Counting in downtown Saint Paul

I had my first book event today—for How Many? at Subtext Books in downtown Saint Paul. Lovely people, great little independent bookstore. You should buy some books from them.

We had a small but loyal crowd that included a three-year old and an eight-year old. The three-year old was charming, as all three-year olds are, and today she answered all yes/no questions in the affirmative. She and I talked about shapes and eggs and money. It was good times.

But I really got to get into the head of the eight-year old.

We discussed the grapefruit page below, and the unsolved mystery of whether there are exactly six grapefruit—the ones we can see directly—or more than that with at least one hiding underneath, possibly reflected in the surface of the bowl.

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We moved on to the next page, which is where the real fun began.

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My eight-year old conversation partner looked carefully, thought for a while, and announced that there must have been more than six grapefruit on the previous page because there are more than six on this page.

I asked, “How do you know?” and it turns out he was visually pairing the grapefruit halves on this page. He used his fingers to show me the pairs he made, but he was having trouble keeping track of their number. So when he came out with more than six pairs of grapefruit halves on this page, that meant there must have been more grapefruit in the bowl.

We flipped pages back and forth several times while sorting this out, and he finally concluded that there were six grapefruit on both pages. Children rarely have math tasks that connect this way, but they expect that the tasks should connect. It was delightful to watch this expectation play out.

Next up was the avocado page.

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He thought for a bit and decided there were “seven point five avocados”. I thought I knew how he knew—same as the grapefruit—but I asked to be sure, and I was wrong.

“Three fives is fifteen, and then divide by two.”

It took a few more exchanges to extract that dividing by two makes sense here because there should be half as many whole avocados as there are half-avocados. Of course this is brilliant and important mathematics, and it arose in the context of making sense of a meaningful counting situation. Also notable is that three fives was a fact he retrieved quickly while three fours (of grapefruit halves) did not seem to occur to him.

The lesson here is that children are brilliant. They build math out of their everyday experiences, and when you offer them opportunities they apply the math they know to make further sense of their worlds.

Another lesson is that my new book—titled How Many?—is out. The best price and free shipping are at Stenhouse.com. If you read it with children, please report back and maybe leave a review at Amazon.

 

Things that give me hope

I am excited to see more and more people working hard to connect students’ informal mathematical thinking to the more formal work of schooling.

The emphasis in the school-home relationship used to be on helping kids do homework (as parodied in these five seconds of the newest Incredibles trailer).

No more! These days there are plenty of projects that seek to stimulate children’s math minds in ways that parallel what we do with literacy.

I’m thinking of the beautiful work of The Museum of Mathematics in New York City, and of Dan Finkel’s Prime Climb and Tiny Polka Dots games. I’m thinking of Malke Rosenfeld’s work, and of Bedtime Math and their associated research at the University of Chicago. and I’m thinking of Table Talk Math.

I’m thinking also of Eugenia Cheng, whose How to Bake Pi does for adults what I want parents to do for kids—show how their natural ways of thinking about their everyday worlds are deeply mathematical.

Some of the momentum for these projects can be traced back to the Cognitively Guided Instruction (CGI) research at University of Wisconsin, which demonstrated that when teachers know the informal ideas about numbers and operations that kids bring to school, those teachers are more effective at helping students learn the formal mathematics of school. The copyright on the first CGI book—titled Children’s Mathematics—is 1999, and it documents research that had been going on for some time before that.

Many of us doing this work now are deeply influenced by this work. Progress on this sort of thing is slow. Time spans are measured in decades, not months or years. But it’s a vibrant space that’s growing. I am optimistic.

Now for the point of today’s post. I want to recommend a delightful new book, Funville Adventures by A.O. Fradkin and A.B. Bishop, and published by Natural Math.

Funville Adventures involves a series of characters in a fanstatical land. Each has a magical power; these powers interact. You think you’re just following some fun and silly adventures on the playground; really, you’re thinking about one of the most important ideas of higher mathematics—functions.

