I then ask them to give me two new numbers — any two positive integers [that are 10 or less, for now] — and the computer will draw a new picture. As each set of new numbers is entered and the corresponding picture is generated on the screen, I ask students to jot down their “I notice, I wonder” in Google Form and to draw a rough sketch of it in their journal. After a few sets of numbers, I ask students to imagine and/or draw a rough sketch of what they *think* the picture will look like **before** I hit the *update* button.

These are the pairs of numbers they’d asked for and their corresponding pictures, listed in the order that was asked.

I love that the kids are asking for…

- 6, 6 after the initial 1, 1
- 3, 8 after 3, 1 (keeping one number the same)
- 8, 3 after 3, 8 (reversing the numbers to see if anything changes)

But, when sets 9, 9, and 7, 1 are asked at the end there, I say to the class, “Hey, what if figuring out this puzzle — which is how the computer draws the picture given two numbers — gets you a million dollars. And you get to ask for sample sketches like you’ve been asking, except that each sample costs you some money! So, make each request worth it. Let it prove or disprove your conjecture. Ask carefully.”

I love the *OHHH*s and *AHHH*s after each picture is revealed. But no one is claiming that he/she had drawn the same diagram. I pause longer for them to write down their noticing and wondering.

I now say, “You may only ask for four more sets of numbers. Remember, make a request that would test your conjecture.”

I ask a normally quiet student. She says, “10, 3.”

Another student wants to know what “100, 5” looks like.

“What about 8, 5?” I reply, “Sure, but draw it in your journal first.” They are fully engaged. Then I say, “Now, share your drawing with a neighbor.”

I ask, “Did anyone sketch the same thing as their neighbor?” They’re shaking their heads, and I say, “That’s pretty crazy! Do you think yours is more ‘correct’ than your neighbor’s?”

I reveal 8, 5.

The last request is 23, 75.

What some of them have written [with minor edits from me]:

When we did the same two numbers the shape didn’t change but when we did different numbers it changed. Why does it divide into little parts within a square when we put 3,8? When we did 8,3 the number switched around. I wonder if the two numbers are dividing to make the shape. How can you figure out the number when it can’t divide easily. My drawing for 8,5 was one whole and 5 little squares. The 23,75 was a little confusing to me.

They’re different, they are the length and width, and when the two numbers are the same it’s just one cube. I notice that if it can be simplified, it is. Example: 6,2 = 3,1. I don’t understand 8,3, 5,9, 23,75 or 8,5. But I did notice that the smaller the parts of the shape are, the lighter shade of blue they are.

I observe that when the same numbers are entered it equals to a blue square. If the first number is bigger than 2 then it will add one more square. I wonder if you double the number for each number will it be the same shape. I wonder why for 3,8 it has one square with three parts. I observed that if you divided the first number by the second it will equal to the number of squares. For 8,5 I didn’t get the right sketch. The sketch was one square with half of a square cut in half, then in one half is has a strip that is cut in half. I wonder why it has half of a square. I think that my answer for how it figures it out is right, but I don’t know how it comes up with that picture for 8,5.

When you do 1 and 1 it doesn’t change because we tried 6 and 6 it didn’t change and if we put 3 and 1 it did change. I saw that when we did any number like 3 and 1 is 3 ones. So I think that all you have to do is divide something by something = the first number that you put in but if you can’t divide by 2 then I’m wrong. I’m not sure that I got this right but this is what I think.

For the first one 1 and 1 I thought it would be a small one by one cube. What threw my off was the 6 by 6 because the size did not change. For the 8 and 5 I drew a big block and and 5 little ones, but my image was wrong. I also wondered if the first number was the amount of shapes that would appear, but I was wrong again. I don’t understand yet. I tried looking for a pattern, but couldn’t find one.

When I tried 8,5, my answer was almost right. I had the one big square right, the half square right, but then I got the little squares wrong. I think that the way the computer does it is dividing the first number by the second number. I am confident that if you put the numbers 10 and 5 in, it will show 2 squares. When using the diagram, the second number will represent the vertical side.

What I’ve been noticing was that if you put the bigger # in the front and the small # last then it would be like a rectangle. I’ve also been noticing that if you put the same #’s it would like keep on drawing a square. So someone said what could (8,5) look like and Ms.Nguyen showed us the drawing and the I notice that nobody got it right. I was expecting something like smaller because the #’s were small they weren’t as big, but at least I tried to get it correct but I drew something a little bit smaller than that. I also wondered why when we put the same #’s together why do they all become a square that’s what I wonder.

For 5, 12 I notice that it is two big squares, two smaller squares, and two tiny squares, I thought it was going to show 1 big block and another big block but that one would be cut off at the bottom or not a whole block. I also notice that the pattern is the first number multiplied by what equals the second number and the number that is missing is the amount of blocks that is created. I thought I knew it but I don’t really get the ones with a bigger number first and the smaller one last. I thought I knew what 9, 23 was going to be but it the result was surprising. It didn’t look at all how I thought it was going to. The website is pretty cool, but one thing I didn’t understand was the placement of the small blocks and what they stand for. Like some were really tiny and some were small but I don’t know what they stand for. But I bet if someone explains it to me I probably will understand perfectly and feel dumb.

