For my students, on their graduation

You might have heard me say before that I believe that my students and I enter into a community of learning together at the beginning of each school year. While it is our tangible goal to study the measure of the earth (geometry) and change in motion (calculus), our intangible goal is to enter into the practice of learning.

We are not here to celebrate because you’re smart. We are here to celebrate what you’ve accomplished because you are committed to the practice of learning. We are here to celebrate the perseverance that you’ve shown, all of the hours of studying and practice that you have put in. We are even here to celebrate the synapses that have fired in your brain every time you’ve made a mistake – every time you’ve learned something new. We are here to celebrate your kindness to us and to each other. And we are here to celebrate the questions you have asked.

In The Falconer – What We Wish We Had Learned in School, Grant Lichtman suggests that “Questions are the waypoints on the path of wisdom”. We are here to celebrate your journey towards wisdom. I know your parents agree that you are well on your way to learning the art of questioning. You have been asking questions since you could talk: Why is broccoli green? Why is 2+2 equal to 4? Why do dogs bark and cats meow? What if my internet goes out on the night of the deadline? Who came up with the number e? What if it snows and we don’t have class tomorrow? What if my alarm doesn’t go off? Why does the unit circle go counter-clockwise?

As your journey continues, we urge you to keep asking questions – to keep learning –to seek peace and defend justice – to live responsibly – but we also want to warn you away from only doing enough to get by.

In his book about ethics, Sam Wells insists over and over that you cannot know what to do and how to act without preparation. Don’t expect to be able to lead later unless you’ve done the hard work of becoming a leader. You can’t sleep now and expect to do the right thing later. So, he tells the story of a surgery that took a tragic turn in an Edinburgh hospital in the 50s resulting in the death of a young child. Later that week two friends were discussing the tragedy, and one of them expressed sympathy for the surgeon who had run into a completely unexpected complication. The other friend disagreed.

I think the man is to blame. If somebody had handed me ether instead of chloroform, I would have known from the weight it was the wrong thing. You see, I know the surgeon. We were students together at Aberdeen, and he could have become one of the finest surgeons in Europe if only he had given his mind to it. But he didn’t. He was more interested in golf. So he did just enough work to pass his exams and no more, and that is how he has lived his life – just enough to get through but no more; so he has never picked up those seemingly peripheral bits of knowledge that can one day be crucial. The other day [at that table] a bit of ‘peripheral’ knowledge was crucial and he didn’t have it. But it wasn’t the other day that he failed – it was thirty-nine years ago, when he only gave himself half-heartedly to medicine. (74)

Our hope for you is for you to be who you are called to be – to find something to which to give yourself whole-heartedly – something about which you are passionate – and for which you learn all of the peripheral knowledge crucial to doing the right thing. (We aren’t suggesting you should never play golf.)

We’ve spent the past several years convincing you that the part plus the part equals the whole. You remember, right? In geometry – the Segment Addition Postulate – If I have a piece of wire that is 4 m long and another that is 3 m long, then together, I have 7 m of wire. It works for angles, and it works for area. When it comes to learners, though, you give us evidence that maybe Aristotle knew more than Euclid: The whole is greater than the sum of its parts. You are better together than you are alone. And we are better teachers and learners because of you.

Take good care of yourselves and keep in touch with us and each other. Don’t ever wonder whether there’s someone who’s cheering for you. We are, and we look forward to hearing about the next part of your journey.


[I shared this with some of my students at a luncheon celebration last month and last year.]

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Posted by on May 18, 2016 in Student Reflection


Collaboration & Perseverance: What Do They Look Like?

I recently wrote about this year’s circumference of a cylinder lesson.

As I was looking through some pictures, I ran across these two from last year’s lesson.

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What do you see in these pictures?

I was struck by what I saw: collaboration and perseverance.

What do collaboration and perseverance look like in classrooms you’ve observed? What about in your own classroom?

How do you create a culture of collaboration in your classroom?

How do you make sure your students know that we want them to learn mathematics by making sense of problems and persevering in solving them?

Thank you to all who share your classroom stories of collaboration and perseverance, so that we might add parts of those to our own classroom stories, as the journey continues.

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Posted by on May 8, 2016 in SMP1, Student Reflection


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Making a Better Question Worse

I recently read a post on betterQs from @srcav with an area question from Brilliant that I added to today’s opener.


I knew something was up when I saw my students’ results.

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My mistyping was a good reminder of the importance of nouns.

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Without showing any results, I sent the corrected question as a Quick Poll.

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(I wonder whether the way the first question was asked prompted the misconception in the wrong answer, but I won’t find out until I have another group of students.)

We are learning to look for and make use of structure.

We are learning to contemplate, then calculate.

