# 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?

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?

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.

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.

# Help wanted: Math On-A-Stick

We are almost ready for Year 2 of Math On-A-Stick at the Minnesota State Fair. Last year’s favorites are back. We have a few delightful new exhibits.

And we need help.

The fair is a bit more than a week away, and we have several shifts that still require volunteers.

If you are in a position to help, please do. Here’s how:

• Bring a friend.
• Recruit, recruit, recruit.
• Email your colleagues (and neighbors and students and….?)
• Forward this post to everyone you know!
• Tell your spouse he/she is gonna need to rally for the cause.

Here are dates and times we especially need help:

Sunday, Aug. 28 5:00—8:15 p.m.

Monday, Aug. 29 2:15—5:15 p.m. and 5—8:15 p.m.

Wednesday, Aug. 31 8:45 a.m.—11:45 a.m., 2:15—5:15 p.m. and 5—8:15 p.m.

Thursday, Sept. 1 11:30 a.m.—2:30 p.m., 2:15—5:15 p.m. and 5—8:15 p.m.

Friday, Sept. 2 11:30 a.m.—2:30 p.m. and 2:15—5:15 p.m.

Saturday, Sept. 3 11:30 a.m.—2:30 p.m. and 2:15—5:15 p.m.

Sunday, Sept. 4 8:45 a.m.—11:45 a.m., 2:15—5:15 p.m. and 5—8:15 p.m.

Monday, Sept. 5 (Labor Day) 2:15—5:15 p.m. and 5—8:15 p.m.

Email me if you have any questions: mathematics.csd@gmail.com

Need help visualizing Math On-A-Stick? Check out our photo album from last year.

Or watch this video.

Or read this post from last year’s Fair.

Whatever you can do to help spread the word, all of us involved in making Math On-A-Stick happen thank you!

# The Summer of Math

Hey parents! Listen closely. Do you hear that? It’s the sound of school letting out for the summer!

You’ve got your summer camps planned, your squirt guns at the ready, and you’re all set to hit the library as many times as needed to keep your kids reading all summer long.

Now you need a plan to keep their math minds active.

At Talking Math with Your Kids, we’ve got you covered.

Announcing The Summer of Math.

A small sample of the fun to be had this summer!

Here’s how it works. You can head over to the Talking Math with Your Kids store, pay for a subscription to The Summer of Math, and all summer long we’ll ship you awesome, fun stuff that will keep you and your 5—10 year old busy playing and talking math.

You’ll color, count, make patterns, designs and shapes. You’ll read together, draw, and challenge yourselves. You’ll notice. You’ll wonder. You’ll play. And when school starts back up in the fall, your kids will remember this as the best, mathiest summer ever.

The details

Each month June—September, we’ll ship you a box that contains a bunch of great stuff—at least one book, at least one related set of mathy things to play with, and at least one special surprise. For example, in June you’ll get one beautiful math coloring book, one terrific activity book, all the supplies you need for both of these, a set of spiraling pentagons (so you can make your own awesome designs like those in the coloring book), and a little something extra we cannot yet divulge.

Plus a newsletter where we’ll share additional ideas, questions, cool math stuff we’ve been doing, and reports you send us of the mathy fun you’ve had this summer.

We’ll ship the first week of each month. One week before we ship it out, we’ll send you an email letting you know exactly what’s coming your way (except for the surprise—that’s always a surprise!) You can let us know if you need to add, delete, or swap anything out. We can easily credit you for things you already have (but it’s not likely you’ll already have much of what we’ve got planned), or substitute something new and awesome for it.

We’ll have a Facebook page where we’ll share our mathy adventures and encourage you to share yours.

What are you waiting for? Click on through and join us for The Summer of Math!

# 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?

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.

# Help Wanted: Math on a Stick

UPDATE:

The updated post about Math On-A-Stick is here.

Here is the Math On-A-Stick page on the Minnesota State Fair website.

ORIGINAL POST:

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.

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.

## Volunteering

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.

## Funding

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.

# How do I help my high schooler?

Jessica Lahey writes for the New York Times parenting blog Motherlode. It is an excellent and wide-ranging resource. Recently I had the opportunity to correspond with her about a math question sent in by a parent. As is the way in journalism, I had more to say than could make it into the piece.

The question (copied below) aims at an older age group (and therefore more advanced mathematics) than the usual Talking Math with Your Kids writing, but the spirit of the advice has relevance at all levels.

The parent’s question:

My husband and I talked to our daughter’s pre-calc teacher about her poor grades.  He said many students hit a wall at this point in math, moving from memorization — apply this theorem to this problem — to more abstract, how can I solve this problem, thinking.  I accepted that because that’s what is happening for her.  What I thought later was, why can’t we find a way to help these many students get over that wall, instead of using it as a tool to weed out less developed brains?

There is a whole collection of lovely questions embedded in this note.

## Arithmetic and mathematics

At heart, the issue is a false dichotomy—between (1) arithmetic/elementary algebra, and (2) mathematics.

We tend to treat arithmetic and elementary algebra as being founded on basic facts rather than on ideas. You can go have a look at the #stopcommoncore hashtag to see this trope play out day after day. But it is more subtly embedded in pretty much every aspect of kids’ mathematical development in this country—from the Shapes books parents bring home from the library, to the flash cards and iPad apps they buy for their children, to curriculum and classroom instruction many experience in American elementary schools…right straight through to the ways that institutions such as mine approach developmental (i.e. remedial) math at the college level.

