Between 2 Numbers

I created a new site, it’s called Between 2 Numbers.

From the About/Contact page:

The inspiration for this site came from John Allen Paulos’ book Innumeracy. I read the book back in 2008, but reading it again this past summer reignited the inspiration and turned it into fruition. In my lesson plan spreadsheet, I started a column of “tidbits” to share with my students; it’s filled with mathematical fun facts, latest news, and stuff you see on here. I teach middle school mathematics, and ratios and proportional reasoning make up large portions of the curriculum, so I’m always comparing stuff, like tossing my flipflop onto this big Danish clog while visiting Solvang, CA, and wondering how big or tall the person wearing such clogs would have to be.

My goal is to have at least 40 entries to match the number of school weeks.

I would love to hear how you would use the site with your students and any other feedback.

And I hope your school year is off to a great start. I get to teach 7th and 8th-grade math this year. We’re on a block schedule for the second year now, and I’m getting used to it. Be well and teach well, my friends!

Posted in General | Tagged , , , , , , , , | 5 Responses

All I Got on Classroom Management

Not enough people write about classroom management in a practical and realistic way. But this does! Helpful tips.

— Michael Pershan (@mpershan) August 9, 2017

Classroom management is hard, and what’s “practical and realistic” for one person could be entirely impractical and unrealistic for someone else to follow. Take a math lesson taught successfully at a parochial school in a class of 12 students wearing uniforms and bring it into an inner-city public school in a class of 37 students wearing skins in shades of brown, black, and yellow, and you might find that you need to tweak the lesson a bit.

You and I are two different people. You and I are two different teachers. What irks me may not even faze you. You and I can tolerate different ranges of decibels emitting from classroom activities. You like foldables, and I like baguettes.

Your students are different than mine. The humans in our classrooms are fussy and demanding and moody and social and shy and are constantly muttering under their breath I-hate-this-shit. Good luck with that lesson on factoring trinomials.

Therefore, any advice on classroom management is making huge assumptions about who you are, what your students are like, and what your admins had for breakfast. I don’t blame you if after reading a how-to book on classroom management, and you feel stuck at step 4 below.

Unfortunately, the final step of “add small details” is what you must do — like practice, fail, practice, fail, shit, practice, shit, fail again — in the classroom!

No book is gonna help you master this thing. I’m sorry. You don’t become a chef by reading cookbooks. You just gotta go into the kitchen, roll up your sleeves, make a big mess, and voila boeuf bourguignon! Or not.

But before you accuse me of writing a post that is completely useless and unhelpful, please allow me to offer just these three tidbits.

Observe your colleague.

Intentionally schedule a time when you may come in to observe a colleague whom you hold in high regard and whom you may ask a thousand follow-up questions. It’s equally important that this colleague reciprocates the observation and gives you constructive feedback. Most likely you know this colleague better than you know any author of some book, and of course, you two share pretty much the same clientele. For optimal efficiency, and if possible, choose a time when your colleague has the same students who seem to be giving you trouble. Kids act differently for different teachers.

Know what you absolutely value above all things.

For me, it’s respect. Not just respect for me the teacher, but respect for all of us in the room. I don’t always handle this gracefully when disrespect unveils itself in my room because it tops my list of all things that make Ms. Win go ape-shit crazy. But because I know this about myself, and being proactive always yields better results than being reactive, I tell my kids of my zero-tolerance for disrespect from the get-go. I say, “I need you to be respectful. That’s the only way we are going to get along in here. Before we can do any mathematics or have any fun in here, we are going to be respectful to each other.” And may God give me the grace to apologize when I am being disrespectful to my students. I find that doughnuts help them forgive me quicker.

Separate the behavior from the child.

