Denise Gaskins hosted the 142nd Playful Math Education Blog Carnival


It’s the final week of the month and so the Playful Math Education Blog Carnival has published again. The host this time is Denise Gaskins, who also runs the project. You can read the 142nd Blog Carnival essay here, and find something interesting or delightful or useful.

If you’d like to host the carnival some month, you can sign up here. Most of 2021 is open, as I write this. I’ve had the chance to host several times. It’s a novel challenge, and something that pushes me to look beyond my usual familiar mathematics reading. You might find it similarly energizing.

Playful Math Education Blog Carnival 141


This is the 141st Playful Math Education Blog Carnival. And I will be taking this lower-key than I have past times I was able to host the carnival. I do not have higher keys available this year.

The Numbers

I will start by borrowing a page from Iva Sallay, kind creator and host of FindTheFactors.com, and say some things about 141. I owe Iva Sallay many things, including this comfortable lead-in to the post, and my participation in the Playful Math Education Blog Carnival. She was also kind enough to send me many interesting blogs and pages and I am grateful.

141 is a centered pentagonal number. It’s like 1 or 6 or 16 that way. That is, if I give you six pennies and ask you to do something with it, a natural thing is one coin in the center and a pentagon around that. With 16 coins, you can add a nice regular pentagon around that, one that reaches three coins from vertex to vertex. 31, 51, 76, and 106 are the next couple centered pentagonal numbers. 181 and 226 are the next centered pentagonal numbers. The units number in these follow a pattern, too, in base ten. The last digits go 1-6-6-1, 1-6-6-1, 1-6-6-1, and so on.

141’s also a hendecagonal number. That is, arrange your coins to make a regular 11-sided polygon. 1 and then 11 are hendecagonal numbers. Then 30, 58, 95, and 141. 196 and 260 are the next couple. There are many of these sorts of polygonal numbers, for any regular polygon you like.

141 is also a Hilbert Prime, a class of number I hadn’t heard of before. It’s still named for the Hilbert of Hilbert’s problems. 141 is not a prime number, which you notice from adding up the digits. But a Hilbert Prime is a different kind of beast. These come from looking at counting numbers that are one more than a whole multiple of four. So, numbers like 1, 5, 9, 13, and so on. This sequence describes a lot of classes of numbers. A Hilbert Prime, at least as some number theorists use it, is a Hilbert Number that can’t be divided by any other Hilbert Number (other than 1). So these include 5, 9, 13, 17, and 21, and some of those are already not traditional primes. There are Hilbert Numbers that are the products of different sets of Hilbert Primes, such as 441 or 693. (441 is both 21 times 21 and also 9 times 49. 693 is 9 times 77 and also 21 times 33) So I don’t know what use Hilbert Primes are specifically. If someone knows, I’d love to hear.

Banner art the Playful Math Education Blog Carnival, showing three carousel figures (a bear, a coati, and a horse). The background is a stream of mathematical symbols in gradients of blue-green and yellow light.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Landscape

I first want to thank Denise Gaskins for organizing the Playful Math Education Blog Carnival. It must be always a challenging and exhausting task and to carry it on for years is a great effort. The plan for the next several hosts of the Carnival is here, and if you would like to host a month, it’s a good place to volunteer.

For myself, you’re already looking at my mathematics blog. My big, ambitious project for this year is The All 2020 Mathematics A-to-Z. Each Wednesday I try to publish a long-form piece explaining some piece of mathematics. This week, I should reach the letter P. If you’d like to suggest a topic for the letters Q or R please leave a comment here. My other major project, Reading the Comics and writing about their mathematical content, is on hiatus. I’ll likely get back to it once the A-to-Z is finished.

One of my newer regular readers is Laura, teacher and tutor and author of the MathSux2: Putting math into normal people language blog. There’s new essays every week.

Grassy field with a long string of cars approaching, and groups of people walking the opposite direction. There is no hint that there's an amusement park anywhere in frame.
Second round of overflow parking at Dorney Park. From a visit we took in August of 2014, you remember, that day everybody in eastern Pennsylvania, north Jersey, and southern New York State decided to go to Dorney Park. All these amusement park pictures are ones I’ve taken and I’m happy to say, truthfully, that they’re all connected to something in the main text.

