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?

Make another conjecture. And another. Does thinking about your conjectures make you wonder about math?

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.

—Denise Gaskins
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.

Language and Math

Still working on my Prealgebra & Geometry Games book. I’m amazed at how much of learning math is really about language.

With the current layout, the book will be 250+ pages long, depending on how the index turns out. A total of 41 games plus four math investigation activities. I’ve defined more than 100 vocabulary words — so averaging more than two new math words per game.

Of course, several of these are words (or at least concepts) students will have learned before. But a large part of prealgebra is consolidating previous ideas and mastering their names (sum, quotient, factor, multiple, etc.) before moving on to apply them in algebra class.

But it’s not all review! There’s a cool game about polar coordinates, a rabbit trail into combinatorics, and plenty of other challenges to keep students learning.

And plenty of clear definitions for adult readers who have long ago forgotten all the math terms they learned in school.

Prealgebra & Geometry: Math Games for Middle School is scheduled for publication in early 2021. Sign up for my newsletter to get updates.

Math Conversation Starter

(Click for larger image.)

What do you see?

Does it make you wonder?

How is perspective art similar to the isometric drawing in yesterday’s post? How is it different?

You may also enjoy:

CREDITS: William Hogarth – “The importance of knowing perspective” (Absurd perspectives), Engraving on paper. From Wikimedia Commons.

How to Draw Minecraft Blocks

Running out of time on my Math You Can Play Kickstarter, so I better get to work on that Kickstarter Special Edition math-art book I promised to all the backers as a bonus reward.

Today I’m working on the Isometric Drawing and Impossible Figures section, because my co-op math classes had so much fun learning how to draw those.

Here’s a starter image on how to draw Minecraft blocks. At first I called them “isometric blocks” — but changing the name to “Minecraft” made the students really excited to learn. I’m not sure whether I like the pencil sketch, or if I should remake the illustrations on the computer…

Key steps:

  1. Make a Y.
  2. Turn it into an M.
  3. Slant down for the bottom.
  4. Slant up for the top.
Student drawings from my co-op classes.

The most common problem for beginners is that they try to make the base straight. They know a block can sit on a table, so the bottom has to be flat, right? But once students get a feel for how it goes, they can really take off and have fun.

UPDATE: The Kickstarter deals have ended, but my playful math books are still available through your favorite online store or by special order at your local bookshop. (Except for the Prealgebra & Geometry Games book, scheduled for publication in early 2021. Sign up for my email list to get the latest news.)

Morning Coffee – 17 August 2020

Morning Coffee image

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.”

—Sunil Singh
Our Fear of Being Lost Devalues The Beauty of Mathematics

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

Behind the Scenes with the Index Edit

Gel pens are wonderful for editing because they show up so well against the manuscript text. Each pass gets a new color. The picture above is the 5th cycle through my Prealgebra & Geometry Games index, this time with purple ink.

When I printed the output from cycle #4, everything looked so neat in three columns with alphabet headers in place. I thought, “This is almost done!”

Moral: Don’t judge an index by how nice it looks.

Still, it’s also not as bad as the marked-up pages make it seem. Only a few of those will be major tweaks. (Like, how did we not notice that the Jack Lyon quote got left out of the reference section?!)

Plenty of work to go, but the end is in sight…

Prealgebra & Geometry: Math Games for Middle School is scheduled for publication in early 2021. Sign up for my newsletter to get updates.

That’s Mathematics

Here’s a bit of fun I found on YouTube. Happy Friday!

Mathematicians and maths educators in order of appearance:

Eddie Woo @misterwootube
Hannah Fry @FryRSquared
James Tanton @jamestanton
Chris Smith @aap03102
Bobby Seagull @Bobby_Seagull
Jo Morgan @mathsjem
David Wees @DavidWees
Matt Parker @standupmaths
Michael Stevens @tweetsauce
Lieven Schiere @lievenscheire
Ben Sparks @SparksMaths
Rob Eastaway @robeastaway
Nira Chamberlain @ch_nira
Ed Southall @edsouthall
Steven Strogatz @stevenstrogatz
Simon Pampena @mathemaniac
Rachel Riley @RachelRileyRR
Alex Bellos @alexbellos
Simon Singh @SLSingh
Katie Steckles @stecks
Craig Barton @mrbartonmaths
Kyle Evans @kyledevans

Still Working on the Index

With the less-than-help of my daughter’s cat, I’ve made it through several cycles of editing my Prealgebra & Geometry index. If you’re curious about the process, you may enjoy my behind-the-scenes peek at How to DIY a Nonfiction Index.

There’s yet a ways to go, but it’s starting to look like what I want. The index includes game listings by category: card games, pencil and paper games, cooperative games, solitaires, etc.

One of the entries is “Games, Complete List of.”

So I counted…

Officially, the book features 41 games that help prepare students for high school math by playing with number properties, mixed operations, integers, algebraic functions, coordinate geometry, and more.

But when I counted all the official game listings plus all the game variations that were different enough to have their own names — 62 games.

Wow!

Prealgebra & Geometry: Math Games for Middle School is scheduled for publication in early 2021. Sign up for my newsletter to get updates.

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.

—Denise Gaskins
Prealgebra & Geometry: Math Games for Middle School

Excerpted from my upcoming book, Prealgebra & Geometry: Math Games for Middle School, scheduled for publication in early 2021. Sign up for my newsletter to get updates.

CREDIT: “Thinking” photo by Sean Kong on Unsplash.