The only thing that I am sure that I know is that when the first number is larger than the second, the shape is wide and when the second number is larger, the shape is tall. Other than that I am very confused.

Then, together as a whole class, they agree on the following;

- When both numbers are the same, then the picture is one square.
- The computer simplifies the two numbers, such that a picture for 6, 3 is the same for 2, 1.
- When the second number is a 1, then the picture shows the first number of squares. For example, 7 and 1 would form a picture of 7 squares, and 100 and 1 would form 100 squares.
- The first number is the horizontal dimension, while the second is the vertical.
- They are all squares.

The dismissal bell is about to ring, and I want to teach forever.

Tomorrow, we’ll spend some time with one set of numbers, like 10, 3 or 8, 5. We’ll dissect the diagram. Play around with a few more. Practice sketching a few. We’ll write out the equations that go with each diagram. I’ll guide them into noticing the size of the smallest square in relation to the two numbers.

I found this investigation at underground mathematics. The site describes itself as having “rich resources for teaching A level mathematics.” From what I understand, “A level” means advanced level mathematics consisting of core modules ranging from quadratic, logarithms, geometric/arithmetic series, differentiation/integration, etc.

Perfect for my 6th graders who continue to torment me with their arithmetic atrocities, such as, 3² = 6 and 5 ÷ 10 = 2.

While the original task is scripted for older and more advanced students, I found in it what I needed to make it rich and appropriately complex for my 6th graders.

Hail, Euclid’s algorithm!

]]>Once a month, however, we use this Core time to do a Community Circle (CC). We’re in our first year of implementation, and because my kids have shown genuine engagement in CC time, we hold it almost every Friday.

I normally pick one or two questions/topics for us to go around and share our answers/thoughts. Questions such as, “If you could be any animal, what would it be, and why?” and “What is your favorite food?” are light-hearted and fun.

But, if the topic gets any deeper than that, then I’m pretty much a wreck.

I don’t know what the hell is wrong with me.

I cried when talking with my students about my father, about my son, Gabriel, about the time when I ran away from home.

I get all choked up at workshops when I speak about a specific student.

Just last week, our school invited Kaiser Permanente to put on a play called, *Someone Like Me*, for our junior high students. It’s about adolescent bullying awareness, and one of the characters had written in her diary that she wanted to kill herself.

The thought of one of my students ever contemplating suicide makes my heart ache, my chest heavy, my head throbs. After the assembly, we were instructed to hold a CC with our students back in our classroom, but I was too emotional to even talk. Luckily, my colleague was there, and I’d asked her to facilitate the discussion.

I don’t want to know what my kids tell their parents when they get home. I wouldn’t be surprised if it went like this, “Jesus, Dad, Ms. Win cried again in our community circle today. And we were just talking about Jell-O.”

Oh, my God, that reminds me. One time — in the evening of our school’s Continuation Ceremony some years ago — a student pulled her father toward me to introduce us, “Dad, this is Ms. Win, she cried just the other day because she was afraid we girls would get pregnant.”

Seriously, I’d signed up to teach math. What is all this crying bullshit? I want to be a badass teacher, and badass teachers don’t cry, for Pete’s sake!

But, there’s still hope for me because one of my student’s moms tweeted this:

]]>The Sutton family took a trip to see the mountains in Rocky Mountain National Park. Linda and her brother, Lee, kept asking,

“Are we there yet?”At one point, their mother answered,“No, but what I can tell you is that we have driven 100 miles and we are about 2/5of the way there.”Linda turned to Lee and asked,

“How long is this trip, anyway?”They each started thinking about whether they could determine the length of the trip from the information they were given.

And I like both methods, especially Linda’s.

Without using a visual, we may have students solve for *x* in the equation (2/5)(*x*) = 100 by multiplying both sides by 5/2.

But I notice two things: 1) Students don’t always remember *why* they are multiplying by the reciprocal, and 2) Students have difficulty showing Linda’s method with an equation like (9/2)(*x*) = 27.

So, I’m having the students think through the problem by answering these two questions:

- If we know that
halves of*nine**x*is 27, then what ishalf of*one**x*? - Now that we know what
half of*one**x*is, what is a*whole**x*?

As we write the fractions, we can keep our focus on the whole number numerator and treat the denominator as if it were a thing, and that thing is not changing.

Another example,

This helps us go back to finding the unit rate in the first step via division, and then find a multiple of that unit rate via multiplication.

Once students make sense of these two steps and become fluent in solving for a whole *x*, then they can work on the not-so-friendly equations — such as (5/6)(*x*) = 4 — because they are more confident and trust the process.

Sure, multiplying by the reciprocal would have solved for *x* in one step, but there’s something uniquely comforting to students when they can first find just *one* part of something.

