A quick plug for Estimation 180

Estimation is more than rounding.

Most of the time we don’t teach this, but it is.

Tabitha (8 years old) had a homework assignment the other night that asked her to imagine she had $100 to spend in a catalog, and to make a list of things she would like to buy from that catalog. She found the latest American Girl catalog and got to work.

There was a table to fill out with three columns.

  1. Description of item
  2. Actual cost of item
  3. Estimate

A couple minutes later she asks, What’s the estimate if it costs five dollars? Should I write $5.01?

She has discerned that estimate means write down a number that is not the exact value.

But that’s not what estimation is about at all. Estimation is about finding a number that makes sense, and not worrying about whether it’s the exact value or not.

The image below seems to be going nuts on the Internet today (despite my exhortations to the contrary! Oh, Internet! When will you learn to listen to me?)


“Is this reasonable?” is a great estimation question. Rounding is one way to answer the question. But if a kid can quickly find a number that makes sense and it happens to be a precise number, then we probably haven’t asked a good estimation question. Rather than mark it wrong because the kid didn’t round, we should ask this kid a more challenging question next time.

What does a good estimation question look like? What would be more challenging?

I’m glad you asked.

Estimation 180. Thinking of a number that makes sense is much more interesting when you have to bring your knowledge of the world to bear.


Is 75 inches a reasonable answer for the difference between the father’s height and the son’s? Is 75 centimeters reasonable?

Oh! And here’s the first in a nice sequence about numbers of pages.

Kindergarten questions

I am spending a bit of time a couple days a week in kindergarten this year. It was part of the now-changed sabbatical plan, but important to me to follow through on.

Today was my first day. It was awesome.

The young ones are working on patterning. AB patterns, AAB patterns, ABB patterns and ABC patterns. I’ll leave the curricular questions for when I know more. Today I’ll take these activities at face value, which is to say: this is the mathematics these children were working on today.

The children were instructed to use square tiles to make an ABC pattern. If you haven’t spent time studying curricular approaches to patterning in early elementary, this means that they were to use the tiles to make something such as this:

Screen Shot 2015-09-17 at 7.12.35 PM

Color is the only variable attribute of the tiles the children were using.

I had several interesting conversations about this task with children today. The one I want to report is the following.

I made this pattern:

Screen Shot 2015-09-17 at 7.17.32 PM

I asked the girl sitting next to me whether I had made an ABC pattern.

Girl: Yes.

Me: How do you know?

Girl: [blank look; long pause]

Me: Show me how you know it’s an ABC pattern.

She carefully points to each tile, saying one letter per tile in the following way.

Screen Shot 2015-09-17 at 7.20.53 PM

She pauses.

Girl: And you need another C.

Learning is messy—beautifully messy.

I left today with two big questions on my mind; each relating to this exchange, and to others not documented here.

  1. What would these children have done if asked—prior to instruction—to make a pattern with their tiles?
  2. How does this kind of patterning work interact with learning to count?

I hope my readers will see that these are not questions I expect to be answered in the comments. I hope you will see that these are big and important questions worthy of wondering about for days, weeks, and beyond. I hope you’ll join me in wondering about these questions, and the consequences of potential answers to them.

I argued a while back that learning is having new questions to ask. I hope you’ll join me on my learning journey.

New year, new job

If you’ve been following along (and honestly, I cannot imagine how anyone could possibly have time to do so!), you are under the impression that I’m on sabbatical leave this year.

There has been a change of plans.

I’ve taken an unpaid leave from my college and am spending the bulk of my professional time on curriculum development work at Desmos as a (nearly) full-time teaching faculty member.


The job actually involves almost no sitting on small children.

Our team is growing (Do you know any awesome designers? Send them our way, please!) We have a lot of great stuff in the pipeline. I’m delighted and grateful for the opportunity to work with this amazing team. It’s well known that I’ve been working with them on an extracurricular basis for some time now (>1.5 years), and this has made the transition super smooth.

I am especially fortunate to be able to set aside a portion of my professional life for ongoing projects that are outside the scope and focus of Desmos (although they are certainly consistent with the overall Des-mission of more and better mathematics for all learners!) I’ll spend a couple mornings a week in a kindergarten most of this school year, for example, and Which One Doesn’t Belong is still slated for a 2016 release from Stenhouse. (We still need to sort out Math On-A-Stick for next summer, but that’s a year away.)