And we are learning how to ask better questions, as the journey continues …


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The Surface Area of a Sphere

The Surface Area of a Sphere

I’ve written before about making sense of the surface area of a sphere. The lesson this year unfolded (unpeeled?) a bit differently.

I’m not sure how students might guess that the surface area of a sphere has something to do with the area of a great circle of the sphere. We talked about what a great circle must be, we used fishing wire to measure the circumference of a great circle of the sphere (orange), and I asked them to estimate how many great circles would cover the orange. You can see the huge variety of responses.

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We cut the orange in half. I showed them the surface of the great circle and the act of “stamping” it onto the orange peeling. Do you want to keep your estimate? Or change it?


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We went from 20 responses below π, 9 infinites, and 3 correct to 13 responses below π, 5 infinites and 7 correct.

I hesitated about what to ask next. We were ready to peel the orange to see how many great circles we could cover and figure out what the surface area formula would be, but I was curious about whether students could make sense of the formula before we did that if I told them what the formula was. So we waited on peeling and I sent another poll.

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68% understood that the sphere’s surface area formula, 4πr^2, meant that 4 great circles could cover the sphere. We peeled it to be sure.


And so the journey to figure out what questions to ask when continues …


#T3Learns Slow-Chat Book Study: 5 Practices for Orchestrating

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After the success of the slow-chat book study on Embedding Formative Assessment we plan to engage in another slow chat book study.

A few years ago, as we embraced focusing our classrooms on the Standards for Mathematical Practice, a number of our community began reading and using the book by Margaret S. Smith and Mary Kay Stein, 5 Practices for Orchestrating Productive Mathematics Discussions.

This book has been transformational to many educators, and there is also a companion book focused on the science classroom, 5 Practices for Orchestrating Task-Based Discussions in Science, by Jennifer Cartier and Margaret S. Smith.

Both books are also available in pdf format and NCTM offers them together as a bundle.

Simultaneous Study
: As our community works with both math and science educators, we are going to try something unique in reading the books simultaneously and sharing ideas using the same hashtag.

We know that reading these books, with the emphasis on classroom practices, will be worth our time. In addition to encouraging those who have not read them, we expect that those who have read them previously will find it beneficial to re-read and share with educators around the world.

Slow Chat Book Study
: For those new to this idea of a “slow chat book study”, we will use Twitter to share our thoughts with each other, using the hashtag #T3Learns.

With a slow chat book study you are not required to be online at any set time. Instead, share and respond to others’ thoughts as you can. Great conversations will unfold – just at a slower pace.

When you have more to say than 140 characters, we encourage you to link to blog posts, pictures, or other documents. There is no need to sign up for the study – just use your Twitter account and the hashtag #T3Learns when you post your comments.

Don’t forget to search for others’ comments using the hashtag #T3Learns.

Need to set up a Twitter account? Start here.

If you need help once we start, contact us (see below).

Book Study Schedule
: We have established the following schedule and daily prompts to help with sharing and discussion. This will allow us to wrap up in early June.

The content of the Math and Science versions line up fairly well, with the exception of the chapters being off by one.

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Daily Prompts

Book Study 2

Contact Information

Moderators will be Jill Gough, Kim Thomas, and Jennifer Wilson.

Please contact if you have any questions.

-Kevin Spry



[Cross-posted on T^3 Learns]


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What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?


As students first started looking, I heard some of the following:

  • positive x axis
  • x is positive, y equals 0
  • they come together on (2,0)
  • (?,y*0)
  • when z is on top of w, z is on the positive side on the x axis


Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?



As students looked for longer, I heard some of the following:

  • The length of the line segment from the origin to Z is the x coordinate of W.
  • w=((distance of z from origin),0)
  • The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

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And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?


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Notice and Note: The Equation of a Circle

I wrote in detail last year about how our students practice I can look for and express regularity in repeated reasoning to make sense of the equation of a circle in the coordinate plane.

This year we took the time not only to notice what changes and what stays the same but also to note what changes and what stays the same.

Our ELA colleagues have been using Notice and Note as a strategy for close reading for a while now. How might we encourage our learners to Notice and Note across disciplines?

Students noticed and noted what stays the same and what changes as we moved point P.

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They made a conjecture about the path P follows, and then we traced point P.

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We connected their noticings about the Pythagorean Theorem to come up with the equation of the circle.

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Students moved a circle around in the coordinate plane to notice and note what happens with the location of the circle, size of the circle, and equation of the circle.

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And then most of them told me the equation of a circle with center (h,k) and radius r, along with giving us the opportunity to think about whether square of (x-h) is equivalent to the square of (h-x).

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And so the journey continues … with an emphasis on noticing and noting.



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Posted by on March 19, 2016 in Circles, Geometry


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