Playful Math Education 142

Welcome to the 142nd edition of the Playful Math Education Blog Carnival — a smorgasbord of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.

Bookmark this post, so you can take your time browsing.

Seriously, plan on coming back to this post several times. There’s so much playful math to enjoy!

By tradition, we start the carnival with a puzzle/activity in honor of our 142nd edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

Activity: Planar Graphs

According to the OEIS Wiki, 142 is “the number of planar graphs with six vertices.”

What does that mean?

And how can our students play with it?

A planar graph is a set of vertices connected (or not) by edges. Each edge links two vertices, and the edges cannot intersect each other. The graph doesn’t have to be fully connected, and individual vertices may float free.

Children can model planar graphs with three-dimensional constructions using small balls of playdough (vertices) connected by toothpicks (edges).

Let’s start with something smaller than 142. If you roll four balls of playdough, how many different ways can you connect them? The picture shows five possibilities. How many more can you find?

Sort your planar graphs into categories. How are they similar? How are they different?

A wise mathematician once said, “Learning is having new questions to ask.” How many different questions can you think of to ask about planar graphs?

Play the Planarity game to untangle connected planar graphs (or check your phone store for a similar app).

Or play Sprouts, a pencil-and-paper planar-graph game.

For deeper study, elementary and middle-school students will enjoy Joel David Hamkins’s Graph coloring & chromatic numbers and Graph theory for kids. Older students can dive into Oscar Levin’s Discrete Mathematics: An Open Introduction. Here’s the section on planar graphs.

Happy Hamilton Day (Belated)

While searching for posts to add to the Playful Math Carnival, I stumbled on a new-to-me math holiday.

Hamilton Day celebrates mathematical discovery — that “Aha!” moment when your eyes are opened and you see something new.

Or something new-to-you. That’s worth celebrating, too.

History of Hamilton Day

Irish mathematician William R. Hamilton was struggling with a tough math problem in October, 1843. It had him stumped. Then on the 16th, as he walked along Dublin’s Royal Canal with his wife, inspiration struck.

He suddenly realized he could look at the problem from a new direction, and that would make everything fall into place.

“And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples … An electric circuit seemed to close, and a spark flashed forth.”

—Sir William Rowan Hamilton

In one of the most famous acts of vandalism in math history, Hamilton pulled out a knife and scratched his new equation into the stone of the Broome Bridge: i² = j² = k² = ijk = -1.

Also by Hamilton

“Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?”

—Sir William Rowan Hamilton
quoted in H. Eves, Mathematical Circles Revisited

Why Celebrate Hamilton Day

“So there’s much to celebrate on Hamilton Day. Beyond its utility, we can appreciate mathematics as a human endeavor, with struggles and setbacks and triumphs. We can highlight the opportunity math affords for daring, creativity, and out-of-the-box thinking.

“Hamilton Day could, in other words, pivot away from Pi Day’s gluttony and memorization, neither of which is part of mathematics, toward the intellectual freedom and drama that are.”

— Katharine Merow
Celebrate Hamilton Day, a Better Mathematical Holiday

How Will You Celebrate?

• Learn about a new-to-you math topic.
• Work on a tough math problem.
• Think about different ways to do things.
• Try a nonstandard approach.
• Talk about how it feels when you learn something new and it finally makes sense.

I’ve penciled Hamilton Day (October 16) into my calendar for next year.

CREDITS: Commemorative plaque photo (top) by Cone83, CC BY-SA 4.0. Hamilton portrait by Unknown artist and “Death of Archimedes” by Thomas Degeorge, public domain. All via Wikimedia Commons.

FAQ: I’ve Ruined My Daughter

My daughter is only eleven, but I’m afraid I’ve ruined her chance of getting into college because she is so far behind in math. We’ve tried tutors, but she still has trouble, and standardized testing puts her three years below grade level. She was a late reader, too, so maybe school just isn’t her thing. What else can I do?

Standardized tests are not placement tests. They cannot tell you at what level your daughter should be studying. They aren’t designed that way. The “placement” they give is vague and general, not indicative of her grade level but rather a way of comparing her performance on that particular test with the performance of other students.

