Warm Ups: Number Talks and More

In my last blog post, I wrote about creating Doing-Math and Thinking-Math classrooms. One small but important ingredient I’ve found helpful for both is a good warmup activity.

The goal of the warmup activity is to get students thinking and active right away. The barrier to entry should be extremely low, so that everyone can able to participate. It’s a chance for students to limber up their minds and possibly get an early success under their belts.

There are lots of great warmups out there, including games like Don’t Break the Bank and simple arithmetic challenges like Target Number. One teacher we work with has been leading some amazing Broken Calculator challenges as warmups, which has been fantastic to see.

But today, I want to talk about a classic warmup, and one that provides one of the best returns on investment in terms of time and energy required: Number Talks.

example-number-talk

Number Talks are one of the best bangs for the buck routines you can incorporate into your math classroom. They’re build number intuition and fluency while giving you insight into how your students think; they support the idea that math makes sense, and you can explain what you see to help it make sense to others.

If you’re not familiar with Number Talks, our how-to guide is here. In a nutshell, the idea is this: pose a simple arithmetic or counting question, along with the prompt to not merely answer the question, but to come up with as many different ways of answering the question as you can. That minor tweak makes the activity more challenging, interesting, and sparks conversations that the students actually get invested in. It takes a few days to help kids learn all the routine, but once they do the benefits are massive.

From teachers who are familiar with Number Talks, the number one request I get is where to find sequences of questions to pose. We’ve written up a bunch of these lately, starting with counting collections of dots, and moving into solving math expressions. If these would be useful for you, please take them for a spin.

There have also been some exciting developments on Number Talk technology recently. My favorites:

  • Fraction Talks (for upper elementary, middle school, and high school)
    This is a fantastic resource with tons of ideas for extending classic Number Talks into higher grades with fraction-based images.
  • Visual Patterns (for upper elementary, middle school, and high school)
    These tend to be accessible as warmups only after students have had some practice with how these growth patterns work. For upper middle and high school students, however, these are an effective warm up, and a place you can see the huge payoff in how the numeracy encouraged by doing Number Talks regularly at the lower grades connects to algebraic thinking in upper grades.

visual-patterns-4

  • Would You Rather
    Leading a Number Talk using a simple comparison question (“Would you rather have 16 dozen dollars or $150?”) can motivate estimation. Using questions like the ones at the link, where certain aspects may be more poorly defined, can be an interesting way to connect mathematical thinking, common sense, and logical argument.
  • Unit Chats
    Unit Chats are a new innovation of Christopher Danielson. These are in a nascent stage of development, as you’ll see if you click the link, but I’m very excited about them. The main idea is to show a picture that contains different choices for units (i.e., avocado halves or avocado wholes vs pits?. You’re not just giving an answer + your strategy; you’re giving a unit as well, which has the potential to change the question and the answer.

avocados

I recommend finding a warm up that works for you and then making it a habit. Keep it short—5 to 10 minutes is optimal, in my opinion—and make sure it gets students thinking immediately. Try to do it 2-3 times a week at least, or every day if you can. It’s a small adjustment to your teaching routine, and one that can pay off in a big way.

One last thought: most curricula start with an instruction for teachers to “show students” how to do such and such a problem. You can run these teacher demonstrations as Number Talks instead. By giving students a chance to think about the problem first, they’ll be that much more primed to learn when the answer—and the explanation—arrives.

The Doing/Thinking/Loving Math Classroom

Summer Staircase 1This summer, we had the opportunity to draw up and institute a wholesale program from scratch, using our own lesson plans and providing the PD and support. Following that program, we began some fascinating conversations about how to articulate our vision of excellent math classrooms in more detail. From 30,000 feet, the direction we want everyone to move feels clear: our goal is to help design experiences that give everyone a chance to fall in love with math. But I feel like we’re due for a new articulation of where we’re aiming.

So what does this loving-math classroom actually look like?

  1. The Loving-Math Classroom is a Doing-Math Classroom
    The central activity in the class must be the doing of mathematics. Our metaphor here is brain as muscle, and classroom as gym. The instructor needs to launch the activity so that everyone can get started (and won’t injure themselves), but fundamentally, the actual “working out” is done by the students. The students must encounter activities that are difficult enough to be challenging but not debilitating. Most of class time (50 – 80%) should be spent actually doing math, preferably individually or in groups. This might mean playing math games (i.e., Pig), working through complex math tasks (i.e. The Power of 37), and also story problems and more plebeian math worksheets, though it can be problematic to rely too heavily on these.
  2. The Loving-Math Classroom is a Thinking-Math Classroom
    In addition to the actual doing of math, which is primary, it’s vital to find moments to take a step back and reflect on the process of doing math. What type of tools are useful to solve problems? How should we organize our data to best find patterns? What are good problem-solving techniques, and which whens are most useful for which kinds of problems? What do we currently know, and what questions do we still have? These times for reflection and discussion are critical for encouraging a depth of thought and development of the habits of mind that are, arguably, the true goal of math education.

