Math Teachers at Play #92


Welcome to the 92nd edition of the Math Teachers At Play math education blog carnival‌—‌a monthly 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 couple of puzzles in honor of our 92nd edition…

Puzzle #1

Pentagonal numbers92 is a pentagonal number, so I was delighted when Lisa Winer‘s (@Lisaqt314) carnival submission came in. Her class spent some time playing around with figurate number puzzles‌—‌including pentagonal numbers‌—‌and collaborated on a blog post about their discoveries.

Click here to find Winer’s own notes about the lesson, along with all the puzzle handouts.

What fun!

Puzzle #2

Or, try your hand at the classic Queen’s Puzzle:

  • What is the maximum number of queens that can be placed on an chessboard such that no two attack one another?

Spoiler: Don’t peek! But the answer is here‌—‌and the cool thing is that there are 92 different ways to do it.

Table Of Contents

The snub dodecahedron is an Archimedean solid with 92 faces.

And now, on to the main attraction: the blog posts. Many 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.

Along the way, I’ve thrown in some videos in honor of the holiday season.

Please: If you enjoy the carnival, would you consider sending in an entry for next month’s edition? Or volunteering to host sometime in 2016?

Early Learning Activities

  • Kids can enjoy making up math problems, but sometimes they can get a bit carried away. Just ask A. O. Fradkin (@aofradkin) about her daughter’s Gruesome Math.
  • Nancy Smith (@nancyqsmith) notices her students struggling with the equal sign in Equality. Strong opinions, and even a few tears. It will be interesting to hear what tomorrow brings…

[Back to top.]
[Back to Table of Contents.]

Elementary Exploration And Middle School Mastery

  • Joshua Greene (@JoshuaGreene19) offers some great ways to tweak an already-wonderful multiplication game in Times square variations. “It was really interesting to see the different strategies that the students took to determining what would go on their boards.”
  • For my own contribution to the carnival, I’ve posted a couple of hands-on arithmetic explorations in A Penny for Your Math.

[Back to top.]
[Back to Table of Contents.]

Adventures in Basic Algebra & Geometry

  • Tina Cardone (@crstn85) experiments with Bar Models in Algebra to help her students think about linear equations. “I did not require students to draw a model, but I refused to discuss an incorrect equation with them until they had a model. Kids would tell me ‘I don’t know how to do fractions or percents’ but when I told them to draw a bar, and then draw 4/5, they could do that without assistance…”

[Back to top.]
[Back to Table of Contents.]

Advanced Mathematical Endeavors

[Back to top.]
[Back to Table of Contents.]

Puzzling Recreations

  • Pradeep Mutalik challenges readers to “infer the simple rule behind a number sequence that spikes up and down like the beating of a heart” in Be Still My Pulsating Sequence.

[Back to top.]
[Back to Table of Contents.]

Teaching Tips

  • How can we get a peek at how our children are thinking? Kristin Gray (@mathminds) starts with a typical set of 1st Grade Story Problems and tweaks them into a lively Notice/Wonder Lesson. “When I told them they would get to choose how many students were at each stop, they were so excited! I gave them a paper with the sentence at the top, let them choose a partner and sent them on their way…”
  • Tracy Zager (@tracyzager) talks about her own mathematical journey in The Steep Part of the Learning Curve: “The more math I learn, the better math teacher I am. I keep growing as a learner; I know more about where my kids are headed; and I understand more about what building is going on top of the foundation we construct in elementary school.”
  • And finally, you may be interested in my new blog post series exploring what it means to understand math. Check out the first post Understanding Math: A Cultural Problem. More to come soon…

[Back to top.]
[Back to Table of Contents.]


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

The December 2015 installment of our carnival will open sometime during the week of December 21-25 at Math Misery? 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!

Free-Learning-Guide-Booklets2Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

Understanding Math: A Cultural Problem


All parents and teachers have one thing in common: we want our children to understand and be able to use math. Counting, multiplication, fractions, geometry — these topics are older than the pyramids.

So why is mathematical mastery so elusive?

The root problem is that we’re all graduates of the same system. The vast majority of us, including those with the power to shape reform, believe that if we can compute the answer, then we understand the concept; and if we can solve routine problems, then we have developed problem-solving skills.

Burt Furuta

The culture we grew up in, with all of its strengths and faults, shaped our experience and understanding of math, as we in turn shape the experience of our children.

Six Decades of Math Education

math on slateLike any human endeavor, American math education — the system I grew up in — suffers from a series of fads:

  • In the last part of the twentieth century, Reform Math focused on problem solving, discovery learning, and student-centered methods.
  • But Reform Math brought calculators into elementary classrooms and de-emphasized pencil-and-paper arithmetic, setting off a “Math War” with those who argued for a more traditional approach.
  • Now, policymakers in the U.S. are debating the Common Core State Standards initiative. These guidelines attempt to blend the best parts of reform and traditional mathematics, balancing emphasis on conceptual knowledge with development of procedural fluency.

Model Math Problems

The “Standards for Mathematical Practice” encourage us to make sense of math problems and persevere in solving them, to give explanations for our answers, and to listen to the reasoning of others‌—‌all of which are important aspects of mathematical understanding.

