The monthly *Math Teachers at Play* (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Click here to submit your blog post

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

**Don’t procrastinate:** *The deadline for entries is this Friday, April 21.* The carnival will be posted next week at Give Me a Sine.

Thank you so much to the volunteer bloggers who have stepped up to carry this MTaP math education blog carnival through the years! I would never be able to keep the carnival going on my own.

If you’d like to join in the fun, we have plenty of openings for months ahead. Read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

While you’re waiting for next week’s *Math Teachers at Play* carnival, you may enjoy:

- Browse past editions of the
*Math Teachers at Play*blog carnival - Carnival of Mathematics
- Carnaval de Matemáticas

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

]]>

Ah, the infinite chocolate bar. If only it could work in real life! But can your children find the mistake? Where does the extra chocolate come from?

Here’s a hint: It’s related to this classic brainteaser. And here’s a video from Christopher Danielson (talkingmathwithkids.com), showing how the chocolate bar dissection really works.

Happy munchings!

CREDITS: Feature photo (top) by Yoori Koo via Unsplash. “Hershey Bar Math” video by Christopher Danielson via YouTube. The infinite chocolate gif went viral long ago, and I have no idea who was the original artist.

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

]]>

Welcome to the 106th edition of the ** Math Teachers At Play** math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 106th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

If you slice a pizza with a lightsaber, you’ll make straight cuts all the way across. Slice it once, and you get two pieces.

If you slice it five times, you’ll get a maximum of sixteen pieces. (And if you’re lucky you might get a star!)

**How many times would you have to slice the pizza to get 106 pieces?**

And now, on to the main attraction: the blog posts. Some articles were submitted by their authors; others were drawn from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, click one of these links.

- Talking Math with Kids
- Exploring Elementary Arithmetic
- Adventuring into Algebra and Geometry
- Scaling the Slopes of High School Math
- Enjoying Recreational Puzzles and Math Art
- Teaching with Wisdom and Grace
- Giving Credit Where It’s Due

Would you like to see YOUR favorite blog post in next month’s playful math blog carnival? Submissions are always open!

I learned most, not from those who taught me but from those who talked with me.

—Augustine

- Rodi describes the first session of a math circle for 6-7 years olds, and talks about how to engage students who haven’t experienced this type of learning before: PROBLEM SOLVING #1: A New Group of Students.

- Susan’s students try to figure out How far did Sam’s plane go?

- Christopher (@Trianglemancsd) notices the power of silence and of conversations in quiet moments: Tens Again.

- Telanna’s (@TAnnalet) students debate how to round off tricky numbers: Where does the number begin and end?

- Katie coins a useful addition to her (and our) math vocabulary: Double perfect squares.

- Kent (@KentHaines) asks, “What topic is equally confounding to my 4-year-old son and my Algebra 1 class?” Halfway.

- Kids say the most delightful things! Eavesdrop on more mathy kid-talk (and share your own tidbits): #tmwyk.

[Back to top.]

[Back to Table of Contents.]

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.

—I.N. Herstein

- Lacy (@playdiscovlearn) explores ways to Build Number Sense thru Staircases.

- Manan (@shahlock) asks, “How do we share fairly without an adult mediating?” Read his four-part series as the 2nd-grade students play with arithmetic, geometry, measurement, and even some intuited Calculus exploring ways to share.

- Do your kids get confused by Roman Numerals? Jim has a conversion calculator and some tips: Why Do We Still Need This Ancient Math Skill? And then check out Manan’s Simple But Evil #5 — Roman Numeral Arithmetic.

- Alexandra’s (@aofradkin) students generate a certain famous sequence: Fibonacci Trees.

- My homeschool co-op math class is enjoying math games. This week, I think we’ll try John’s (@mathhombre) metric measurement game: Michigan Smith.

- Michael (@mikegold1950) explains the deep connections between different methods: Thirteen Ways of Looking at Multiplication (with apologies to Wallace Stevens).

- Simon (@Simon_Gregg) talks fractions: Fracton Talk Spotlight 02.

- Nat (@NatBanting) shares Marie’s (@MarieMcMB) fantastic fraction-sense game: Fracton Talk War.