What does this look like in real life? Our Summer Staircase classroom had a four-part structure:

  1. Warm-up (5-10 min) — keeping with the gym metaphor, a quick game or exercise to get the mathematical thinking started. Number Talks are a prime example.
  2. Launch (5-10 min) — this is the only time in lesson structure that doesn’t explicitly include doing or thinking math: the time when the teacher is explaining something new. In general, the teacher should streamline all explanation to the minimum required to allow students to work on their own. Sometimes the launch is time for a mini-lesson, or to pull together ideas from previous lessons. Sometimes it’s simply to demonstrate a new game or activity. In general, the goal of the Launch is to get them to the starting line, not the finish line.
  3. Work (30 – 45 minutes) — this is the period when the students actually do the math. For the summer, we used a station model, since younger kids usually have trouble focusing on a single activity for a long period of time. Typically, stations included a math game, a new problem or task, and an activity aimed at helping kids practice to master a technique or get comfortable with a mathematical representation. Starting in 3rd or 4th grade, and certainly by middle and high school, it’s more reasonable for students to be able to focus on a single, more complex task for the duration of this time.
  4. Wrap Up (5 – 15 minutes) — this is when the doing-math class has the opportunity to become a thinking-math class. The Wrap Up allows students to articulate their conjectures, counterexamples, arguments, and questions from class. It’s also where the teacher can underline deeper lessons learned, in line with the Common Core Math Practices, for example.

We found that virtually everyone could create a doing-math classroom. All it takes is the teacher belief that student activity is the central function of the classroom, combined with good materials, which we provided. Creating a thinking-math classroom is trickier; it takes more artistry and more practice.

______________________________

Aside: Last year, we wrote an introduction for our 2015 Summer Staircase curriculum that outlined what we dubbed five principles of extraordinary math teaching that later become my TEDx Talk:

  1. Start with a question
  2. Give students time to struggle
  3. You are not the answer key
  4. Say yes to your students’ thinking and ideas
  5. Play!

It’s clear that these are all urging teachers toward a doing/thinking/loving math class as well. Principles 1-3 are designed to get kids doing and wondering as quickly as possible, and to avoid short-circuiting the doing/thinking process. Principle 4 is about expanding from doing to thinking. Principle 5 is about creating the atmosphere that actually makes it all jell.

______________________________

So here’s the goal for us now: we need to articulate and fully formulate a vision, first of the doing classroom that teachers can make their own by a series of small, manageable steps, and second, of a pathway from the doing to the thinking classroom. Here’s a small beginning to that process:

Doing-Math Classrooms ideally involve a Warm Up, as brief a teacher Launch as possible, and then most of the class time devoted to the students working on a deserving problem or task.

Thinking-Math Classrooms are Doing-Math Classrooms that additionally include Wrap Ups for reflection and development of higher order skills, in line with the Common Core Math Practices.

Loving-Math Classrooms are Thinking-Math Classrooms that, at least occasionally, blow kids’ minds.

We’ll try to go forward with sketching a fuller vision of all of this. In the meantime, I’d love to hear your feedback.

App Review: Exploring the Core

I just downloaded a new app that I think will be helpful reference! It’s called Exploring the Core. (At last check, it was $2.99.) It is, essentially, an easy-to-use encyclopedia of Common Core math content standards. It includes a list of the standards, by grade, and a glossary to help you understand what all the terms mean. So, handy, though not much added value there.

But what is really special is the skills section of the app, which gives a map by major content area of the skills students will need to know. Say we click on Operations and Algebraic Thinking.