There can be many different reasons for a low score. I’ve listed a few of them in my post In Honor of the Standardized Testing Season.

Here’s the full quote:

Audrey seemed, for once, at a loss for words. She was thinking about the question.

I try to stay focused on being silent after I ask young children questions, even semi-serious accidental ones. Unlike most adults, they actually take time to think about their answers and that often means waiting for a response, at least if you want an honest answer.

If you’re only looking for the “right” answer, it’s fairly easy to gently badger a child into it, but I’m not interested in doing that.

—Thomas Hobson
Thank You For Teaching Me

CREDITS: “Pismo Beach, United States” photo by Tim Mossholder on Unsplash.

How to Build Math Literacy

Here’s the full quote:

We all know reading a book each day to our child develops their love of literacy… well, playing games is the equivalent in maths.

Through playing card games and board games (just short and sweet ones) children develop problem solving, counting and so many other skills.

Imagine if every time you play a game you say, “Let’s do some maths.” What a positive association your child will develop with maths!

—Ange Rogers
Instagram post

Discover more creative ways to play math with young children at the Number Doctors blog.

CREDITS: “Falling dice” photo by Riho Kroll on Unsplash.

Morning Coffee – 31 August 2020

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to read with your morning coffee this week:

• David Butler’s post Twelve matchsticks: focus or funnel presents an interesting puzzle. But even better, it opened up a rabbit hole of thought-provoking posts about how to talk with children — or anyone.

“The approach where you have an idea in your head of how it should be done and you try to get the student to fill in the blanks is called funnelling. It’s actually a rather unpleasant experience as a student to be funnelled by a teacher. You don’t know what the teacher is getting at, and often you feel like there is a key piece of information they are withholding from you, and when it comes, the punchline feels rather flat.”

—David Butler
Twelve matchsticks: focus or funnel

Morning Coffee – 24 August 2020

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to read with your morning coffee this week…

“We are all mathematicians. We all have the power to notice, describe, and generalize patterns. You have all had this ability since birth. If we believe this then every day we must plan lessons that allow students to act as mathematicians. We must put something in front of our students to notice. We must put something in front of our students to describe, to generalize.”

—Sara VanDerWerf
What is Math? What do Mathematicians do?

How Mathematics Works

The full quote, as it appears in my new book:

Make a conjecture. A conjecture is a statement that you think might be true.

For example, you might make a conjecture that “All odd numbers are…” How would you finish that sentence?

Can you think of any way to test your conjectures, to discover if they will always be true?

This is how mathematics works. Mathematicians notice something interesting about certain numbers, shapes, or ideas. They play around and explore how those relate to other ideas. After collecting a set of interesting things, they think about ways to organize them. They wonder about patterns and connections. They make conjectures and try to imagine ways to test them.

And mathematicians talk with one another and compare their ideas. In real life, math is a very social game.

Prealgebra & Geometry: Math Games for Middle School

Excerpted from my new book, Prealgebra & Geometry: Math Games for Middle School. Look for it at your favorite online bookstore.

CREDITS: “Three girls counting” photo by Charlein Gracia on Unsplash.

Morning Coffee – 17 August 2020

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to read with your morning coffee this week:

“A strategy is how you mess with the numbers, how you use relationships and connections between numbers to solve a problem. A model is a representation of your strategy, the way the strategy looks visibly. Modeling your strategy makes your thinking more clear to others because they can see the thinking and the relationships that went into your process.”

—Pam Harris
Strategies vs. Models

• Do you have preschool or elementary students? Michael Minas has a great collection of games on his blog. Easy to learn and full of mathematical thinking.

“Doing mathematics like this deprives students of, well, let’s be honest, mathematics itself. We need to get to the answer faster. We need to move on. No time to stumble around rabbit holes. There is a curriculum to cover.”

CREDITS: Feature photo (top) by Kira auf der Heide via Unsplash. “Morning Coffee” post format inspired by Nate Hoffelder at The Digital Reader.

Only by Thinking

The full quote, as it appears in my new book:

When we give students a rule, we give them permission not to think. All they need to do is remember our instructions.

But it is only by thinking — by struggling their way through mental difficulties — that our students can build a foundation of mathematical knowledge strong enough to support future learning.