asked negative questions like "Why would you want to use part of your summer vacation time to travel to ______ [fill in the blank with St. Louis/Philadelphia/Jenks, OK/Pasadena] in the summer for professional development?" and we ignored the negativity of anybody in our home districts dumping on "Common Core math" or "all that fancy-schmancy group work nonsense."

We took a deep reflective breath and said a holy "yes" to being deliberate about our teaching practice and taking the risk of investing our whole selves into it. And THAT, as I constantly remind my students, is the essence of "luck."

Wednesday, August 19, 2015

DELETED SCENE: Harry Potter and the Logical Statement — GEO (statements, compound statements, and truth tables)

I'm absolutely slammed for time, but I wanted to share this script I wrote for my Geo students, which was a surprise hit.

I'm taking a page this year from the patented Sam Shah "Here kids, teach yourselves this stuff" pedagogical method, which really deserves a lot more credit than it gets.

I had students act out this "script" which is, of course, from a confidential deleted scene from the first Harry Potter movie. The studio insisted I return all copies after the class. Students delighted in finding the many "problems" with the script and we laughed about all the many reasons why this scene was obviously deleted.

When you release students to launch the task, you want to slam a ruler on the table and yell, "ACTION!"

Students had a lot of fun and got their intro to symbolic logic. Mission accomplished.


The file is available on the Math Teacher Wiki:

Deleted Scene: Harry Potter and the Logical Statement

Friday, July 31, 2015

Have Students Introduce Themselves to Talking Points — Algebra 1 Day 2-ish

With Talking Points, I keep finding that the more I push control down into student groups, the better they self-regulate and dive into the material.

So here is a self-guided intro to Talking Points for Algebra 1 students.

Also, with blind students in the classroom, it becomes even more important for equity that student groups speak and listen equitably to ensure inclusion. So with Talking Points, in addition to handouts, I can give a blind student the Word document on a flash drive, they can plug it into a Braille reader (which allows them to read a line at a time), and everybody is off to the races.

The ever-growing Google Drive folder for new sets of math Talking Points is at

Friday, July 24, 2015

NCTM and The Math Forum Join Forces

Well, here at Twitter Math Camp (#TMC15),  this happened today:

Personally, I am thrilled for my good friends at The Math Forum, who have contributed so much to our extensive worldwide professional learning community. But I also want to witness what a milestone it is for TMC that this is the venue at which the merger was announced.

Five years ago, we were a positive but isolated group of individuals connected by Twitter and by our math teaching blogs. Today, our little conference was the platform for an important piece of news in the math education world.

I have said this before — TMC and the MTBoS (the Math Twitter Blog-o-Sphere) are not a flash in the pan. They represent a paradigm shift. We are a movement. 

They and The Math Forum are living proof that the "market" does not want what focus groups or policy committees think is the safest generic middle course to follow.

They are proof that what is needed — desperately needed — is a community of individuals committed to embodying a better and more sustainable set of principles in our teaching practice and in our professional development lives:
  • Honor the actual work of mathematics teaching that is going on every day — not some sanitized generic ideal that is so removed from reality it cannot be valued.
  • Step forward and be that community you wish you could find. As the great psychoanalyst and cantadora Clarissa Pinkola Estès has written, "if you build that community, people will mysteriously show up, announcing that this is exactly what they have been looking for all along."
  • Witness and celebrate each other's amazing accomplishments in the classroom, even though the power structure and outside forces refuse to accept the good that we do every day. Cheer each other on. This is about "growing up" as a profession and as a community and accepting that true grown-ups do not wait for permission to do what they know what needs to be done. True grown-ups see what needs to be done and say, "Oh, I see. I'll do it."
  • Recognize that this is a movement — and that a movement is what is needed.  We have serious problems, but we have phenomenal capacity to respond to what needs to be done. It is easy to stop a few people, but it is impossible to stop a thousand. Remember the motto of #OtterNation:

  • Don't take "no" for an answer. As @TrianglemanCSD said in his keynote address today, "Find what you love, and do more of it in your classroom."