Map in Exploring the Core

We’ve got an easy-to-scan map of skills by grade and type. And the types are easy to understand: word problems, fluency, equations, graphing, etc. Click on one and you get a page that shows you the target, the standards that match it, and mathematical images that explain it, along with the conceptual knowledge beneath it. skill in exploring the coreAnd here’s the real beauty of Exploring the Core: the examples are mathematically rigorous, clear, connected to a solid conceptual understanding, and feature useful diagrams/manipulative ideas to use when teaching. How easy would it have been to just say, know your sums and differences up to 10, and apply up to 20 with this standard? By applying the images, the app suggests the kind of thinking numerically fluent students should actually be doing to build fluency. If they don’t have everything memorized yet, you can use ten frames and manipulatives or pictures to help get your kids there.

And that’s what makes this app so useful. I think Exploring the Core will prove helpful for me and the teachers I know, as a reference for the CC standards that fills in what’s between the lines with mathematically accurate images.

2

Breaking and Remaking Lesson Plans

I just had an interesting lab session with a school district we’re working with this year. The teachers (K-2) want to extend the scope and dynamics of their lesson plans, and make sure they challenge their students. But the teachers also don’t have much free time, are, in many cases, new to the curriculum, and also feel that they need to be in alignment with the curriculum, for various administrative reasons. (Although the expectation from admin is that the curriculum will be supplemented. But still.)

The implications for me were that rather than demo one of my favorite Math for Love games or lessons, I wanted to model how one might break and remake a lesson plan. I wanted a process that would be quick and dirty, so teachers could implement ideas quickly; after all, they might have only a short time to review a lesson plan from the curriculum before teaching it.

The lesson that was given to me to demo was flawed in some pretty intense ways. For one thing, it was a lesson for 1st graders that was meant to be teacher-led instruction for a full hour. (Let that sink in.) The goal of the lesson was to have students connect story problems and subtraction equations. It included, in the middle, almost as an afterthought, the instruction to let students come up with a few of their own ideas for story problems, which the teacher would transcribe on the board and solve altogether.

Here’s the process I wrote down, which I think does a pretty good job of imitating the sieve I pass lesson plans through in my own mind.

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I thought this was a pretty good draft, and a good introduction to breaking and refashioning lesson plans to fit your classroom better. I’d appreciate feedback, especially if you have some favorite strategies my list is missing.

For this particular lesson, I launched with a related warmup (Target Number), and built an example story problem with the class. In fifteen minutes, we had accomplished what the lesson had said should take forty minutes, and then I released the kids for what we decided should be the main event: writing and solving their own story problems, and challenging each other with the ones they’d written. By launching quickly and letting the students generate their own work, we’d also succeeded in raising the ceiling of the task, and many of the students showed that they were comfortable with much larger numbers than the class had explored up to that point, adhering, as it had, to the curriculum.

And more than that, it was fun. The students were proud of their work, and stayed engaged, even though the material, without their input, wouldn’t necessarily have held their interest. Here are some of the student’s problems.

student-crafted-story-problems1 student-crafted-story-problems2 student-crafted-story-problems3

After 15-20 minutes of students dreaming up and solving their own problems, we had plenty of time to wrap up and pose even more difficult problems, involving one-digit numbers everyone was comfortable with, but more complicated stories, involving addition AND subtraction.

Many teachers don’t have the time, expertise, or permission to go too far outside their curriculum. But the good news is that most lesson plans in your curriculum are built around some good idea, even if it’s only a kernel at the center. But there’s a filter you can put these lessons through that accentuates the good while straining out the bad.

You’ve seen my draft above. Try it out, and let me know how it works for you, and if there are changes that should be made to it.

Thoughts on story problems

Story problems! They are the great bugaboo of math class, the problems everyone remembers hating, On the other hand, when educators think of “real-world” math, useful math, or motivated math, story problems are where they want to go. And this instinct makes sense. Story problems should be a fantastic resource in the classroom—a chance for reading comprehension, making sense of problems, modeling, and more—but instead they’re dreaded and ineffective.

The seedy underbelly of story problems was stunningly revealed recently in this video by Robert Kaplinsky.

Let’s take a minute and consider what happened here. Three quarters of the students who saw this question attacked it with a kitchen-sink strategy: just do some arithmetic with the numbers and maybe you’ll get it right. Why don’t they do what the other quarter did and say, “this doesn’t make sense”?

It’s possible that they don’t want to disappoint the questioner, and that they figure doing something is better than doing nothing. It’s possible they didn’t feel comfortable asking questions or expressing confusion. But look at that last girl describing why she decided to divide; this strategy doesn’t come from nowhere. These kids are doing story problems as they’ve been taught to do them.