Features

A friend knowing me well shared the Stand-Up Maths video Why is there no equation for the perimeter of an ellipse? The friend knew me well. I once assigned the problem, without working it out, to a vector-calculus class. The integral to do this formula is easy to write. It’s one of the many, many integrals that can’t be done. Attempting to do it leads to fascinating formulas, as seen in the video. And also to elliptic curves, a major research topic in mathematics.

Christian Lawson-Perfect, writing at The Aperiodical, looked at The enormous difficulty of telling the truth about escalators with statistics. Lawson-Perfect saw a sign claiming the subway station’s escalators worked 95% of the time. What did that mean? Defining what it means to have “escalators working” is a challenge. And it’s hard to define “95% of the time” in a way that harmonizes with our intuitions.

Also, at the risk of causing trouble, The Aperiodical also hosts a monthly Carnival of Mathematics. It’s a similar gathering of interesting mathematics content. It doesn’t look necessarily for educational or playful pieces.

I do not have a Desmos account. It’s been long enough since I had a real class that I haven’t yet joined the site. This may need to change. Christopher Sewards posted a set of activities in Permutations and Combinations which may be useful. There’s three so far and they may be joined by more. This I learned through Dan Meyer’s weekly roundup of links.

Meyer’s also made me aware of TheCalt, a mathematics tournament to be held the 17th of October. They’re taking signups even now. Here’s a page with three sample problems for guidance.

Sarah Carter similarly attempts a Monday Must-Reads collection at the MathEqualsLove blog. Given the disruptions of this year this was the first in the series in months. This collects a good number of links, many of them about being interesting while doing online classes.

Photograph of a model of the Montana Rusa roller coaster, under glass, and watched over by a Alebrije-styled dragon-ish figure. It sits on the platform for a roller coaster.
Montaña Rusa, in Mexico City, seen in 2018. It’s one of three Möbius-strip wooden roller coasters in existence.

Helene Osana writes Mathematical thinking begins in the early years with dialogue and real-world exploration. This is an essay about priming the mathematical thinking for the youngest children, those up to about five years old. One can encourage kids with small, casual activities that don’t look like education.

The Reflective Educator posted Precision In Language. This is about one of the hardest bits of teaching. That is to say things which are true and which can’t be mis-remembered as something false. Author David Wees points out an example of this hazard, as kids apply rules outside their context.

Simon Gregg’s essay The Gardener and the Carpenter follows a connected theme. The experience students have with a thing can be different depending on how the teacher presents it. The lead example of Gregg’s essay is about the different ways students played with a toy depending on how the teacher prompted them to explore it.

Also crossing my desk this month was a couple-year-old article Melinda D Anderson published in The Atlantic. How Does Race Affect a Student’s Math Education? Mathematics affects a pose of being a culturally-independent, value-neutral study. The conclusions it draws might be. But what we choose to study, and how we choose to study it, is not. And how we teach it is socially biased and determined. So here are thoughts about that.

The last several links describe things we know thanks to modern psychology and neuroscience studies. Nicklas Balboa and Richard D Glaser published in Psychology Today Three Habits That Reduce Conversational Success. There are conversations which are, effectively, teaching attempts. To be aware of how those attempts go wrong, and how to fix them, is surely worth while.

Ben Orlin, of the popular Math With Bad Drawings blog, wrote Democracy isn’t math. But it isn’t NOT math. He contributed recently to David Litt’s Democracy In One Book Or Less. The broad goal of democracy, the setting of social rules by common consensus, might not be mathematical. When we look to the practical matters of implementing this, though, then we get a lot of mathematics. I have not read Litt’s book, or any recently-published book, so can’t say anything about its contents. I bet it includes Arrow’s Impossibility Theorem, though.

Photograph of the lift hill for a wooden roller coaster, taken with the sunlight gleaming on the tracks. Standing traverse to the end of the coaster in front is another roller coaster, a steel coaster with a more scaffold-like structure and seen in silhouette against the blue sky.
Lift hill for Thunderhawk, Dorney Park’s antique wooden roller coaster. Behind it, if I’ve got this right, is Steel Force, a much taller steel coaster. Photo from August 2014.