[Back to top.]

[Back to Table of Contents.]

It is better to solve one problem five different ways, than to solve five problems one way.

—George Polya

- Alex (@msmathman) creates washi-tape angle puzzles to make his students think: Angles, Triangles, and the Start of Geometry in 6th Grade Math.

- Shellie (@themathmentors) details a game to replace review worksheets: How to play the four quadrant game.

- Ben (@benorlin) pushes students to understand Lines Beyond y = mx + b.

- Mike (@mikeandallie) and sons explore the sum of squares: Connecting arithmetic and geometry.

- Nat (@NatBanting) leads his students into Experiencing Scale in Higher Dimensions.

[Back to top.]

[Back to Table of Contents.]

Mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that—by some mysterious agency—capture patterns of the universe around us?

—Ian Stewart

- Lisa (@Lisaqt314) shares a cool update to a classic student project: A Great Conics Project Using #Desmos.

- Have you seen Manan’s (@shahlock) “Simple But Evil” blog series? What fun! Simple But Evil #4 — Find The Vertex Of A Parabola.

- Inna and Yoni (@oneonepsilon) ask, Is i More Imaginary than -1?

- Dan (@normalsubgroup) points out how playing with an easier-to-remember, simpler version of a formula can be both more instructive and more fun for the beginner. But how might students develop the initial formula? Start with Don Cohen’s Infinite Series…

- Alexa (@AlexaLim22) interviews mathematician-author Eugenia Cheng: To Infinity and Beyond.

- Ben (@benorlin) has comic fun with Limericks for Mathematicians.

- And don’t miss the 143rd Carnival of Mathematics.

[Back to top.]

[Back to Table of Contents.]

What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for beauty.

—Paul Lockhart

- Jon (@MrOrr_geek) and daughters experiment with cutting paper Magic Rings, and they’d like you to guess what will happen.

- Malke (@mathinyourfeet) is making more math at body scale: Build big. I’d love to try this with my homeschool co-op math kids!

- Brian (@bit_player) shares a new Sudoku-ish logic puzzle and ponders The uniqueness constraint.

- Dan (@mathrecreation) explores the surprising variety of patterns you can form with Truchet Tiles.

- John (@mathhombre) shares an awesome collection of links: Math x Art.

- And for my (@letsplaymath) own contribution to the carnival, here’s a collection of ideas for informal mathematical art: Dot Grid Doodling.

[Back to top.]

[Back to Table of Contents.]

One thing to keep in mind is that mathematics is a story and that teachers are story tellers. If you can bring the story of mathematics to life then you will have a much better chance of reaching all your students.

—Scott Baldridge

- Graham (@gfletchy) posts the latest video in his wonderful
*Making Sense of Math*series: The Progression of Early Number and Counting.

- Megan (@Veganmathbeagle) picks up a new hobby and finds out it’s Always About Math – Eventually.

- Crystal (@Tri_Learning) shares how her homeschooling family is Using Narration to Evaluate Math Learning.

- Joe (@JSchwartz10a) takes a look at better ways to meet students at their own levels:
*To Each According To His Need.*

- Lane (@LaneWalker2) offers time-tested advice for motivating math students: When Passion is Not Enough.

- Katrina (@Kschwart) highlights one key to effective teaching: How Kids Benefit From Learning To Explain Their Math Thinking.

- Marilyn (@mburnsmath) tackles Preparing and Planning: How I Get Ready for Teaching a Math Lesson.

- Simon (@Simon_Gregg) reviews Mike Flynn’s (@MikeFlynn55) book
*Beyond Answers.*

- Michael (@mikegold1950) shares the wisdom of master teacher Herb Gross: Applying the Platinum Rule.

- Tracy (@TracyZager) warns us about The little phrase that causes big problems.

[Back to top.]

[Back to Table of Contents.]

*Route 106 Pentagon*image by Mitchazenia via Wikimedia Commons.- The Pizza-Cutting Puzzle is a recreational math classic, but this variation of it comes from Kjartan Poskitt’s Murderous Maths website.
- Art photos are from the Public Domain Collection at the Metropolitan Museum of Art.
- The quotations are from my
*Dot Grid Math Journals*series.