We know this already. But there are adult assholes out there. My hypothesis is that we teachers — and parents and society at large — have allowed the adorable children in our care to morph into assholes without effective intervention. What happened? I don’t know exactly, but I do know that we tend to let misbehaviors slide for fear of hurting the kid’s feeling or that it’s not the best time to deal with it. Well, I always try to immediately find the time and will immediately tell a kid that her actions/words are not okay because while the humans are young, their behaviors are especially removed from their true selves. I’m reprimanding the behavior, but I’m keeping the kid. Being a parent for 22 years and a teacher for 26, I conclude that children are highly manipulative. Not because they think it’s a desirable trait, but because they can’t drive and have no income, being manipulative is their survival mode of choice.

A few years ago I was sitting in the lodge at CMC-North in Asilomar when someone recognized me from my Ignite talk and came over to chit chat. He was lamenting the frequency in which his students were asking to use the restroom pass. I asked some related questions to learn more before I realized that he wasn’t lacking classroom management skills as much as just lacking a good lesson.

So, there’s that, step 0 as I call it — find that great math task.

Posted in Teaching | Tagged , | 13 Responses

Long Live the #MTBoS!

Dan Meyer is calling for the retirement of the hashtag #MTBoS. He’s suggesting and going with #iteachmath or #iteachmaths instead.

I get it. I get it that #iteachmath means I TEACH MATH. I get it that #MTBoS could mean anything. I get it that Dan’s intention is no more than to simply make the math community more inviting and inclusive. After all, Dan is among the first handful of math bloggers who helped transform my teaching, and he continues to inspire and challenge my craft. I’m grateful for Dan’s work.

What I don’t get is folks saying #MTBoS is exclusive and alienating and cliquish.

This kind of statement actually hurts my feelings, like someone just said something negative about my children without ever having met them. I feel hurt and insulted because the #MTBoS that I am grateful to be a part of and the people who are in it are anything but “alienating” and “cliquish.”

That’s because I know of the ENORMOUS amount of TIME, WORK, and LOVE that had been poured into making #MTBoS what it is today.

Don’t know what #MTBoS is? Please start here:

When you’re done there, please see here:

It takes ONE click to learn what #MTBoS is.

Anna Vance’s tweet exactly expresses how I feel.


Posted in Teaching | Tagged | 32 Responses

Euclid’s Algorithm

I show my 6th graders this image, pointing out that this picture represents the two numbers 1 and 1 that I’d entered at the top.

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 OHHHs and AHHHs 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;

  1. When both numbers are the same, then the picture is one square.
  2. The computer simplifies the two numbers, such that a picture for 6, 3 is the same for 2, 1.
  3. 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.
  4. The first number is the horizontal dimension, while the second is the vertical.
  5. 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!

Posted in Course 1 (6th Grade Math), Math 8, Problem Solving | Tagged , , , , , , , , | 10 Responses

Community Circle

We have Core time with our homeroom students — mine are 8th graders — for 3o minutes at the end of each day. Core is our SSR time, but ELD students are in ELD during this time.

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:

Posted in Teaching | Tagged , , | 3 Responses

Solving an Equation with a Fraction

From CPM:

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 2/5 of 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:

  1. If we know that nine halves of x is 27, then what is one half of x?
  2. Now that we know what one half of 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.

Posted in Algebra, Course 1 (6th Grade Math), Math 8 | Tagged , , | 6 Responses


Lillian naturally comes to mind when I plan a lesson. She eats it up. She goes above and beyond. She’s thoughtful and appreciative. She shares her thinking with the class in mindful and modest doses. She smiles quietly at my jokes. I wish I had a copy of all her math work — especially her written reflections — because each and every piece holds the joy of my teaching. Here’s one:

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?

Posted in Teaching | Tagged , , , , , | 17 Responses

Be Brave or Be Desperate

I had lunch with a former colleague — I”ll call her Laura — whom I hadn’t seen for some 20 plus years. She’s now retired and was in my town for a wedding.

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.

Posted in Shallow Thoughts, Teaching | Tagged | 6 Responses

These Twenty Things

I get to wake up without an alarm clock going off for the next 14 days. I will eat leftover grilled ribeye steak for breakfast and Cheerios for lunch. I will drink IPA for dinner and go to my mailbox in my pajamas. I will take a luxurious bath. 