Yet true to the nature of most of the projects I discussed above (and to the nature of this blog), it doesn’t matter whether you know about the relationship between the story and the mathematics. If you do, that’s great. If you don’t but are curious, there’s an addendum for that, and if you just want to stay at the level of the story, you’ll exercise your math mind thinking about the relationship between growing and shrinking, the relationship between doubling and halving, and why flipping upside down has no sibling.

Funville Adventures should be in every Talking Math with Your Kids-friendly library. I supported it as a Kickstarter; I’m a big fan of A.O. Fradkin’s blog. The book is on sale right now. More info and reviews here.

How Many? An invitation to #unitchat

Make Math Playful is an unofficial slogan here at Talking Math with Your Kids. An important part of play is that there is not one right answer. Through Which One Doesn’t BelongI showed a way to make geometry playful. Now with How Many? I’m working on a way of making counting playful.

The idea has grown out of the TED-Ed video I did a while back, and the more I play with it, the more I see it in the world around me. My goal is to help parents, teachers, and especially children see it too.

Most counting tasks tell you what to count. Whether it’s Sandra Boynton’s adorable board book Doggies, or Greg Tang’s more sophisticated The Grapes of Math, the authors tell you what to count—or even count it for you.

How Many? is a counting book that leaves possibilities open and that seeks to create conversations. Creativity is encouraged. Surprises abound.

The premise is simple. Every page asks How Many? but doesn’t specify what to count. Each image has many possibilities.

An example. How many?

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Maybe you say two. Two shoes. Or one because there is one pair of shoes, or one shoebox. Maybe you count shoelaces or aglets or eyelets (2, 4, and 20, respectively). The longer you linger, the more possibilities you’ll see.

It’s important to say what you’re counting, and noticing new things to count will lead to new quantities.

Another example. How many?

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A few possibilities: 1, 2, 3, 4, 6, 12, 24, 36. What unit is each counting? Maybe you see fractions, too. 2/3, 4/6, 3/4, 1/12….others? What is the whole for each fraction? The number 3 shows up more than once—there are three unsliced pizzas, and there are also three types of pizza. Are there other numbers that count multiple units?

All of this leads to two specific invitations.

Let me come talk with your students.

(It turns out my schedule filled very quickly, and I’m no longer seeking new classrooms to visit right now—thanks to everyone for your support!)

If you are within an hour of the city of Saint Paul and work with children somewhere in the first through fourth grades, then invite me to come test drive some fun and challenging counting tasks with your students. I have set aside November 17 and 18 and hope to get into a variety of classrooms on those two days. Get in touch through the About/Contact page on this blog.

Join the fun on Twitter.

I’ve been using, and will continue to use and monitor, the hashtag #unitchat, for prompts and discussion of fun and ambiguous counting challenges. Post your thoughts, your own images, the observations of your own children or students, and I’ll do likewise.

How Many? A counting book will be published by Stenhouse late next year.

Tessalation: A great new book

Tessalation is a terrific new picture book by Emily Grosvenor. The story involves a little girl whose mother needs a bit of peace and quiet, so sends her outside to play. While outside, Tessa (get it?) notices shapes fitting together without gaps everywhere she looks.

I helped sponsor Tessalation on Kickstarter this spring, and our hard copies arrived last week. Naturally Tabitha (9 years old) and I read it together right away.

Here are some of the things Tabitha, Griffin (11 years old) and I noticed and discussed while reading it, and afterwards:

  • The turtles are delightful.
  • While they are somewhat different turtles from the ones we’ve played with around the house for the last year, they have an important characteristic in common—two noses and two tails come together in both tessellations.
  • There are tessellating leaves that look an awful lot like some shapes I’ve made and we’ve played with a number of times. We saw kites and hexagons and triangles in the leaves just as we have in the pink quadrilaterals below.
  • We wondered whether this object counts as a tessellation. (It’s not from the book, but Tessa set a great example for us to notice and ask about tessellations in our world.)