And that rounds up this edition of the ** Math Teachers at Play** carnival. I hope you enjoyed the ride.

The next installment of our carnival will open sometime during the week of April 24-28 at Give Me a Sine blog. If you would like to contribute, please use this handy submission form. Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Past posts and future hosts can be found on our blog carnival information page.

We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the ** Math Teachers at Play** blog carnival, please speak up!

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

]]>

But as with any paperback book, these have one problem. How do I use them without cracking the spine?

When we exercise, we need to warm up our bodies with a bit of stretching to prevent injury. In the same way, we need to warm up a new book to protect it. The process is called “breaking it in.”

It only takes a few minutes to break in a paperback book:

Never force the book but help it limber up gradually, and it will serve you well.

Because my journals are *working* books, I take the breaking-in process a bit further than shown in the video:

**(1)** Set the book on its back and follow the process above. Press down each cover, but not completely flat — let it bend at the fold line, about 1 cm from the actual spine. Then press a couple pages at a time, alternating front and back, down flat on each cover.

**(2)** Flip through the pages of the book forward and backward to limber them up.

**(3)** Repeat the steps of the video. This time, gently lean the main part of the book away from the part you are pressing down. Aim for a 130–140 degree angle.

**(4)** Flip through the pages again. Even roll the book back and forth a bit — curving the cover and pages as if you’re trying to fold the book in half — to encourage flexibility.

**(5)** Repeat the breaking-in process one more time. This time, fold each section back as close to 180 degrees as it will go.

And you’re done!

The pages will still curve in at the fold line, where they connect to the spine of the book. You want that because it makes the book strong. But now they’ll also open up to provide a nice, wide area for writing or math doodling.

]]>

Yesterday, I mentioned my new series of paperback dot grid notebooks, and I promised to share a few ideas for mathematical doodling.

Doodling gives our minds a chance to relax, wander, and come back to our work refreshed. And though it goes against intuition, doodling can help us remember more of what we learn.

Math doodles let us experiment with geometric shapes and symmetries. We can feel our way into math ideas gradually, through informal play. Through doodles, our students will explore a wide range of mathematical structures and relationships.

Our own school experiences can make it hard for us to teach. What we never learned in school was the concept of playing around with math, allowing ideas to “percolate,” so to speak, before mastery occurs, and that process may take time.

—Julie Brennan

I like to doodle on dotty grid paper, like the pages in my math journals, but there’s *No Purchase Necessary!* You can design your own printable dot page at Incompetech’s PDF generator, or download my free coloring book (which includes several pages of printable dot and graph paper).

To make a faceted mathematical gemstone, start with any shape you like. Then build other shapes around it. What do you notice? Does your pattern grow outward from its center? Or flow around the corner of your page? How is each layer similar, and how is it different?

Arbitrary constraints can lead to mathematically interesting doodles. For instance, create a design out of 45-45-90 triangles by coloring exactly half of every grid square. How many variations can you find?

Play a symmetry puzzle game. Draw a line of symmetry and fill in part of the design. Then trade with a partner to finish each other’s doodles.

Make more complex symmetry puzzles with additional reflection lines.

- Who can talk about mathematical doodling without mentioning Vi Hart? If you’ve never seen her “Doodling in Math Class” video series, you’re in for a treat!

- For more 45-45-90 triangle-tile designs, check out Dan’s “Doodling with Froebel”…

- …and the Math Munch blog post “Truchet, Truchet, Truchet!”

- Enjoy Cindy’s beautiful “Parallel & Perpendicular Art” and make some of your own.

- See if you can draw a rotational-symmetry design, like Don’s “Order 4” graphs.

- Play around with number-patterned doodles like Spirolaterals.

- Follow Michelle’s directions for “How to Draw a Celtic Knot Pattern.”

- Or experiment with the more flexible rules in John’s “Knot Fun” lesson.

- And my latest obsession: the “ultimate” tutorial series on Celtic Knotwork, which explores the link between knots and their underlying graphs.