Find what you love. Do more of that. [#tmc15]

Here is the text of the keynote I planned to give at Twitter Math Camp. Actual product may have varied substantially in content (but not in spirit) from the typed original.

If you’ve never seen me talk in a large group you’ll have to imagine the energy, cadence and passion you see in my ShadowCon talk brought to this longer form.

Thanks to the Twitter Math Camp organizing committee for inviting me to talk, and to the whole community for supporting my work over the years. It means a lot; I hope to return the favor many times over.

Lisa Henry’s introduction:

Christopher Danielson teaches and writes in Minnesota. You may know him through his documentation of his children’s mathematical antics on Talking Math with Your Kids, through his exhortations not to share bad “Common Core” homework assignments on Facebook, or through his shapes book Which One Doesn’t Belong?

Of course you may not know him at all. But if you do, you’ll recognize that he holds little sacred besides the responsibility we take on in this profession to foster the growth of young minds.

He is currently working on a teacher guide for Which One Doesn’t Belong? to be published by Stenhouse in the spring. He encourages each and every one of you to promote the heck out of Common Core Math for Parents for Dummies. Most of his time this summer is devoted to bringing Math On-A-Stick—a new event to support children and caregivers in informal math activities—to the Minnesota State Fair, and he continues to work with Desmos on developing online networked classroom activities such as Polygraph and Function Carnival.

This is a very American talk about teaching. From what I’ve learned about teaching in other countries with robust educational systems—Singapore, Finland, Japan, Germany, and so on—the U.S. is unique in its tradition of sink-or-swim for teachers.

We equip new teachers with a modest set of tools and experiences, and we say Do the best you can with what you’ve got!

At the policy level, we understand that this is a disaster. But in the American fashion we try to legislate and standardize our way to improvement. We issue pacing guides and measure fidelity to adopted texts. And of course we measure teacher quality by testing students in an effort to standardize learning.

In this sense, the message of my talk today is a very American one. My message is this: Find what you love. Do more of that.


Viewed one way, this is advice to teachers trying to survive and to serve their students well in the era of NCLB and high-stakes testing. (There are probably several such folks in our midst today.)

And that will be a valuable takeaway, but it’s not the heart of my talk. The heart of my talk is more forward-looking and hopeful.

Sending minimally prepared teachers into the field, leaving them to figure it out on their own, and then evaluating whether they have—these things we do well in this country. If you believe that quality comes from sorting out the bad apples, then we’ve built a good machine for this, and the major impediment to improving it is the unions. I assume that this is familiar rhetoric.

What we don’t do well is orient and induct teachers to a community of professionals. We don’t structure our communities to draw on the diverse strengths and passions of its members. This is something that I understand those other nations I mentioned do much better than we do.


For me, the group assembled here, together with the ones who would be here if they could, and many more of the teaching professionals we interact with—whether regularly or sporadically—for me, this group is a community. A community that grows and changes in response to the contributions of its members. It’s not a community that agrees on everything—no community can while remaining honest, open and vibrant. Instead, disagreements offer healthy opportunities for the community and its individual members to grow.

So the hopeful vision in my message (Find what you love. Do more of that.) is that in identifying where your heart is in this profession, you can strengthen your voice and focus your efforts as you contribute to and help shape this community. What our larger American educational system does poorly—foster a professional community that grows and responds to the diverse strengths of its members—the MTBoS does quite well.

So I put two big questions in front of you:

What do you love?

How can you incorporate more of that in what you do? (In your classroom and in your community)

When I ask, What do you love? I don’t want to hear that you love…

Rectangles or stats or 3-Act Lessons or Spirals or Technology or Groups.

I want you to dig deeper. Those things embody what you really love. Whatever you are truly passionate about is bigger than these things. If you can say why you love rectangles or stats or whatever, you’ll be closer to the kind of thing I have in mind.

So now I’ll tell you what I love and how that—in the context of what I have to do—helps to guide my teaching and my contributions to our community.

This is so nerdy. I really hope this is a safe space for this.

I love ambiguity.