Tuesday, July 14, 2015

The Primacy-Recency Effect: a conversation with Jennifer Carnes Wilson (episode 1)

Dear Jennifer,

Thanks for engaging with me on David A. Sousa's  The Primacy-Recency Effect article. I too like to reread it and think about it around once a year, so your tweet was most timely for me.

There is a Freudian slip-style of typo in his very first sentence that has always struck me as encapsulating the entire debate he has provoked:
When an individual is processing new information, the amount of information retained depends, among other things on what it is presented during the learning episode. (emphasis mine)
Clearly he means to say "when" rather than "what," but for me, that question of "when" versus "what" lies at the very heart of the debate about student discovery of new ideas. Is it more important when students encounter a new idea or how they encounter it? If I have students tinker and investigate for too long at the beginning of the class period, I risk missing their window of greatest receptivity and retention.

On the other hand, if I start right away by framing the big idea, I harness their optimal moment of receptivity and retention, but am I doing so at the risk of their autonomy?


Tuesday, June 30, 2015

What it means to be a part of a learning community - Tribal Elder Edition

One interesting moment from the Oregon Math Network Conference this past week:

Bill McCallum was leading a large-ish session on building a culture of collaboration through jointly investigating student work.

Fawn (@fawnpnguyen) and I were sitting side by side at a table off to the side, each of us prepping madly for our next sessions. But neither of us could resist the lure of student work. We set our own presentations aside and pulled up the examples of middle school student work on Fawn's computer.

The task for the teachers in the room involved making sense of middle school students' written-out interpretations of different possible takes on how to simplify the expression

                  7 – 2 ( 3 – 8x)

Being experienced teachers of middle school math students, Fawn and I were both immediately captivated.

"Look at how this student identified right away that the value being distributed is a negative 2 — not just a 2," she said. "They noticed that part of it right away."

I nodded.

I noticed the student's language, which indicated a little mid-process magical thinking about the how to distribute multiplication over subtraction: "...because you use order of operations";  "you always do the problem inside the parentheses first"; "...but then "it's a problem that you've got  – 2 on the outside and – 8x on the inside."

"The student is using these phrases as magical incantations," I said. "The rules are still spells to him or her." Fawn agreed.

We both recognized these pieces of productive struggle from our own students's journeys. We dissolved into flow as we started talking about different ways to provoke authentic insight and discovery in our students. This is what is fun about getting together with kindred teacher spirits. It gives us the chance to share a deep kind of noticing that happens automatically during the school year, when we are trying to avoid drowning in the sheer overwhelming volume of student work.

While we'd been lost in analyzing, noticing, and wondering — and unnoticed by us — Bill had stepped closer to eavesdrop on our conversation and to join in the fun. At a certain point, he stepped right into the flow of conversation, offering his own noticings and wonderings about the students' wordings and insights. Several times we all burst out laughing — not at the student's work but at our own pure delight in it. Even after all this time, we can all still be captivated by adolescent mathematical thinking.

Well into his late 80s, Michelangelo was often heard to repeat the motto, "Ancora imparo" — "I am still learning." That is a concise summary of the delight that all teachers feel when we get the chance to sit together as a part of a learning community and think about teaching and learning together. This is the best teaching and learning reminder I know, and I always feel blessed when I have one of these flashes of self-remembering during one of these moments. So I wanted to capture this one.

Saturday, April 25, 2015

Impromptu Twitter master class on homework strategies

On Friday afternoon Anna (@borschtwithanna) tweeted out this call for conversation:

What followed was a virtual master class on handling homework in different settings. It's the sort of conversation that the #MTBoS excels at, pulling in thoughtful responses from teachers at every level of practice.

I've been thinking a lot about homework because I am overhauling my homework strategy for next year.  This blog post is my attempt to capture my strategy overview, along with pointers to resources that have helped me think through what I need to do.