- Finally, check out my Math Doodling board at Pinterest.

**Before you start doodling:** How to Break In Your New Math Journal.

Feature photo (top): *Sommermorgen (Alte Holzbrücke in Pretzfeld)* by Curt Herrmann, via Wikimedia Commons. [Public domain]

]]>

The problem is, I’m not a naturally organized person. I like making lists and plans, but sticking to them is tougher. And I’ve never found a planner or organizational system that I could follow for longer than two weeks at a go. That is until I heard of bullet journaling.

But journaling requires a journal — a notebook of some sort. And I couldn’t find any that I liked. Either the pages were too narrow and felt cramped, or the thing didn’t fit even in my oversized purse. Or the fancy, hardcover binding made it heavy to lug around. Or there weren’t enough pages to last more than a few weeks. Or the lines were too dark, or too widely spaced.

Never quite what I wanted.

So I decided to make my own.

I started with dot-grid pages for flexible layouts and for doodling. I scattered some of my favorite math and education quotations through each book. And then I added several of my most flexible geometric coloring pages (based on Islamic tessellation designs).

And I had so much fun I couldn’t stop with just one. So let me introduce my *Dot Grid Notebook with Coloring Pages* series:

With 170 roomy pages, each book gives you plenty of space to record memories, plan projects, and keep track of tasks. The dot grid makes it easy to draw graphs or diagrams. Take notes, jot down ideas, copy your favorite quotations, or doodle to your heart’s content.

- Light gray dots at 5 mm spacing provide guidance for flexible page layouts.
- 11 geometric coloring pages allow a multitude of artistic possibilities.
- 31 favorite quotes offer a vision for creative math education.
- 6 × 9 inch (about 15 × 23 cm) pages are wider than many journals, yet still fit comfortably into a large purse or bag.
- Paperback binding makes the journal sturdy but lightweight. Carry it anywhere!

**Prevent cracked spines:** How to Break In Your New Math Journal.

**The ebook edition features all 124 quotations** (31 from each journal) about mathematics, education, and problem solving. Read through for your own pleasure, post them by your workspace, or use them as writing prompts for yourself or your students.

Yes, all of the ebooks are the same, so there’s no point in buying more than one. And at Amazon, if you buy a paperback journal, you can download the companion ebook for free!

Of course, you can use them for bullet journaling. That’s why I originally created the books, because I couldn’t find planners that fit my personal style. My bullet journal is basically an anthology of To-Do lists, bound together so they don’t get lost in the clutter. It’s the only planner system I’ve been able to stick with for more than two weeks at a go.

Or you could use the dotty pages for a commonplace book. That’s my favorite kind of journaling. Like a magpie, I collect shiny tidbits from books, websites, conversations overheard, and more. Passages. Definitions. Poems. Recipes. Proverbs. Things I’m wondering about. Cute kid sayings. It all goes into the mix.

And math puzzles, of course! Below, I’m playing my way through Paul Lockhart’s *Measurement.* I use the cloud-like labels in the outer margins of each page for keywords that identify what I’m writing, because someday I’ll need to skim back and find an old note.

But where dot grid pages really excel is at *doodling* — I’m sure you noticed the faceted design filling the lower half of my journal page above and the gem almost overrunning my February calendar. So watch for tomorrow’s blog post featuring a variety of ways to create your own mathematical doodles.

Best wishes, and happy mathing!

P.S.: Do you have a blog? If you’d like to feature a *Dot Grid Journal* review and giveaway, I’ll provide the prize. Leave a comment below, and we’ll work out the details.

]]>

“Beauty in mathematics is seeing the truth without effort.”

“There’s something striking about the economy of the counselor’s construction. He drew a single line, and that totally changed one’s vision of the geometry involved.

“Very often, there’s a simple introduction of something that’s not logically within the framework of the question — and it can be very simple — and it utterly changes your view of what the question really is about.”

CREDITS: Castle photo (top) by Rachel Davis via Unsplash. “A Mathematical Fable” via YouTube. Story told by Barry Mazur. Animation by Pete McPartlan. Video by Brady Haran for Numberphile.