The spaces between the certainties are much more interesting to me than the certainties themselves. Ambiguity can provoke wonder, surprise, reflection and clarification.

I’ll share with you how I incorporate this love in the work I do in the classroom and in our community. Let’s start with a video.

This video lacks ambiguity.

Children are smarter than this.

Children can handle ambiguity, which is what I have found so appealing about the Which One Doesn’t Belong framework. (Megan Franke and Terry Wyberg mentions)

In case you aren’t familiar with it, I’ll give you the rap I give kids.


Let me tell you what children say in response to this richer set of shapes. 


[do that]

As they talk about these things, I summarize, paraphrase, probe, review and restate. By the time we’re done, we have a list of properties of shapes. Which of these properties are they supposed to use in deciding which one doesn’t belong? Which of these properties are important? Which ones matter? It depends.

Here’s another set of shapes. 


Sometimes ambiguity comes from studying new objects for which we don’t have a repertoire of vocabulary.

Which properties are important? Which ones matter?

We don’t know yet when we’re looking at a new class of objects. But when we do agree that a property is important, we can name it.

So all that vocabulary which is associated with geometry—that vocabulary isn’t important on its own. Instead it points to important properties, or to properties which somebody has deemed important.

(A non-math example: at some point, it became clear that this growing collection of math teachers online needed naming. The community’s historians are still working on the lineage of the phrase mathtwitterblogosphere and the abbreviation mtbos, The point is that the name exists because there was a thing that needed naming)

Here is what kids do with the ambiguity of the spirals.

[say what that is]

One last WODB example.


The shape in the lower right is the one that provokes discussion. (get to vertices)

Returning to the theme of what you have to do…

In my capacity as a College Algebra teacher, I have to teach rational functions.

Lots of rules and vocabulary and certainty are associated with standard textbook treatments of rational functions. There are asymptotes (vertical, horizontal, oblique…), rules to go along with locating these. There are zeroes and intercepts and symmetries and on and on…

So I played Polygraph: Rationals with my students.


The design of Polygraph is that it puts you in the position of needing to describe to somebody else what you see before you have a shared vocabulary for it—as with WODB. But now importance has an implicit definition. A property is important if it’s useful for distinguishing between functions, and for helping your partner to do that too.

At the beginning of the game ,we shuffle the functions to suggest to you that location in the grid isn’t important. But maybe you think it is. So you ask about it. And you’re likely to get burned.

Orientation, by the way, is important in this context. The difference between 1/x and -1/x is one of orientation.


So orientation doesn’t matter in plane geometry, but it does matter in coordinate algebra.

And the graph on the left is a function, but the one on the right isn’t.


Orientation matters.

But these are both squares.


(Tangentially related…How far can you turn a parabola and have it still be a function?)


I’ll finish with an example from the community. Meg Craig cares about kindness and empathy. She wrote about this on her blog recently. She urged us all to remember that We as teachers are all trying, to the best of our ability, to have students reach the best of their ability.  If you haven’t read this post, you need to.


That kind of thing has a lasting impact on our community. Just the other day, I got worked up about fences. 


There is so much wrong here. Unambiguously wrong, and I called it out. Then I was thinking later that day. OK. That was angry Triangleman. What could kinder gentler Triangleman do?

Well, what do I do in my classroom? What do I do at home with my children? In both of these contexts, I educate patiently. I accept people for where they are and I try to help them see new perspectives; to think differently about things than they do right now.

What does that look like on Twitter? I don’t know. It really is such a great medium for ranting. But I do know that the reason I’m asking myself this is that MathyMeg brought her strengths and passions to our community.


I told you to find what you love, and to do more of that. I told you that I love ambiguity. Maybe you’re the opposite of me. Maybe what you love is certainty in mathematics. How can you help your students appreciate and understand the unique nature of mathematical truth (different from all other disciplines)?

Maybe your heart is with the beauty of geometric forms, or the rhythmic regularity of patterning. Maybe you love how statistics can inform us as we strive to make equitable decisions in an unjust world.

Truth, beauty, regularity, fairness…

Each of these is a more important grounding for your classroom perspective than a pacing guide or textbook sequencing. But none of them is antithetical to these either. What you love can be found in what you have to do.