As we wrap up 2016, I’d like to humbly share these 20 things that I have done — or will/want to do — and suggest that you too may want to do some of these things as a human and as a teacher.

  1. Find a reason to make caramelized onions. You can add it to your favorite pasta sauce or mashed potatoes.
  2. Call a parent to let her know how much you appreciate having her kid in your class. Maybe the kid is struggling in your class, but nonetheless, he is kind and laughs at all your jokes.
  3. Listen to country music to realize that your pain ain’t so bad after all — not the country from Garth Brooks and Tim McGraw or any of them pretty boys — I mean outlaw country music from David Allan Coe and Merle Haggard and Willie Nelson.
  4. Lie to your students that they were always on your mind during winter break, then let them hear Willie Nelson’s Always on My Mind.
  5. Watch La Maison en Petits Cubes by Kunio Katō. It won the Academy Award for Best Animated Short Film in 2008.
  6. Make hot chocolate for each kid in your favorite class. Seriously. (Point out to your non-favorite classes that they’d done a poor job in sucking up to you, hence going forward, they ought to try harder.)
  7. Stop consuming products with the label “lite” on it. Sure, it might mean 1/3 fewer calories and 1/2 less fat, but did you know it also means 1/10 of the taste?!
  8. Ask your students, “Did you know that diarrhea is genetic?” Let them ponder that for a few seconds, then say, “Yeah, it runs in your jeans.”
  9. Buy the latest book from #MTBoS: Tracy, Christopher, Mike, John and Matt, Edmund, Malke. (I’m sure I’m missing some people. Please help me out.)
  10. Treat the entire 180 days of school as flu season, spray bleach on everything in your classroom. Avoid the students’ eyeballs.
  11. Finish reading The Sound of Gravel. (For God’s sake, make time to read a non-nerdy book!)
  12. Lie — yes, again! — to your students that you’d graded all their papers over winter break. Then know that you’re fucked and must skip dinner [and life] to grade papers like a squirrel on crack that evening.
  13. Make something from scratch that you’ve never made before, like a baguette. If it comes out looking and tasting like shit, toss it immediately and buy frozen. (Ashli‘s number will be on speed dial as I attempt this.)
  14. Remind students that kindness trumps everything you do in your classroom.
  15. Be kind to yourself. Buy that item you didn’t get for Christmas from your favorite person who is now no longer your favorite. If you sleep next to this person, scream, “I hate you!” in the middle of the night like you are dreaming, except you aren’t.
  16. Connect with your students. Stand up for them. Speak up for them. Difficult decisions aren’t so difficult when we all put children first.
  17. Go to church, go to counseling, go to a friend. Reach out to someone because talking about stuff helps. Writing stuff down helps too. But it’s best to meet up with that person because a good hug is worth the drive.
  18. You are part of a team. Find the rest of your team and collaborate and share strategies and seek solutions. Leave the whiners and downers in the teachers’ lounge.
  19. Let’s not make a list of New Year’s resolutions. It’s like the goddamn pacing guide, sets us up for failure every time. Just repeat #15 above — minus the psycho screaming part, do that just once. Okay, twice. Definitely not more than three times.
  20. Critique the effectiveness of your lesson, not by what answers students give, but by what questions they ask.
Posted in General, Shallow Thoughts, Teaching | Tagged | 20 Responses

Quality Question Metrics

If I drive 60 miles per hour, my journey will take 4 hours. How long will my journey take if I drive 80 miles per hour?

Paulina volunteered, “I did sixty divided by eighty, that equals point seven five, or three-fourths. So, it would take three hours.”

When I asked Paulina why she divided 60 by 80, or what the quotient 0.75 meant, she struggled to tell me her reason. Nor could she explain how she deduced that “three-fourths… so, it would take three hours.”

We have to keep asking why-why-why all the time. Our job is to help students ask better questions. One of my question quality metrics that gets high marks is if a student can ask a question that causes the class to say Oh-shit-I-did-not-think-of-that! 

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