]]>

“Spring cleaning has made my desk look worse than before. Nobody feels like studying. The kids would rather be outside, and their mom would rather take a nap. If I line everyone up on the curb in the morning, do you think the yellow bus will take them?”

Homeschool burnout — it’s a perennial problem. If you’re suffering from lethargy and can’t face another day of school work, here are some ideas that kept me going long enough to graduate almost-five kids (my “baby” finishes homeschooling this spring!):

**(1)** Re-read the homeschooling books on your shelves, or get some new ones from the library. Write down your favorite quotes as you read. Try to read about one a month, to help get your enthusiasm back. And then read at least one new homeschooling book per year to help you stay inspired.

**(2)** Connect with other homeschoolers. Meet with friends for tea, or have a Mom’s Night Out while Dad babysits. Talk about substantive things, like educational philosophy — what you like about homeschooling, and what you’d like to change. Share your dreams for your children. Remind each other why you’re doing this.

**(3)** Attend support group meetings. I find that after so many years, I let the meetings slide. I think, *I already know everything they are going to say*. But being with other homeschoolers is encouraging. And if you find out that you can help a new homeschooler with advice, that gives you a boost, too.

**(4)** Find one or two forums where you can become one of the resident experts, and answer posts as often as you can. As with number 3 above, being able to give advice (and being appreciated for it) can give you the energy to keep on going.

**(5)** Go to a homeschooling convention, if you get the chance. The speakers are stimulating, and you may find some new book or tool that sparks your imagination.

**(6)** Do school anyway. It may seem impossible when you’re stuck in the doldrums, but once you get going, you may find it easier. The light of understanding in a child’s eyes can give Mom quite a lift!

**(7)** Try something completely different. If you have always used a textbook program, then set it aside for a month and just read library books. If you have read lots of great literature, then try some hands-on projects, or get out those science experiments you keep putting off, or visit all the museums within a two-hour radius, or… I’m sure you can think of something that has been lingering on your good-intentions list. I never could stand to teach the same old thing every year, and none of my five kids got exactly the same education. Happily, there is always another way to approach any homeschooling topic. How about Gameschooling?

**(8)** Figure out what your students are able to do on their own, and let them do it. Encourage them to develop as much independence as possible.

**(9)** Use some of your children’s independent time to learn something new for yourself. Have you always wanted to try painting, or crochet, or woodworking? Be an example of life-long learning.

**(10)** Start (or join in progress) a group class or co-op. You may be able to trade around with some other families: you teach history and others teach math or cooking, or whatever arrangement fits for you. This is especially helpful for those time-consuming projects that always seem to get put off, like art or science experiments.

**(11)** Try some of these intensely practical Tips For Coping With Homeschool Burnout.

**(12)** And are you a Christian homeschooler? Then pray! Your Father knows what you need, and Immanuel is with you always. Try praying your way through 1 Corinthians 13 (or this homeschooling version).

If you have any other ideas for beating the burnout blues, please share!

Homeschooling is not always peaches and cream. If anyone promised you that, they lied. But be assured that it homeschool burnout is not a terminal condition. You will recover your joy in sharing your children’s education.

I learned one thing from every story I’ve ever read: adventures never run smoothly.

And what greater adventure could there be than to introduce your child to all the wonderful things in God’s world?

CREDITS: “Scream” photo (top) by greg westfall via Flickr (CC BY 2.0).

]]>

I told her bar models themselves are not the goal. The real question for parents and teachers is:

- What can you do when your child is stumped by a math word problem?

To solve word problems, students must be able to read and understand what is written. They need to visualize this information in a way that will help them translate it into a mathematical expression.

Bar model diagrams are one very useful tool to aid this visualization. These pictures model the word problem in a way that makes the solution appear almost like magic.

It is a trick well worth learning, no matter which math program you use.

https://www.youtube.com/watch?v=HKsYDzQK8Zw

“Visualization is the brain’s ability to see beyond what the eyes can see, and we can develop visualization in many ways.”

https://www.youtube.com/watch?v=I6Ipio8JntU

“A bar model is a way to represent a situation in a word problem using diagrams — in particular, using rectangles.”

https://www.youtube.com/watch?v=i7LAHc1qvig

“This is one of the ideas that children learn in mathematics: the use of diagrams to represent quantities, especially quantities which are unknown.”