In school, many teachers teach kids how to solve story problems as a sort of code. There’s a protocol:

  • Step 1. Underline the numbers
  • Step 2. Circle the important words, such as plus, minus, sum, product, difference, quotient, together, and, more, less, etc.
  • Step 3. Create an equation using the numbers and the operations corresponding to those words. (If the operation is subtraction or division, we’ll subtract the smaller number from the larger, or divide the smaller number into the larger)
  • Step 4. Solve the equation. That’s probably your answer.

This kind of rubric for solving story problems is self-defeating. We’re basically turning the intuitively sense-making project of reading a story into another kind of encoded math project, devoid of meaning. There’s a subtle line here, because teachers want to help, and underlining or noticing key words isn’t inherently a bad thing. But to begin by sweeping aside all pretense of meaning in favor of a mechanical process is bad mathematics.

Not surprisingly, it ends up being self-defeating as well. Story problems get trickier as students get older, and when they do, these kinds of mechanistic strategies backfire big time. The fact that most of the eighth graders in the video still seem to approach problems in this way spells trouble for their future in math.

So what’s to be done? Story problems have been around for millennia, and though they’ve often felt a little contrived, I don’t think they’re going anywhere. And truly, they are a relatively untapped resource. How do we use them to better effect?

Here are some ideas. I’d love to hear yours too.

Idea 1. Use story problems, but don’t teach a rubric to solve them.
Drop the story-problem “strategy” and focus on helping kids draw pictures or models of the situation instead.

Idea 2. Create story problems that resist basic strategies.
Interestingly, it’s not that hard to write story problems that can’t be cracked with the rubric-based story-problem-solving strategy. I’ve seen these pop up on high-stakes tests at the end of the year, leaving teachers to feel cheated that the test was gamed with questions designed to fool their students. But what if all the questions always resisted the basic hack of getting the answer without understanding the problem?

There are a number of ways to create hack-resistant story problems. The simplest is to create problems that don’t follow the same basic structure, but can be better solved by understanding the situation or drawing a picture. Here’s an example of this kind of problem from our Summer Staircase curriculum. This is a sheet for 2nd graders. Simply adding more steps to the problem makes these virtually impossible to solve without understanding what’s happening. On the other hand, if you draw a picture or build a model, they’re quite straightforward.

a-trip-to-the-bakeryGet the full worksheet here.
Idea 3. Create story problems with real interest as well as complexity.

The name “story problem” suggests a story. Why not tell a real story? Folks like Marilyn Burns and Greg Tang have been writing math story books for some time now (see a long list at living math.net). What’s nice is that these aren’t too hard to write, they get huge buy-in from the students, and they combine the fun of story time with a much deeper thoughtfulness regarding the math. Here’s one I wrote for our Summer Staircase curriculum, suitable for 3rd/4th graders.

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Notice that the engagement created by a good story allows for a much greater complexity in the mathematical modeling. What’s the relevant info for each question? The meaning can’t be lost, and kids are tuned into the meaning because it’s a real story. Some teachers gave kids a chance to draw their own version of the monster, creating an opportunity for an interdisciplinary lesson, involving reading, math, and art.

the-monster

Even if we skip the pictures, we get reading comprehension combined with mathematical meaning. The downside is that creating these kinds of stories is more work. But I can imagine a collectively-produced library of them.

Here are a few more examples of these long-form story problems.
Story Problem – The Ant and the Grasshopper
Story Problem – The Kite

Idea 4. Switch to Video
The 3-act math lesson is another way to grab attention and focus on meaning-making. There’s a lot to be said for this format (and a lot has already been said by others). Check out a lovely example here: The Cookie Monster.

Using video or rich images as a launch can be great. Really, though, this is a different animal than story problems, so I won’t focus on it here.

Idea 5. Use story problems as launches to complex tasks
This is another idea that stretches the very idea of a story problem. Consider a problem like the indecisive director problem.

View Fullscreen
This is a great project, but it resists a straightforward solution, and requires less of the modeling and simple arithmetic of the story problems above, and is more about digging in to a much deeper problem. I’m a huge fan of complex tasks, and personally think they should be much more represented in math class, but this feels like a different animal to me too.

Idea 6. Have students write their own story problems
There’s nothing like standing on the other end of a process to understand its inner workings. A student who finished all the problems in The Kite asked what she should do next, and I suggested she write her own question. Here’s what she came up with—I liked it so much I converted it into a challenge problem for the other kids. (Check out the lesson to see.)