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All in all, Tessalation is perfectly aligned with the Talking Math with Your Kids spirit. It creates a richly structured and playful space for parents and children to notice things and to converse. The language is fun. The images are beautiful. Tabitha and I highly recommend it.


Quick notes: Tessalation will be a component of August’s Summer of Math box. It’s not too late to sign up! Also, we’ll soon have a Tessalation/Tiling Turtles combo pack available. You can order the book right now from Waldorf Books, and e-books from Amazon.

 

Talking Math with Your Kids update

As spring approaches, it’s time to update readers on what’s going on behind the scenes at Talking Math with Your Kids.

The blog

The pace of posting has slowed way down in recent months. Rest assured that we’re still talking math around the house, and that my dedication to helping others do the same remains strong. I have lots to write, but not much time to write it because…

Math On-A-Stick

Two years ago, I began to wonder how to expand the work of this blog beyond the parents who have the time, technology, and inclination to read blogs.

One year ago, I pitched an idea for this to the Minnesota State Fair.

And last summer we inaugurated what is now an annual event: Math On-A-Stick. Planning is under way for year two, with help from the Minnesota Council of Teachers of Mathematics, The Math Forum, the National Council of Teachers of Mathematics, the Minnesota State Fair, and the Minnesota State Fair Foundation.

The number one question at the Fair was Where can we buy the turtles?

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At the time, the answer was “Nowhere”. We had asked permission from their designers, Jos Leys and Kevin Lee, only to cut them for Math On-A-Stick. Soon afterwards, I got permission from Jos to make and sell these turtles. I also got permission from Kevin who adapted Jos’s design for laser cutting using his own software (which is a ton of fun, and which you can buy from him) Tesselmaniac.

The store

The Talking Math with Your Kids Store, at talkingmathwithkids.squarespace.com, opened late last fall with tiling turtles as the main offering. It is now stocked with a number of things to support parents and children in math activities and conversations—Pattern Machines, Tiling Turtles, Spiraling Pentagons, a gorgeous coloring book, and more on the way soon.

Click on through and have a look if you haven’t done so yet.

A book

I recently submitted the final manuscript for Which One Doesn’t Belong? A Better Shapes Book. There will be both a home/student edition, and a companion guide. It is being published by Stenhouse this summer.

More

The big ideas continue to flow, and further collaborations are in the works. Keep an eye on this space. In the meantime, you can expect a few new posts in the coming weeks as my attention shifts from book-writing mode.

And don’t forget to follow the fun on Twitter at the #tmwyk hashtag, where people share young children’s beautiful ideas and questions on a daily basis.

Building a better shapes book [Which One Doesn’t Belong?]

IMPORTANT NOTE: The moment alluded to below has arrived! Which One Doesn’t Belong? is now available from Stenhouse as a student book (awesome for home reading, too!) and a teacher guide.

As a result, I have removed all links to the version I was previously distributing free.

There are many shapes books available for reading with children. Most of them are very bad. I have complained about this for years. Now I have done something about it. Most shapes books—whether board books for babies and toddlers, or more sophisticated books for school-aged children—are full of misinformation and missed opportunities. As an example, there is nearly always one page for squares and a separate one for rectangles. There is almost never a square on the rectangles page. That’s a missed opportunity. Often, the text says that a rectangle has two short sides and two long sides. That’s misinformation. A square is a special rectangle, just as a child is a special person. After years of contemplation, I had a kernel of an idea the other night. The kids are back in school before I am, so I had some flex time available. One thing led to another and voilá. A better shapes book. (Links removed—see above note.)

How to use this book

On every page are four shapes. The question is the same throughout the book—which one doesn’t belong? For example, which shape doesn’t belong in this set? 1 If you are thinking, “It depends on how you look at it,” then you’ve got the idea.

  • The bottom left shape doesn’t belong because it’s not shaded in.
  • The top left shape doesn’t belong because it only has three sides, while the others have four.
  • The top right doesn’t belong because it is the only square.
  • The bottom right doesn’t belong because it’s the only one resting on a side.