So my message to you is simple. Name what you love—be explicit about what makes your mathematical heart sing; what resonates in the depth of your teacher soul—and look for it in every corner of your professional life. Share with your students and share it with us, your colleagues and your community.

Fruit snacks

Kellogg’s has issued Froot Loops fruit snacks in the shape of digits. (Side note: Cheez-Its need to get on board with this! There have been Scrabble tile Cheez-Its for years. We want numbers, operations and relational symbols!)

Naturally I bought some.


Tabitha (8 years old) asked—as she does in these scenarios which occur with great frequency—Are you just buying that because it’s mathy?

Yes, sweetie. Yes I am.

But how to put them to use?

After many rejected ideas, here’s my favorite.

The task

Here are the contents of one pack.


That’s 5, 2, 9, 1, 3, 2, 4, 3, 9. Their sum is 38.

I’m setting the over/under on the sum of the next pack at 41. Do you want the over or the under? Why?

Play along with your questions and answers in the comments.

I’ll open the pack on Wednesday, May 27.

Help Wanted: Math on a Stick

The following is cross-posted from Talking Math with Your Kids.

I want to tell you about a vision of a beautiful thing, and I want to ask you to help make it happen.



Math on a Stick logo by Emily Bremner Forbes, who makes beautiful things. Many thanks, Emily!

Math on a Stick will be an annual event at the Minnesota State Fair (12 days of fun ending Labor Day!) that engages young children (4—10 years old) and their caregivers in informal mathematics activity and conversation using the Fair as a context.

  • Parents will push children on a protractor swing so that together they can notice the angles and fractions of a circle the children travel through.
  • Parents and children will use beautiful tiles to make shapes and intriguing patterns.
  • They will comb the fairgrounds looking for groups of many different sizes, asking questions such How many mini donuts are in a bag?, How many sides does the Agriculture-Horticulture building have? and Why is it so hard to find a group of 17?
  • They will notice the rotational and reflection symmetry in a wide variety of plants and flowers, then copy these symmetries by making a paper flower to take home.

Math on a Stick has four components:

  1. The Math-y Midway
  2. The Garden of Symmetry
  3. The Number Game
  4. Visiting mathematicians and mathematical artists.

Find out more about each of these below.

The major question now is whether Math on a Stick happens for the first time this year or next. The organizing body is the Minnesota Council of Teachers of MathematicsThe Math Forum is by our side. Max Ray and Annie Fetter from the Math Forum plan to come to Minnesota to help run the event. The Minnesota State Fair and Minnesota State Fair Foundation love the idea. We just need to convince all parties that it is possible to pull this off in the coming three months, and we need to locate the funding to make it happen.

Your Call to Action

We’ll need help with three things:

  1. Volunteer hours this summer, before the Fair
  2. Volunteer hours during the Fair
  3. Funding

Of course I expect that most who heed this call will hail from the great state of Minnesota, but I encourage others to consider scheduling a visit. This will be a wonderful event, and the Minnesota State Fair is truly a grand spectacle.


Before the Fair, we’ll need help finding and creating the things that will make the event go.

During the Fair, we’ll need help staffing the event. It runs 9 a.m. to 9 p.m. August 27—Sept. 7. We’ll have have about four shifts a day and we’ll require multiple people staffing each shift.

If we get Math on a Stick up and running this summer, one of our first orders of business will be to establish our volunteer website. Please check your summer calendars, pencil us in, and keep an eye on this blog for more information.


If you (or someone you know, or an organization you are involved with) are in a position to help fund Math on a Stick, get in touch with the Minnesota State Fair Foundation to let them know you’d like to help make this happen. Our overall budget is on the order of $20,000.

The specifics

Here are specifics on the four components of Math on a Stick.

The Number Game

The major activity at Math on a Stick is The Number Game. Adapted for math from the Alphabet Forest’s Word Game, children and parents are challenged to find groups of every size 1—20 at the fair. Examples: A corn dog has 1 stick, a cow has 4 legs, the Ferris Wheel has 20 carts.

Players receive a form they carry with them around the fair to record their findings, and can return with a completed form to claim a ribbon. Additionally, players can email, tweet, and post to Instagram, their Number Game fair photos. These are curated by Math on a Stick volunteers and posted to a public display that resets each day so that collectively State Fair attendees recreate daily a new visual answer guide to the Number Game.