I’ve written a series of blog posts that explain bar model diagrams from the most basic through to solving multistep word problems. Check them out:

- Penguin Math: Elementary Problem Solving 2nd Grade
- Ben Franklin Math: Elementary Problem Solving 3rd Grade
- Narnia Math: Elementary Problem Solving 4th Grade
- Hobbit Math: Elementary Problem Solving 5th Grade
- Solving Complex Story Problems
- Solving Complex Story Problems II

I’ve started working on a book about bar model diagrams, and I’d love to hear your input. Have you tried using them? Do they help your children? What questions do you have?

CREDITS: Videos and quotations from Dr. Yeap Ban Har’s YouTube channel. “Girl doing homework” photo (top) by ND Strupler and “math notebooking equal fractions” by Jimmie via Flickr (CC BY 2.0).

]]>

This month’s post features algebra tips, geometry proofs, Fibonacci rabbit trails, math art, and much more.

Click Here to Go Read the Carnival Blog!

Past carnivals are still full of mathy treasure. Check them out:

]]>

Walliman says, “To err is to human, and I human a lot. I always try my best to be as correct as possible, but unfortunately I make mistakes…”

- Can you find three mistakes in the map?

Check your answers in the description on Walliman’s YouTube page.

If you enjoy this video, you can purchase the poster (or T-shirt, coffee mug, tote bag, etc.) at Red Bubble.

Map of Mathematics poster by Dominic Walliman via Flickr (CC BY-NC-ND 2.0).

]]>

People assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.

Apply logic to that statement.

The conclusion simply isn’t valid.

My mathematical understanding is stuck in the early-to-mid 17th century.

After reading several intriguing quotes about the Riemann Hypothesis, I was overcome by curiosity. I looked it up. The Riemann Hypothesis is a string of nonsense syllables surrounding one magic phrase: *non-trivial zeros*. Those words create surreal images in my brain.

I cannot reliably remember *pi *past three digits. Four if you count the decimal point.

In my world, groups are friends who hang out together. People who are good at math talk about groups, and I will sometimes almost believe that I am close to understanding at least part of what they mean. Then it all slips away again.

To me, combinatorics sounds like something done by a less-than-respectable woman in studded-leather underwear and spiked heels.

The story I want to tell involves combinatorics, but only the G-rated kind.

I have forgotten most of the mathematics I ever learned. Some of it I never understood, so it passed away painlessly, without regrets. Other math I did enjoy at one time, but it perished from extended lack of use. Most of calculus is that way. I mourn its loss.

Even in the math that I normally teach — and therefore that I *should *be good at — I occasionally stumble into chasms of appalling ignorance.

My story begins with one of these.

If, in reading my blog, you discover more evidence of mathematical ineptitude, please deal gently with me. I know I am not good at math. I am just a dabbler, but I’m eager to learn.

You may be wondering, if I am not good at math, then how dare I teach it, or blog about it, or offer advice to others?

I love mathematics. I can’t stay away from it. Like Isaac Newton’s boy at the beach, I want to grab every ocean-splashed pebble I can reach. My reach does not extend very far, and my stones are not as beautiful as his, but they are my treasures nonetheless. I understand them.

And there is one thing I am relatively good at. When I understand something, I can see how to explain it to others. Usually several ways, in multiple representations. For me, this is the definition of understanding: to be able to see connections and illustrations, elaborations and parables.

This is what makes me a teacher.

Which brings me (at last!) to my story.

One of the parents from my MathCounts class brought in a combinatorics problem, and it stumped me. I was forced to invoke the *Teacher’s Emergency Response*: “I don’t know. Let me do some research, and I will get back to you.”

Here is the problem, for those who are curious (from the 2006-2007 MathCounts Handbook, Workout 9):

Four people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction.