Maybe you have different reasons for some of these. That’s great! The only measure of being right is whether your reason is true. With an infant, you can use this book like any other shapes book. Look at each page together. Point at each shape and talk about it as you snuggle. With a young child, ask which one doesn’t belong and why. Most pages in the book have at least one shape that a young child can identify as not belonging. Join the conversation by pointing out a different shape that doesn’t belong for some other reason. With an older child, challenge yourselves to find a reason for each of the 44 shapes in the book. There is no answer key. This is intentional–to encourage further discussion, and to encourage you to return to the book to try again. I have tested the file out on the Kindle app on my iPad, and it looks good. I made one printed copy and prefer it to the e-version because I can leave it out for browsing and we can touch the shapes without accidentally turning the page.

The legal details

I owe thanks to Terry Wyberg at the University of Minnesota, who regularly plays the “Which one doesn’t belong?” game with numbers in professional development sessions; to Megan Franke at the University of California, Los Angeles, who adapted the old Sesame Street game “One of these things is not like the others?” and to my online colleagues including but limited to Justin Lanier, Megan Schmidt, Dave Peterson, Matt Enlow and Andy Rundquist.

Some additional prompts

The following Which One Doesn’t Belong? prompts are yours to use in your classroom or home. If you’d like to share them more widely, please link people here. Thanks.

A book recommendation

Sue VanHattum is a fellow community college teacher and a friend of the project. She cares deeply about math, about parents and kids, and about bringing those three things together for fun and for learning.

She has compiled and edited a book, titled Playing with Math filled with the writing of other wonderful people. Honestly, some of my favorite writers about math and teaching are this compilation. If you don’t have time to seek out amazing writing about math learning on the web, Sue brings Fawn Nguyen, Kate Nowak, Paul Salomon, Malke Rosenfeld, Avery Pickford….so many talented writers and teachers to you in one neat package.

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She is crowdfunding the publication of this book. Any contribution helps make the book a physical reality. For 9 bucks, you’ll get an electronic copy. For 25 bucks, you’ll get a hard copy once it is produced. For 5 grand, she’ll come lead a math playtime with your group!

I have put it on my summer reading list. You should too.

Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers has over 30 authors, who each tell their delightful stories of sharing their enthusiasm for math with others. It was lovingly compiled and edited by a teacher whose passion is to share the love of math with as many people as she can.

The Read-Aloud Handbook of math

Jim Trelease’s The Read-Aloud Handbook is lovely and very helpful for parents wanting to immerse their children in the world of written and spoken language, stories and books.

I aspire to creating the math version of this; the Read-Aloud Handbook of Math in a sense.

Here is how he began.

The dearth of accessible material inspired him to write and self-publish the first edition of The Read-Aloud Handbook in 1979. “I self-published because I never thought any of the major publishers would be interested in it. At that point, ‘reading aloud’ was too simple and not painful enough to do the child any good. At least, that’s what many educators thought,” he says in hindsight. But that mindset would soon change.

His book is now in its seventh edition and has sold nearly 2 million copies.

Wish me luck, OK?

Talking Math with Your Kids on Kindle!

Someday there will be a full-sized paper version of a Talking Math with Your Kids book (Hear that publishers? Wanna talk? You can find me at the About/Contact page.)

Until that day, there is now a mini-version (15,000 words; roughly three chapters, $4.99) available on Kindle (and readable on other devices with the Kindle app).

Tabitha is thrilled with the news!

It is aimed at parents of children from 3—9 years of age. Parents of older or younger children will likely be able to extend the ideas to their own situations, too.

Go have a look, won’t you? Share widely and let me know what you think.

Table of contents is:

  1. Introduction
  2. Counting and other adventures in number language
  3. Adding and subtracting: Two peas in a pod
  4. Conclusion
  5. References and further reading

About 1/3 of the conversations in the book have been previously documented here and/or on my blog Overthinking My Teaching. The rest are new to readers. Also, there is a ton of new content summarizing research and the mathematical development of children in parent-friendly ways.

Early response has been awesome.

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