Overall HW strategy for next year

There are four pillars to my homework strategy for next year:

  1.  HOMEWORK LAGS CLASSWORK: I am implementing  Henri Picciotto's strategy of having the majority of each night's homework "lag" the current classwork focus of the day; 
  2. EACH DAY'S INTRO TASK IS TABLE GROUP REVIEW: the first ten minutes of class will be for students to discuss/review/help each other out on the previous night's homework problems together in their table groups; 
  3. REFRAMING THE ROLE OF THE TEACHER: I will take questions on particularly troublesome homework problems, but I will take whole-group questions only; and 
  4. COLLECT, STAMP, & GRADE HW PACKETS EVERY 2 WEEKS: I collect, stamp, and grade homework packets every two weeks, grading for completeness of effort (every problem in every night's problem set attempted). 
Here are my elaborations on these pieces.


Henri Picciotto's blog post on this is a classic. There are so many valuable things about having the majority of each night's homework "lag" the current day's classwork. First and foremost, it ensures that most of every night's homework is accessible to every student every day. Secondly, it helps with heterogeneous classes. It gives students multiple at-bats for each kind of homework problem, keeping things meaningful for everybody. New material challenges proficient students, while those who are still working toward proficiency get multiple opportunities to work toward master.


The first ten minutes of class are the time for students to get help on homework — and by "help" I mean helping each other first. A big function of math homework in my view is to help each student cultivate an autonomous and independent approach toward their own struggles with their own problems.

A.H. Almaas describes the problem like this: "Many people... unconsciously act out a desire to be 'saved' by a teacher. But if a teacher 'saved' you, you would lose something. You would lose the value of struggle" (Diamond Heart Book One, p. 123).  In my view, the first ten minutes of class are the place where I expect students to begin to shift their mindset about the value of struggle. If a problem is not pushing beyond their Zone of Proximal Development, I expect them to develop the habit of resolving their own confusions themselves.


In keeping with #2, I want to be conscious and intentional with framing my role in their lives about resolving their own problems with problems. Almaas describes this reframing better than anybody I know:
So there are two ways to approach the teacher. One approach is to hope the teacher will take away your problems; the other is to use the teacher, not with the expectation that she will take away your problems or offer solutions or "make it better" but that she will give you a little push in your struggle. (DH Book One, p. 124). 
This is why I love Dan Anderson's (@dandersod) description of the "mass confusion" rule: unless a problem causes mass confusion, students have to work out their problems independently and help each other out on during that first ten minutes of class.


This has been the most surprisingly successful thing I've done this year. Students have to turn in a stapled packet of their "Home Enjoyment" problems every two weeks. I collect it, I stamp it, and I give them a grade for it every two weeks. Every problem must have been attempted. They should show effort to have sought a resolution for problems they didn't understand the first time through.

This has been a great accountability practice for students. It's an easy, easy 'A' grade every two weeks plus it gives them that push to keep themselves on track and not fall behind on homework. In a high-achieving school filled with motivated students, I expected not to have to do this, but in fact, my experience has revealed the reverse: students appreciate that little push toward accountability. It triggers their automatic reflexes in a way that supports their autonomy.

It also takes me very little time. I am basically just stamping packets, looking for effort and gaps, and rendering a grade whose default is 100% unless stuff is missing or late. I take off a 10% late fee per day. That is usually the only penalty students incur. I've been astounded by how being an old-school hard-ass about this has simplified and streamlined the process.

I hope this is helpful!

Sunday, April 19, 2015

The deeper wisdom of the body in math class

This post is for Malke.

As I was eavesdropping on a recent conversation between Lani Horn (@tchmathculture) and Malke Rosenfeld (@mathinyourfeet), I received a pointer to an article summarizing recent research that shows that kids with ADHD actually need to squirm in order to learn.

This makes sense to me. The deeper wisdom of the body is usually overlooked in thinking about teaching, learning, and assessment in mathematics. And yet, it can provide a vital link for our students in claiming their mathematics as well as their humanity.