Did someone say “shapes book“?
Helping teachers and parents to rethink this is a huge chunk of the work that I do in many settings—not least the blogs. (And quick plug….Common Core Math for Parents For Dummies is out this month.)

There is a portion of the population that can make sense of a big picture by putting together lots of individual details. This portion of the population tends to be more successful in math and science because they can function in math courses as currently conceived. But many, many people shape their understanding the other way—by keeping the big picture in focus and working out the details when they come up.

## Kinds of understanding

One way to think about this is with a navigation metaphor. There’s a lovely classic math ed article by Richard Skemp titled “Relational and Instrumental Understanding” (warning: paywall) that works with this metaphor. I’ll give you the short, updated version.

Think about the difference between the directions you get from a GPS as you’re driving, and the directions you get from a person who knows both you and the city you are navigating very very well. GPS tells you when to turn and which direction, always getting you where you are going. But you don’t learn your way around a city by following GPS directions. The knowledgeable person, though, will remind you about the restaurant you walked to the other day, and other landmarks, road names and places that you’re familiar with. The knowledgeable person helps you to expand your knowledge of the city by talking about it in ways that involve relationships. GPS only gives you the next step (and a crappy map).

Image from wikipedia
Let’s say you’ve been driving around Scranton, PA for a week following GPS directions. Does your lack of knowledge of the city reflect on your capacity to learn urban navigation skills? I would say “No”. It is more reflective of the fact that you never received any instruction in how the city is laid out. And it’s the same with math. That the young lady in question is struggling to navigate this particular curriculum transition doesn’t reflect on her as a learner so much as it reflects on the difference—which should not exist, but which the teacher cites—between how math has been taught to her in the past and how it will be taught moving forward.

GPS directions are not the city anymore than standard algorithms are mathematics. Each is an efficient and convenient way to get a particular job done. But neither one should be expected to support learning the big picture in large portions of the population.

## Gender and mathematics

OK. One more thing. Anecdotally, I see this kind of transition hitting girls and young women especially hard. I and others think it is related to the kinds of gender disparities we see in math and science—surely not the only contributing factor, but not an unimportant one either. My father once commented that I have more Tabitha examples on my Talking Math with Your Kids blog than Griffin examples. There are a few reasons for that, but on the whole I’m OK with the idea that my blog over-represents the mathematical thinking of girls. I only wish I had more young children of color in my immediate sphere of daily living—I’d over-represent their stories in a heartbeat!
In short, there are serious equity issues embedded in the story the parent tells. We like to pretend that mathematics—being an objective science—is immune to these issues, but it just isn’t so.

Here are a couple of things I’d advise:
(1) Get specifics. What are some specific examples where this student needs to understand the big picture, but is stuck trying to memorize? “Hitting a wall” is sort of the opposite of actionable advice. This family needs to understand exactly what is going wrong.
One possible example of what’s happening, relevant to precalculus, is the unit circle. The unit circle is one of the foundational tools for building from right triangle trigonometry to the trigonometry functions (sine, cosine, tangent, etc.) that are essential for work in Calculus, engineering and the sciences. Many people (teachers and students alike) treat the unit circle as a series of subway stops. See the image at this link (and pretty much any image in a Google search for unit circle). Each point indicated on that circle has two values associated with it, and students proceed to memorize their way towards “understanding” the unit circle.
But this view leaves many students unable to think about the unit circle as a dynamic, changing object (see this graph—press the play button on aand pay attention to the relationship between the height of the moving point and the developing graph). This is an important view of functions in general—they express the relationship between two changing variables. A student who memorizes the subway stops on the unit circle may struggle with a question such as, “On what intervals is sin(x) increasing?” If the student is just thinking about the individual points, there is no increase or decrease, there are only points. A student who sees the unit circle as a dynamic object can think this way: As the angle increases, the point goes counterclockwise around the circle. Sine is the height of that point. The height is increasing from the bottom of the circle (–π/2, as one example) to the top of the circle (π/2). Then it descends…

In such a case, finding resources for thinking about the unit circle dynamically would be important. And this is much more actionable than “she has hit a wall of abstraction”. The teacher and tutors should be able to help find such resources. Getting specifics will help focus the student’s efforts.
(2) Focus on the big things. This is closely related to the first thing, but a bit more broadly based. See if the teacher can describe the (say) five most important things to learn in his course. I have written an article in Mathematics Teaching in the Middle School  (warning: paywall) that argues that keeping our eye on the big ideas of rate of change (slope) and functions is important to success in calculus, but sorely neglected in favor of a thousand algebraic skills throughout middle school and high school. I don’t believe every decision we make in 6—12 curriculum should be based on an assumption that Calculus is the summit all kids should scale, but if this young lady is in Precalc, it’s because she’s moving on to Calc eventually and she’ll need these two ideas as building blocks for that work.
If the problem is that she is trying to memorize her way to understanding, then having some small number of things on which to focus—and having names and descriptions for them—will make things easier for her than having vague descriptions of the need to abstract or problem solve.

# 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? 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.