Having kids write story problems is usually a great idea. The dangers are that it becomes another unmotivated exercise, and that their story problems may not be appropriate for others to solve—it can be tough to write a story problem of the appropriate difficulty! That said, there are great opportunities when students are involved in the back end of producing problems as well as solving them.

What’s your take on story problems? It seems that story problems are an untapped resource, and with the right approach, they could be leveraged in all sorts of powerful ways. I’m still hopeful about producing a free library of good story problems in the style of The Monster. I like the prospect of combining the meaning-making involved in reading comprehension and mathematics.

Thoughts?

In Praise of the Open Middle with Pyramid Puzzles

Sometimes you hear the perfect word to describe something you were already doing, but didn’t quite realize you were doing. My most favorite recent case of this came from openmiddle.com.

We’ve all heard of open-ended problems, the genuine, intriguing problems that can take us on journeys to unexpected answers. A lesson based around an open-ended problem can be a beautiful thing, and done right, there’s hardly a more exciting educational experience you can have.

The trouble is, it doesn’t always go so well. Guiding a class through a discussion and exploration of an open-ended problem requires tremendous insight and understanding on the part of the teacher, and even the most experienced teachers sometimes have them go awry. Will this student’s comments lead in a productive direction, or just muddle the issue, and leave us all frustrated? Is this problem so hard that we can’t hope for more than a partial understanding of it, or is there some key idea that we’re all missing? When you have to make assessments on the fly, things don’t always go well.

One trick to coax these explorations in productive directions is to game them just a little bit. The teacher can go in with a plan, or an arc of where they expect the lesson to go. So they can “discover” something like the fact that the midpoints of quadrilaterals seem to form parallelograms and ask, “that doesn’t always happen, does it?”, all the while knowing that it is, in fact, true, and with a couple of good ideas about how to prove it. This little bit of lesson planning can greatly improve the chances of an individual lesson feeling like a success, even if it gives up a little bit of the total free of open ended lessons.

Enter the Open Middle. The idea of Open Middle problems is that the beginning (the problem) and the end (the answer) are both clearly well-defined, but the middle of the problem—the part where you look for the solution—is wide open. The middle is where you cast around for ideas, structures, tools, or just blind guesses about what’s going on. Because Open Middle problems have such a well-defined structure, there’s much less of a chance of them going off the rails. This makes them a fantastic tool for teachers who might be taking their first steps into less regimented math teaching, or who don’t feel like taking a chance on an exploration yielding a broad discussion that doesn’t go anywhere. (Read more about Open Middle problems here.)

I produced a number of these Pyramid Puzzles recently, and I think they’re great examples of Open Middle problems. it’s relatively easy to adjust their difficulty as well.

Pyramid Puzzle 1.

Each number must be the sum of the two directly below it in the pyramid. Fill in the blanks.

pyramid-puzzle-1

This is a puzzle that seems easy, but the solution is just out of reach. We can put a 7 above the 2 and 5, but what then? There’s no obvious next step, and our best guess is to take a stab at it, and see what happens. Some people like to work from top to bottom, and try taking a guess of what two numbers might go below the 30 (say, 12 and 18?), while some prefer to work up from the bottom, putting a number in the blank and going from there.

Suppose I put a 10 in the bottom blank. Then I’d have this:

pyramid-puzzle-2

A 10 on the bottom leads to a 51 at the top, which is a problem. But it’s also a useful mistake, since now I know the number in the bottom must be smaller than 10. One wild guess gives me traction, and it’s only a matter of time before I solve this problem.

Pyramid Puzzle 2.

Each number must be the sum of the two directly below it in the pyramid AND no number can appear more than once. Fill in the blanks with positive integers so that the top of the Pyramid is 20.

Bonus: Could the 20 at the top of the pyramid be replaced with a smaller number and the pyramid still be solved? Show how if it’s possible, or show why it’s impossible.

pyramid-puzzle-3

This is a much subtler puzzle, and solving the bonus problem in particular requires you to delve into the workings of these puzzles. Is there some kind of theory that can tell us what the maximum number at the top of the pyramid can be, given the rules that we’re filling them only with positive integers, and never the same one twice? What about differently-sized pyramids?

pyramid-puzzle-4

 

This is clearly the smallest number that can be atop a pyramid 2 stories high.

 

 

pyramid-puzzle-5

 

And this seems to be the smallest number that can be on top of a pyramid three stories high.

 

 

 

So what’s the smallest for four stories high, or more?

We’re in the territory of open-ended problems now, which is where I always seem to end up. But that’s the exciting thing about mathematical exploration: there are always questions pointing in directions you’ve never gone before.