At home, I worked through the problem and got an answer that I recognized as patently ridiculous. I worked it another way and got the same answer. I left the problem on my desk and went to bed.

I am not Maria Agnesi. No one solved the problem while I was asleep.

When I tried again the next morning, my wrong answer came back like a summer fly determined to sit on my forehead and rest its wings.

Online, I checked the MathCounts website. They host a forum for coaches, which may contain a discussion of this problem. But I was not an official coach, and the forum is closed to the general public. I did belong to another [no longer active] forum, however, where I often gave math advice to struggling homeschool parents. On that forum, someone who is better at math than I am was running a diagnostic workshop. You bring the problem, and he would teach you how to solve it.

Well, I had a problem. Was I brave enough to share it? These people thought I was good at math. This was going to be embarrassing.

I humbled myself and submitted the problem. The “professor” suggested an approach I hadn’t tried. I misinterpreted his suggestion and set off on a wild goose chase, only to find my familiar answer waiting at the end of the trail. The professor asked specific, pointed questions. I saw that his questions went straight to the heart of my problem. I couldn’t answer them. I explained my reasoning step by step, showing the most logical way to derive my wrong answer.

There it was — my ignorance on display, naked and quivering, ready for dissection.

The professor had pity on me, pointed out the step where I had gone wrong, and gave me the correct step. I could see that his method worked, but it sat like a fig leaf over my still-shivering ignorance.

Why would his step work when mine would not?

How could I know what to do the next time a combinatorics problem came up?

I was too tired to think. A nasty germ had dropped into my life and made itself at home. I thanked the professor for his help and went back to bed.

Sometime during the night, as I tossed around unable to sleep, I saw it all. I understood both the *how* and the *why* of the professor’s solution. I knew the prerequisites, the things a student would have to master before even attempting the problem. I saw how to explain the key insight that broke through confusion. I sketched all the diagrams and calculations on my mental chalkboard. I *could* teach this problem.

Victory tasted sweet.

As soon as I felt well enough, I asked the professor to find me another, similar problem. I wanted to make sure I could generalize my insight and apply it in a new context. But I had no doubt of my success.

I had found a new beach pebble for my collection, and I would not let it get away.

This is what learning math feels like.

Next weekend, we will probably hear plenty of talk about “the Agony and the Ecstasy” of the Big Game. I say, football is nothing compared to mathematics.

This post is my too-late entry for Week Four of the #MTBoS #MtbosBlogsplosion blogging challenge. It’s an expanded reblog of an article that originally appeared in 2007.

CREDITS: Feature photo (top) by One Laptop Per Child. Spiral Fractal by Kent Schimke. Child on Beach photo by Dennis Wong. Dice photo by Ella’s Dad. Embarrassed Lion photo by Charles Barilleaux. Stones on Beach photo by Moyan Brenn. All via Flickr (CC BY 2.0). “Pieces of Math” poster from Loopspace (CC-BY-NC-ND).

]]>

“I used to think my job was to teach students to see what I see. I no longer believe this. My job is to teach students to see; and to recognize that no matter what the problem is, we don’t all see things the same way. But when we examine our different ways of seeing, and look for the relationships involved, everyone sees more clearly; everyone understands more deeply.”

[Feature photo (above) by jenn.davis via Flickr (CC BY 2.0).]

]]>

This month’s post features measurement games, algebra activities, paper folding, math podcasts, the secret to avoiding commitment, a variety of number puzzles, and much more.

Click Here to Go Read the Carnival Blog!

Do you write an education or family blog? Classroom teacher, math coach, homeschooler, parent, college professor, unschooler — anyone interested in helping kids play around with math? Please consider volunteering to host the MTaP blog carnival for one month.

We still need volunteer hosts for most of 2017.

You choose the month that fits your schedule and decide how much effort you want to put in. Writing the carnival can take a couple of hours for a simple post — or you can spend several days searching out and polishing playful math gems to share.

If you want more information, read the MTaP Math Education Blog Carnival home page. Then let me know which month you want.

]]>

Video from the Global Math Project.

And here are some additional answers.

Ask your kids the question: “What Is Math?”

I’d love to hear what they say!

]]>