The Math-y Midway

A protractor swingset, tables with fun tessellating tiles, and images from Which One Doesn’t Belong? and a (forthcoming) counting book to play with and discuss.

The Garden of Symmetry

Flowers are grown in planters along a path. As you walk from one end of the path to the other, you pass flowers with increasingly complex symmetry. Grasses (with one line of symmetry) are near one end. Irises are a bit further along (with three rotational symmetries), and sunflowers are near the far end (with MANY symmetries). Visitors to the Garden of Symmetry are invited to carry a tool consisting of two small mirrors taped together to investigate symmetries in the garden and the interpretive signage.

Visiting mathematicians and mathematical artists

An activity area is set aside for a daily visit from a mathematician or mathematical artist. Each provides engaging, hands-on math activities during a scheduled period each day. We will draw upon talent from Minnesota, as well as nationally (budget allowing).

For full details on the event, have a look at our Math on a Stick white paper.

Hit me in the comments with any questions you have.

Get in touch with me through the About/Contact page on this blog.

Please help us build this thing. It’s going to be great!

The Twin Cities Shapes Tour

I recently put out a call for K—2 classrooms in which I could talk shapes with students. As a result each of the next several Mondays (Presidents’ Day excluded), I will be in a different early elementary classroom somewhere in the Minneapolis/St Paul metro area.

Last week I was at two schools: Dowling in Minneapolis and Echo Park in Burnville. I talked with one kindergarten class, three first grade classes and four second grade classes. I have learned a lot.

In particular…

Young children find composing and decomposing shapes to be much more compelling than adults tend to. They nearly all saw the bottom-right figure here as being a square and four circles. Adults can see that, of course, but we are more likely to think “not a polygon”.


On that note, I am now quite certain that we spend way too much time having young children sort polygons from non-polygons. That bottom-right shape has many more interesting properties than that of not being a polygon.

For example, a class of second graders on Friday were variously split on the number of “corners” that shape has. Is it 0, 4 or 8? Second graders can understand each other’s arguments for and against these possibilities.

These arguments can lead to the reason that mathematicians use vertex instead of corner. “What exactly is a vertex?” is a much richer and meatier mathematical question than “How many vertices does this shape have?” But if that latter question only comes up with respect to convex polygons, then it is unproblematic and not interesting for very long.

So imagine for just a moment that the lower-right figure has 8 vertices (and it wouldn’t be too difficult, I now believe, to get a classroom full of second graders to agree to this perspective, whether it agrees with the textbook definition of vertex or not).

Now kids can work on stating exactly what makes a vertex.

And what makes a vertex is going to be awfully close to what makes a point of non-differentiability (large point at apex of figure below).

Screen Shot 2015-02-08 at 4.35.50 PM

I’m telling you: in twenty minutes with second graders, we can get very close to investigating things that are challenging for calculus students to describe. My point is that second graders are ready to do some real mathematics, and that sorting polygons from non-polygons is not the road to it.

Other things I found interesting:

• When kids give us something close to the answer we expect, it is easy to fool ourselves into thinking they understand. Example: on the page below, one boy said about the lower left shape that “if you tip your head, it’s a square.” A couple minutes later, it occurred to me that there might be more to the story. I asked whether the shape is a square when your head isn’t tipped, or whether it only becomes a square when you tip your head. He confirmed that it’s the latter.

2• Another second grade class was unanimous that the one in the lower right doesn’t belong because it’s not a square. When I asked “is the lower left now a square, or does it only become a square when you tip it?” the class was evenly split. This was surprising to both me and the classroom teacher.

• Diamondness is entirely dependent on orientation in the mind of a K—2 student.

• The 1:1 correspondence of sides of sides to vertices in polygons is not at all obvious to young children. I sort of knew this but saw it come up again and again in our work.

• A first grader said that the spirals below didn’t belong with all the other shapes we had seen that day because “you can’t color them in”.


Even the unshaded ones that had come before could have been colored in, you see. These spirals you cannot color in even if you try. What a brilliant and intuitive way into talking about closed figures—those that can be colored in.