Find more of our Pyramid Puzzles here, at our lessons page.

Check out openmiddle.com for more examples of Open Middle problems.

Pazuju – the great new puzzle you never heard of

I remember years ago reading about Sudoku. Already popular in Japan, the reviewer predicted that it would be featured in newspapers as regularly as crossword puzzles. Since then it indeed had, as predicted, a meteoric rise.

Math educators tend to be fans of Sudoku and similar puzzles (especially KenKen). Logic puzzles motivate the same sorts of thinking we use when we solve math problems: if I do this, then what’s the result? What can I learn if I assume the opposite of what I think is true?

Still, I’ve never been a huge fan of Sudoku. I find the puzzle a little dry, and don’t usually bother with it.

Cut to a few months back, when a publisher contacted me about a new puzzle called Pazuju. Pazuju is a kind of marriage of Sudoku and Tetris, including both logical and geometrical elements. It was hard for me to imagine.

Pazuju

The publisher sent me a sample of the book, and I tried one out. Then another. And suddenly, I was hooked. What I found was a puzzle that had the same logical appeal of Sudoku, but with more variety and interest. In my opinion, it’s quite simply a superior puzzle.

Will Pazuju be featured in newspapers everywhere, and eclipse Sudoku? I don’t know; it’s always hard to know what will rise to the top. But frankly, I think it deserves to be played everywhere Sudoku is. If you find yourself with a little time to spend on puzzles, I highly recommend this one. (I’m especially partial to the smaller 6 x 6 versions, which are generally quicker, though can still be quite subtle to solve.)

You can play Pazuju puzzles for free here. There’s also a book and an app available.

Math Circles for Teachers

Growing Squares.

We’ll thrilled to be offering math circles for elementary and middle school teachers in partnership with the Washington Experimental Math Lab at the UW. With their generous support, these unique professional development meetings are absolutely free. Clock hours will be available.

You can sign up by filling out this survey:

Elementary Math Teacher Circle

Middle School Math Teacher Circle

If you know teachers who would be interested in joining us, please spread the word!

More details:

Meeting place will be at UW main campus – specific location TBA
Meeting time 5:00 – 7:00, or possibly a little longer, to accommodate dinner
Participation is free!

Our dates are as follows, most of which are the third Tuesday of each school month:

Sep 20
Oct 25
Nov 29
Jan 24
Feb 28
March 28
April 25
May 23

Questions? Contact Kristine Hampton at kristine.l.hampton@gmail.com

Summer Staircase Retrospective Part 1

Summer Staircase 1

We recently wrapped up our most ambitious project ever, and as data on it starts to roll in, I thought I’d take a moment to share.

Summer Staircase stats:
—19 schools
—57 teachers
—2500 students (plus or minus)

This summer, we produced a math curriculum for Seattle Public Schools Summer Staircase, a six-week program in Seattle to prevent summer slide and, ambitiously, help kids like school.

Our job was to write a 6-week math curriculum for three grade bands: Kindergarten, 1st/2nd grade, and 3rd/4th grade (these are the grades the students completed in 2015-2016). We also trained the teachers in the spring, and provided support during the summer.

The curriculum was built on the idea that play is the engine of learning. We featured games, explorations, and story problems that were actually stories. Our goals were high engagement, differentiation, critical thinking, and productive disposition. This last one was maybe the most important goal. We wanted students to leave the program believing in the project of education, especially mathematical education.

I’ll share more about the details of the program soon, especially as we begin to parse the data. Browsing through the teacher feedback, I feel like we’re onto something incredibly exciting. The overwhelming response from teachers and site leaders—the summer “principals”— was that something great was happening in these classes.

Math for Love Summer Staircase Teacher Recommendation

Math for Love curriculum recommendation

For example, of the 30 out of 57 teachers who have so far filled out a survey:
100% would recommend teaching math in summer staircase to a colleague
97% would recommend our curriculum to other schools or districts

At first glance, the students seemed to improve on every measure—pre/post assessments, teacher observations of their math understanding, perseverance, sense-making, and argument skills, and even their enjoyment and engagement in math. I’m deeply excited about how using a play-based curriculum can engage an enormous range of students in an experience of mathematics that is both more fun and more rigorous. I’ll be sharing more about the details of the program, the types of lessons we wrote for the curriculum, and the outcomes as we crunch the numbers.