I was thinking about this on Friday afternoon when I tweeted out the following:
I love the dulcet tones of compasses, rulers, & pencils during a Friday afternoon constructions quiz. #geomchat
Malke tweeted back:
I love that they are doing all that by hand. And that there are dulcet tones. :)
I responded:
Geo has affirmed my belief in the life made by hand. Huge benefits to Ss [students] from physically constructing their understanding.
I had another in a series of lightbulb moments this past month about what How People Learn says about externalizing our understanding.
And ever the good online learning partner, Malke tweeted back:
have you blogged about it?
So here I am.

In How People Learn, the authors talk about how we can use skits, presentations, and posters in group work to help students externalize their emerging understanding. This makes sense to me. In order to learn something, one first has to notice it, and that means developing a metacognitive self-awareness of the process and how it’s going.

Over the last few years, I have found that teaching students to use foldables, INBs (Interactive Notebooks), guided note-taking, and physical constructions is another extremely rich field of helping students to externalize their emerging understanding — only in these cases, they are externalizing their understanding through physical, kinesthetic processes — not just through talk, listening, and presentation processes.

The physical dimension is a good grounding for conceptual understanding. Teaching students how to literally use their tools can be a multidimensional process of making their learning both physical and tangible. Flipping open a flap or a page in a composition book is a physical manifestation of the process of retrieval or comparison or evaluation. Likewise, the process of using patty paper as a tracing medium to externalize the concept of superposition and projection of a figure to confirm congruence is a way of helping students to slow down their speeding monkey minds and to become present with the mathematics that are right in front of them.

When I tuned in to the clatter of compasses, rulers, and pencils on Friday, I really noticed how deeply engaged my students were with the geometry they were working on. Their body postures indicated how deeply immersed they were in the experience of flow: set your compass opening to an appropriate width and draw an arc across the angle you want to copy. Stab the endpoint of the segment where you want to create a new copy of your original angle and swipe the same arc there. Go back to the first angle. Refine your compass opening so that it now matches the width between the intersections of your arc and the original angle. Shift your paper, stab at the lower intersection of your copied angle-in-process and swipe an arc that will intersect with the arc you just drew there. Drop your compass; pick up your straight edge. Carefully draw a line to connect the endpoint of your target segment with the intersection of the arcs you have drawn, completing the terminal side/ ray you need to draw. Drop the straight edge; position the patty paper over your original angle and use your straight edge to trace it. Drop the straight edge and carefully slide your traced angle over your constructed angle. Does it match your figure perfectly?

The concentration etched on their brows matched the precision of their work on the page in front of them. Bisect an angle. Construct the perpendicular bisector of a segment. Construct a parallel line through an external point by using it to define an angle that you can copy.

Hopping from one stone to the next, you can cross an entire river. By placing one foot on the Earth after another in a pattern of glide reflections, you can complete a journey of a thousand miles or more.

That is one of the lessons of deductive and spatial reasoning at the heart of any good Geometry course. Noticing that it is happening in my classroom — really happening through physical, mental, and whole-hearted engagement — is one of the greatest blessings of being a teacher.

Tuesday, April 14, 2015

40 days and nights of standardized testing: a reflection

And on the 289th day, God made brownie mix. He divided the mix into the batter & the topping. And God saw the brownie mix, that it was good. And God said, Let the brownie mix be cheap and abundant and distributed to all the grocery stores so that all those who suffer, may make brownies. And the baked brownies He called "one serving," and he forbade the posting of the nutrition facts. And there were brownies, later with toppings, a comfort food.

Saturday, April 11, 2015

Resonance in our work as teachers — a love story

I went to the neighborhood farmer’s market this morning. I wanted to seek out Q and find out where she is going to college next year. I picked my way through the booths, looking for the almond grower’s stand where she works each week. After a few booths, she spotted me. She bounded over and gave me a huge hug. “I’m not selling almonds any more. I’m the assistant manager of the farner’s market now.”