At the Continuation ceremony last year, Lillian delivered a succinct and grateful valedictorian speech.

A month ago, on March 11, I got an email from her.

I was looking at old pictures on your Twitter and in my camera roll, and I could totally see how much I loved your class. I was tearing up. I’m moving up to Math 3 Honors next year, yet I’m not sure I’ll ever be as excited about math as I was in your class. My current class is something of speed and prior knowledge… Not my favorite environment for growth, but you live and you learn to deal with it.

To this day, I remember so many little things about your classes. You truly changed the way I saw the world. I think my intense activism and political vocalness is in part your doing. I use my voice because you gave me one. I’m not a shy little sixth grader anymore. I’m beginning to come into my own as a badass bisexual intersectional feminist. I’m learning, and you pushed me to do so. There’s a lot of work for me to do on myself and the world around me. Maybe my first pattern equation wasn’t so far away (You told me “just because her equation is right, yours isn’t any less right”).

I miss being her teacher. I miss watching her persevere and hearing her explain her thinking in number talks.

Then, last week on April 7, late in the evening, I saw this video of Lillian posted on Twitter by her friend, Sam. I asked Sam for a copy and got Lillian’s permission to share it here.

I cried hard. Not because her poem is eloquent and powerful and makes me so goddamn proud, but because her message is all too real and urgent. The expectations placed on students by parents and teachers — on top of self-expectations — can be and *are* enormous.

We talk a good talk — about respecting the child and letting her learn at the speed of learning, about persevering and playing with mathematics, about nourishing critical and deep thinking in problem-solving, about ensuring access and equity, about cultivating a voice grounded in truth and heart.

But I’ll be the first to admit that I don’t always walk the walk. I’m bound to a system that requires me to issue a grade at the end of the quarter. I have to do this for each child four times a year. Because that’s just how it is.

At what cost?

]]>I got into my car, entered the restaurant’s address into my phone, Google Maps said I’d arrive at 12:05 PM, and I was upset for thinking I lived closer. I texted Laura to let her know I’d be 6 minutes late. She texted back, “No worries!” (I hate being late, it’s rude and arrogant.)

I instantly recognized her. Of course, she was wearing an Oregon Ducks sweatshirt. We hugged, and the waitress showed us to an empty booth. Laura reminded me that she still needed to get a pair of TOMS after lunch because of the blisters she got from walking all day yesterday in her new shoes. I then reminded her that 20 years ago, we were in Nordstrom for her to buy new underwear because she was too lazy to do laundry.

She handed me two gifts wrapped in *The Sunday Oregonian COMICS *— one dated November 8, 2015, the other July 3, 2016.

We both regretted that neither one us thought of the *Dammit! Doll*. I mean, Jesus, just look at its mishappened head and scraggly yarn hair. I could have made *that*.

I really wanted to order a thick juicy burger because this place could put together a great thick juicy burger. But Laura said she wanted to order something healthy, so I opted for the turkey sandwich instead. (Who goes out and orders a turkey sandwich when it’s readily available in your own fridge at home?!) Then Laura ordered, lo and behold, a goddamn burger with two strips of bacon! (For a split second, I wanted to tell her that there was a burning car outside right behind her, so when she turned to look, I could steal her bacon.)

When we were colleagues in Oregon, Laura was teaching math, and I was teaching science. We became friends on Facebook just this past year, and she learned from there that I’ve been teaching math and giving talks at conferences. She used the word “brave” to describe my speaking at conferences. She said it at least three times, “You’re so brave.”

I told her I was terrified each and every time that I accepted a keynote or featured speaker assignment. She looked puzzled. I told her that the honor of being invited was always bigger than who I was, so it was hard to say no. And I didn’t *want* to say no. My father, a math teacher his entire career, would want me to accept. I’d told my own three children that doing the easy stuff ain’t worth their time, so I accepted because I could hear my own voice preaching. I accepted the invitation to speak because I wanted to bring the voice of classroom teachers and students to the forefront. There are stories to be told, and they are fresh and alive.

I’m not brave, I’m desperate. I’m desperate in wanting to share what’s happening right now with the 100 students on my rosters. But I’m terrified that I might get their stories wrong. I’m terrified that I may inadvertently amplify our small successes and diminish our big failures. I must get it right — the-truth-and-nothing-but-the-truth-so-help-me-God kind of right — or I’ll die a miserable death. Back in December, at CMC-North, Dan Meyer had invited me and two other teachers, Shira and Juana, to be part of his keynote. I’m grateful to Dan for the invitation and was excited that we *teachers* got to share with a wider audience.

So, if you’re in the classroom, I hope you’ll consider speaking at conferences and workshops because the number of students you currently have is the number of reasons you have to say YES. Consider co-presenting with someone if it would be your first time; it is less scary that way. My first gig was with this tall white dude. Be brave, or be desperate, I think one is just a glorified form of the other.

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