But for now, I wanted to share some of the less tangible growth that teachers reported seeing this summer:

“In their pre-tests students were only writing the answer to each question. I didn’t see any work or thought on their papers. In the post-tests, every students showed thinking and pictures or used counters. They were excited to show their strategies. That’s really exciting to see as a teacher!”
“I think they all (or most of them) learned that they could love school and learning could be fun. This was awesome.”
“For many of them, the enjoyment of math was the single best growth they could have had. They definitely deepened their math understanding of certain concepts. Learning how to win and lose games graciously was huge for them.”
“They all shared a valuable experience in learning with each other (and the teaching staff- we, too learned a great deal), and being a part of the collective and individual student growth!”
“They grew in confidence, grew in their ability to talk about math and share ideas, and growth was evident as they played games and treated one another with respect.”
“[This program] renewed my faith and love in learning and teaching.”

And this from a site leader:

“The kids were so happy and had fun, and LOVED telling me how they figured something out. So even if the scores didn’t move that much, I know that the kids will return to school with a greater acceptance of math, and without that, “I hate math,” attitude.”

Will the scores move? We’ll find out. But I’m very, very hopeful. I think we’re on the cusp of something big.

An Interview with Emily Grosvenor, author of Tessalation!

Tessalation Cover Tessalation! is a new children’s book about a little girl who discovers tessellations in the outdoor world. I backed the project when it was on Kickstarter earlier this year, and my book just arrived. Both the drawings and the writing is beautiful, and it is, to my knowledge, the only book for kids about tessellations.

Emily Grosvenor, the author of Tessalation!, and I have been interviewing each other about tessellations, the mathematical projects that stick with us from elementary school, and big words for little kids. A lightly edited version of this conversation is below.

The website for the book is here.
Tessalation! is also on Kindle here.

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Dan: Why did you choose to write a book about tessellations?

Emily: I’d say tessellations chose me. I’m a magazine writer and memoirist, but one day I got it in my head that I was going to write a children’s book and sat down to do it. I think most parents have this compulsion — when you’re reading a dozen every night you start to see the world in picture book ideas. From there, I thought about what I loved as a child and what I love now. I had discovered tessellations in a 4th grade gifted class and remembered having a blast making them. But I also love tessellated pattern as an adult. All I had to do was look around my home and its patterned curtains and pillows and floors to see I had an obsession. All that remained was the challenge of creating a book where tessellations were organic to the story. Patterns are soothing to look at.

D: Your description of the love of patterns cuts close to my own feelings. I used to imagine the imaginary ball that would ricochet around the room just right to hit some chosen spot, like a trick pool shot. Tessellations are similar—the shapes that align perfectly so foreground and background flip back and forth. There’s a visual beauty there that’s easy to fall in love with. For me, mathematical love is like that: the recognition that ideas fit together with the same sort of elegance and precision as shapes, so that everything meshes perfectly. There’s a deep satisfaction and also a sense of awe that can arise when all the pieces fit. Elation, you might say.

I’m curious about your original encounter with tessellations in fourth grade. Do you actually remember the lessons you did with tessellations? What about that experience grabbed you?

E: Yes, I do remember. Fourth grade was a seminal year for me. Nearly everything I’m passionate about had its seeds in fourth grade: tessellations, Germany (our country of study, I later majored in German and worked for the German Embassy in Washington, D.C.), poetry (our teacher made us memorize a poem a month and I still know three of them and write my own), novels (I specifically remember a book report I did on Mischievous Meg) and Oregon (I think Emily’s Runaway Imagination is the real reason I moved across the country as an adult). All I remember about the tessellation lesson was being shown how to make my own using a square. I made some seals leaping out of waves which, admittedly, was not a very compelling tessellation. But I think it stuck most because of the setting. I was shy and gifted class gave me a small group setting where we could do creative projects and where I felt comfortable enough to participate. It was also distinctly hands-on. Nearly every lesson I remember from elementary school had that quality. And it involved art. Dreamy, imaginative kids absolutely need to make art. I did not connect tessellations to math at all.

What do you think determines whether or not lessons “stick?”

D: That’s a great question, and one I’m often preoccupied with. I remember certain lessons and experiences from elementary school too, and I do think there are some hallmarks that distinguish the memorable ones from those we forget.

I think having control over the experience is critical to making an experience memorable. This is why art can be so compelling: we get to be the actor, the doer. We choose where the line goes, what the colors are, and even if we can’t always make it look like what we imagined, we’re still in charge. Mathematical experiences can involve the same kind of creative ownership, and when they do, they stand out.