I gave her another hug. “Congratulations! I’m so proud of you! What a wonderful promotion.”

“I miss you,” she said plaintively. “So does R. We talk about you all the time.”

This was a good reminder to me that we really have no idea how long or how deep our impact and reach as teachers will go. Now I know I need to find my way over to his Taco Bell / KFC Store to touch in with him.

I asked, “So where are you going to school next year?”

Her face fell. “I’m going to City College. I got rejected from all my schools.”

I hugged her again. “That’s OK. City College is awesome. You will go there for two years and transfer to the UCs or CSUs. This isn’t a defeat — it is only a setback. You are going to get where you need to go and you are going to do great. I believe in you.”

“I miss you so much,” she replied, wrapping her arms around me again.

This was all a good reminder for me. Sometimes our most important job as teachers is to show up and to be adults in their lives who are as not-full-of-shit as we can possibly be for them. We provide some much-needed continuity as adults — continuity that can be lost to them when a parent dies or moves away or goes to prison for an extended term. It is so important for all kids to have lots and lots of adults in their lives who love them in our own unique ways. We help to witness their suffering and to encourage their courage. As I write this right now, my heart hurts for her. I know how much she wanted an elite college and how disappointed she must be feeling. But I also know and believe in her unique giftedness and heroism, and I am grateful that I got a chance today to witness and reflect that back to her at a moment when she really needed it.

I will keep showing up at the farmer’s market and at KFC and believing in them both because that is an essential part of my work and my calling as a teacher. And even though Bill Gates and Arne Duncan are utterly clueless about this essential part of my work as a teacher, I will continue under all circumstances anyway. It is a great gift to get to be guerrilla bodhisattvas in our students’ lives, just as our teachers were for us.

And so I wanted to document and honor this unspoken, unappreciated, unmeasured, and undervalued part of our work as teachers on this Earth. May we all continue to be present with an open heart for all our students everywhere throughout space and time.

Friday, April 3, 2015

If it is in the way, it is the way: the only true path to a growth mindset

I believe that helping our students to find their way a growth mindset is so important it must become one of the pillars of our math teaching, but I also believe that the primary ways our leading experts are pushing right now are so misguided I can no longer stay quiet.

The bottom line is this: if you believe that a learner can simply let go of their fixed mindset just because you tell them to, then I have a bridge to sell you. I believe that the positive intentions behind this initiative are leading students to develop new ways of hiding their true selves in math class, and I can already see this approach leading to even worse forms of self-abandonment and closed-off-ness that are only going to make the whole situation much worse.

So this is my plea for us to all stop trying to coerce students into a growth mindset and instead to start developing a more mindful approach to helping students engage with a growth mindset.

Carol Dweck and Jo Boaler have done more than anyone else to popularize the idea that adopting a growth mindset is the way to go in math, but I believe that the ways they are trying to spread the gospel of a growth mindset are both harmful to students and doomed before they begin.

They are doomed because they amount to lecturing and shaming students about their defense mechanisms — an approach they would never take in the actual teaching of mathematics. A fixed mindset is a set of conditioned habits, and you can't change a habit just by force of will.

The reality is that a fixed mindset is a defense mechanism — an unconscious set of adaptive survival behaviors that evolve within a person's sense of self as a defense against what it perceives to be a threat from the outside. In the math classroom, that threat is often the threat of failure, of annihilation, of humiliation. It doesn't matter what you or I perceive the threat to be. It doesn't matter whether you or I perceive the threat to be real or not. Simply put, a fixed mindset about math — as is a self-identification as a "non-math person" —is a defense mechanism. It's not about you.

Please repeat that last part after me: a student's own personal fixed mindset about math is NOT about you.

It's the psyche's way of protecting the soft, vulnerable center of the student's own self from what it perceives to be a threat to the continued existence of its organism.