But the rigid way in which the Common Core standards have been imposed and the ever-increasing emphasis on standardized tests seem likely to sabotage any hope of peace in the Math Wars.

What Does It Mean to “Understand Math”?

Math-HomeworkThrough all the math education fads, however, one thing remains consistent: even before they reach the schoolhouse door, students are convinced that math is all about memorizing and following arbitrary rules.

Understanding math, according to popular culture‌—‌according to movie actors, TV comedians, politicians pushing “accountability,” and the aunt who quizzes you on your times tables at a family gathering‌—‌means knowing which procedures to apply so you can get the correct answers.

But when mathematicians talk about understanding math, they have something different in mind. To them, mathematics is all about ideas and the relationships between them, and understanding math means seeing the patterns in these relationships: how things are connected, how they work together, and how a single change can send ripples through the system.

Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Theories emerge as patterns of patterns, and significance is measured by the degree to which patterns in one area link to patterns in other areas.

Lynn Arthur Steen

Understanding Math, Part 2: What Is Your Worldview? Coming soon…

CREDITS: “Thinking” photo (top) by Klearchos Kapoutsis via Flicker (CC BY 2.0). “Math on a Slate” (middle) by Pranav via Flicker (CC BY 2.0). “I Can Model Problems” poster by Nicole Ricca via Teachers Pay Teachers. “Math Homework” photo (bottom) by tracy the astonishing via Flickr (CC BY-SA 2.0).

LPM-ebook-300This is the first post in my Understanding Math series, adapted from the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…

Free-Learning-Guide-Booklets2Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

Everyone Can Learn Math

Here’s a new video from Jo Boaler at

Boaler’s Four Key Research-Based Messages

There is a huge elephant standing in most math classrooms, it is the idea that only some students can do well in math. Students believe it, parents believe and teachers believe it. The myth that math is a gift that some students have and some do not, is one of the most damaging ideas that pervades education in the US and that stands in the way of students’ math achievement.

—Jo Boaler
Unlocking Children’s Math Potential

A Wealth of ResourcesBoosting Math screenshot

The YouCubed site is full of encouragement and help for families learning math.

— and plenty more!

Free-Learning-Guide-Booklets2Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

Math Calendars for Middle and High School Students


High school math teacher Chris Rime posted three wonderful review calendars for middle and high school students on his blog.

The links at Chris’s blog will let you download editable Word docx files. If you’re cautious about internet links and prefer PDF, here you go:

Chris writes:

There are no explicit instructions about process being more important than the answer on these, so you’ll need to stress that in class.

I remind students that everyone already knows the answer to each of the questions, and that one of the things we’re practicing is explaining our reasoning…


And if anyone else has a math review calendar to share, for any grade level, please add your link in the comment section below.

Free-Learning-Guide-Booklets2Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

The Math Student’s Manifesto


[Feature photo above by Texas A&M University (CC BY 2.0) via Flickr.]

Note to Readers: Please help me improve this list! Add your suggestions or additions in the comment section below…

What does it mean to think like a mathematician? From the very beginning of my education, I can do these things to some degree. And I am always learning to do them better.

(1) I can make sense of problems, and I never give up.

  • I always think about what a math problem means. I consider how the numbers are related, and I imagine what the answer might look like.
  • I remember similar problems I’ve done before. Or I make up similar problems with smaller numbers or simpler shapes, to see how they work.
  • I often use a drawing or sketch to help me think about a problem. Sometimes I even build a physical model of the situation.
  • I like to compare my approach to the problem with other people and hear how they did it differently.

Continue reading The Math Student’s Manifesto

Ruth Beechick on Teaching

[Feature photo above by Samuel Mann (CC BY 2.0) via Flickr.]

Here’s one more quote from homeschooling guru Ruth Beechick. It applies to classroom teachers, too!

Everyone thinks it goes smoothly in everyone else’s house, and theirs is the only place that has problems.

I’ll let you in on a secret about teaching: there is no place in the world where it rolls along smoothly without problems. Only in articles and books can that happen.

you can

— Ruth Beechick
You Can Teach Your Chile Successfully (Grades 4-8)

Free-Learning-Guide-Booklets2Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

Math Debates with a Hundred Chart

Euclid game
Wow! My all-time most popular post continues to grow. Thanks to an entry from this week’s blog carnival, there are now more than thirty great ideas for mathematical play:

The latest tips:

(31) Have a math debate: Should the hundred chart count 1-100 or 0-99? Give evidence for your opinion and critique each other’s reasoning.
[Hat tip: Tricia Stohr-Hunt, Instructional Conundrum: 100 Board or 0-99 Chart?]

(32) Rearrange the chart (either 0-99 or 1-100) so that as you count to greater numbers, you climb higher on the board. Have another math debate: Which way makes more intuitive sense?
[Hat tip: Graham Fletcher, Bottoms Up to Conceptually Understanding Numbers.]

(33) Cut the chart into rows and paste them into a long number line. Try a counting pattern, or Race to 100 game, or the Sieve of Eratosthenes on the number line. Have a new math debate: Grid chart or number line — which do you prefer?
[Hat tip: Joe Schwartz, Number Grids and Number Lines: Can They Live Together in Peace? ]

Free Learning Guide Booklets

Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.