Looping this back to tessellations, I taught a class for 2nd and 3rd graders on tessellations a little while ago. It eventually centered on the question of whether you can create a polygon with any number of sides that still tessellates. Can you make a 17-gon that tessellates? A 31-gon? A 99-gon? The problem is really fascinating. It starts with the more artistic project of just finding examples that do tessellate, and there’s a fair amount of coloring and decorating polygons to make cool looking designs. Eventually, we start discovering certain “moves” that change the number of sides of our polygon in a predictable way, and still produce something that tessellates: joining two polygons together, for example, or adding a bump to an existing tessellating shape. Suddenly we have families of tessellating polygons with 4, 8, 12, 16, 20, etc. sides, for example. And then there’s a very clear goal: how do we get those missing ones?

I don’t know for sure, but I suspect this problem, and problems like it, are memorable. There’s a clear answer, but constructing the argument (and all the tessellations) calls for a tremendous amount of choice and autonomy from the students. The fact that the products are so pretty is nice. And the arguments are just as beautiful, even though they’re harder to draw pictures of.

Reading Tessalation! with my nieces

But back to the book! Do you have a vision of how it will be used? A good book for bedtime reading, or for classroom use, or to launch a rainy Sunday craft project at home, or for a walk in the woods? Or is it for all of the above?

E: Tessalation! definitely works best in the classroom or as an introduction to an afternoon activity at home. I am already getting emails from teachers who are using the book as a jumping off point to a discussion of patterns, tessellations and pattern-making. One of my backers hasn’t got the book yet in her hands but printed out her digital PDF for just that.

I specifically designed the book to have entry points for several age groups. For younger kids, such as preschoolers, just finding Tessa within the tessellations is excitement enough. My three-year-old, Griffin, points out tessellations wherever we go.
Older kids will want to make their own and may respond to some of the other, more complicated ideas in the text. For example, on one page I write about bees dancing where to imbibe. What does imbibe mean? Why are they dancing?
Children 3-8 have a natural obsession with animal life and the outdoors and love learning about how nature works. Conventional wisdom in children’s books holds that you use the simplest words possible, but if I want my children to know that the back of a turtle is called a carapace, and it works with the rhyme, I’m going for it!
As for getting kids outside, I have a strong fondness for the international nonprofit Hike It Baby and have been developing a Tessa Hike to do at Hike It Baby meetups. Basically, you read the book, gather objects on your hike, and then make patterns with them in the parking lot. Fun!

D: One more question for you. Tessalation! include lots of long words. Why use advanced words in a children’s book? And are there implications for math learning here?

E: I’ve long bristled at the thought that young people can’t handle big words. The publishing market is designed within evermore defined reading age groups, and understandably so. A teacher or parent who goes looking for a story or book kids can connect with will inevitably search out one that has the themes, language and subject matter appropriate for that age.

I will always be a fan of the simple story well told in simple language — the Llama Llama books are still big in our house — but I also know that children will take ownership of words that they hear. So when my 3-year-old tells me he just found a tessellation on the bottom of his new shoe, it is because he has been exposed to the word and the visual of a tessellation.
Here’s another 4th grade story for you. I remember doing a book report on Mischievous Meg by Astrid Lindgren and using the word “fabricate” to describe the main character’s penchant for storytelling. The teacher marked it with a little note that said: “Your word?”
I was highly insulted — was she asking if my mom had helped or suggesting that that couldn’t be my word?
One of my all-time favorite children’s books is Bubble Trouble by Margaret Mahy. Here’s an example, from the climax of the book’s action:
“But Abel, though a treble, was a rascal and a rebel,
fond of getting into trouble when he didn’t have to sing.
Pushing quickly through the people, Abel clambered up the steeple,
With nefarious intentions and a pebble in his sling!”
When my older son was young, he simply basked in the rhythms of the language, but as he got older he started asking what the words meant. With language, exposure is important. How would you rather learn a word — spoken by your parents, in your favorite book, or on a worksheet in Middle School?
As for Tessalation!, the reason it sat in a drawer for a year and a half was because I got some early feedback that it was a book for highly literate children, and I might think about making it more accessible to all readers. I sat on that for a while. It’s valid criticism. But in the end, I plowed ahead with my original goal, which was to make the cleverest book I could. Kickstarter was a good way to connect with people who respond to that.
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Thanks, Emily Grosvenor, for a wonderful conversation, and best of luck with the book!