The only thing that matters in all of this is how the learner perceives the threat for him- or herself. And a fixed mindset in the learning of mathematics is a (misdirected) protective function that has arisen inside the learner as a way of keeping that learner safe from harm — often harm that you or I, as a teacher, represent.

About 40 years ago, Eugene Gendlin (the great psychotherapist from the University of Chicago) teamed up with psychologist Carl Rogers (who pioneered the humanistic or client-centered approach to psychotherapy) to investigate the question of why some people are able to make permanent and lasting change through therapy while others cannot. What Gendlin discovered was that those who make progress are the ones who are able to direct their inner "focusing" on their own subtle, internal bodily awareness or "felt sense" — a felt sense that opens the door to finding their own self-directed resolution of the problem about which they felt stuck. In his books starting with Focusing, Gendlin documented and popularized a simple yet powerful six-step process which could be taught to individuals to help them access their inner felt sense, and to work with it to bring about a "felt shift" out of their stuck place and into a freer and more authentic relationship with their triggering situations.

This process takes time and patience and psychological and emotional maturity and generosity of spirit that few of us get trained in via the usual teacher training and professional development pathways. But this is the only truly non-coercive way to support students in developing an authentic growth mindset about mathematics.

The only successful way to work with defense mechanisms — the only way that has been shown to bring about long-term inner change, either in a therapeutic or in an inner development context, such as mindfulness — involves empowering learners to gently and non-coercively notice their own defense mechanisms when they pop up.

The choice to leave behind a self-identification as a "non-math person" MUST come from inside the learner him- or herself. It cannot be imposed from the outside, no matter how well-intentioned that coercion might be.

This is what my Ignite talk at CMC-North Asilomar 2015 was about this past December. I hope this will help others to make sense of how we can best support our students on this inner development path.

Tuesday, March 24, 2015

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.

Saturday, February 28, 2015

Desmos plus INBs — Conic Sections Edition

One of the things I have always been frustrated with is the crappy way example graphs look in student notebooks.

Well, no more.

For my conic sections notes sessions in Precalculus, I'm using Desmos-created graphs with all their equational and slider glory.

Here's how I ensure kids have readable, meaningful examples in their INBs:

I've created some modified graphs of the Desmos parabola graphs — one with a vertical axis and one with a horizontal axis.

I take a screen shot of the equation drawer PLUS the graph and paste it into an Omni Graffle document. For those of you playing our home game on the Mac, that's:

  • Press Cmd-Shift-4 to enter screen grab mode
  • Select the region of the Desmos window that you want to use as your graphic (this pastes it directly onto the Mac OS X clipboard)
  • Paste into a blank Omni Graffle document (from )
  • Resize to fit your needs, then
  • Select, copy, and paste as many times as you need to create the master for your tiny handout
I arrange them 3-UP on the photocopy master so the tiny handout will fit onto a standards notebook/INB page.

Here are the files for my photocopy masters:
Chop, glue, annotate.

I recognize this is totally old school, but everything old is new again.

Thursday, January 29, 2015

Swan-style matching task — matching polar graphs with polar equations (Precalculus)

This was a winner. By turning a perfunctory matching activity into a group task, I created a rich mathematical investigation that achieved in one period what two days of lecture and guided practice failed to come close to.

I just grabbed these graphs and their equations from our textbook, enlarged them using Omni Graffle, and provided the markers, scissors, glue sticks, and half-sheets of poster paper.

A motivated class did the rest.

We followed with a SILENT gallery walk and a whole-class debrief. I told them to find the best ideas they could adopt from other table's posters.

It was not until I brought these into the Math Office and laid them out on the floor that several of us noticed the most original, most insightful, and deepest conceptual learning aspect of poster number 3 below, which is left as an exercise to the reader.  :)

Here's the activity file (on the Math Teacher Wiki).

And here are three of the posters the class created. Click on the photos to zoom.

Poster 1  

Poster 2

Poster 3

What do you notice?

Tuesday, January 20, 2015

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.