Anyone attempting to teach this year is having a heck of a time. Sarah Carter offered Goals for the 2020-2021 School Year – PANDEMIC STYLE as an attempt to organize planning. And shared her goals, which may help other people too.

Emelina Minero offered 8 Strategies to Improve Participation in Your Virtual Classroom. Class participation was always the most challenging part of my teaching, when I did any of that, and this was face-to-face. Online is a different experience, with different challenges. That there is usually the main channel of voice chat and the side channel of text offers new ways to get people to share, though.

The National Centre for Excellence in the Teaching of Mathematics offered Two Pleas to Maths Teachers at the Start of the School Year. This is about how to keep the unusual circumstances of the whole year from encouraging bad habits. This particularly since no one is on track, or near it.

S Leigh Nataro, of the MathTeacher24 blog, writes Learning Math is Social: We Are in This Together. Many teachers have gotten administrative guidance that … doesn’t … guide well. The easy joke is to say it never did. But the practical bits of most educational strategies we learn from long experience. There’s no comparable experience here. What are ways to reduce the size of the crisis? Nataro has thoughts.

Enlightenment

Now I can come to more bundles of things to teach. Colleen Young gathered Maths at school … and at home, bundles of exercises and practice sheets. One of the geometry puzzles, about the missing lengths in the perimeter of a hexagon, brings me a smile as this is a sort of work I’ve been doing for my day job.

Starting Points Maths has a page of Radian Measure — Intro. The goal here is building comfort in the use of radians as angle measure. Mathematicians tend to think in radians. The trigonometric functions for radian measure behave well. Derivatives and integrals are easy, for example. We do a lot of derivatives and integrals. The measures look stranger, is all, especially as they almost always involve fractions times π.

Small swing ride, in mid-ride-cycle, with the swings lifted into the air and tilted. There are only two people on the whole ride and they are barely visible against the silhouette of background trees.
(Children’s) swing ride at Seabreeze Park in Rochester, New York (2019). It was a cool day when we visited.

The Google Images picture gallery How Many? offers a soothing and self-directed counting puzzle. Each picture is a collection of things. How to count them, and even what you choose to count, is yours to judge.

Miss Konstantine of MathsHKO posted Area (Equal — Pythagorean Triples). Miss Konstantine had started with Pythagorean triplets, sets of numbers that can be the legs of a right triangle. And then explored other families of shapes that can have equal areas, including looking to circles and rings.

Sarah Carter makes another appearance here with New Puzzle: Only ‘Takes’ and ‘Adds’. This is in part about the challenge of finding new puzzles to make each week. And then an arithmetic challenge. Carter mentions how one presentation is quite nice for how it teaches so many rules of the puzzle.

Cassandra Lowry with the Australian Mathematical Sciences Institute offers Finding the Maths in Books. This is about how to read a book to find mathematical puzzles within. This is for children up to about second grade. The problems are about topics like counting and mapping and ordering.

Lowry also has Helping Your Child Learn Time, using both analog and digital clocks. That lets me mention a recent discussion with my love, who teaches. My love’s students were not getting the argument that analog clocks can offer a better sense of how time is elapsing. I had what I think a compelling argument: an analog clock is like a health bar, a digital clock like the count of hit points. Logic tells me this will communicate well.

YummyMath’s Fall Equinox 2020 describes some of the geometry of the equinoxes. It also offers questions about how to calculate the time of daylight given one’s position on the Earth. This is one of the great historic and practical uses for trigonometry.

Games

To some play! Miguel Barral wrote Much More Than a Diversion: The Mathematics of Solitaire. There are many kinds of solitaire, which is ultimately just a game that can be played alone. They’re all subject to study through game theory. And to questions like “what is the chance of winning”? That’s often a question best answered by computer simulation. Working out that challenge helped create Monte Carlo methods. These can find approximate solutions to problems too difficult to find perfect solutions for.

At Bedtime Math, Laura Overdeck wrote How Do Doggie Treats Taste? And spun this into some basic arithmetic problems built around the fun of giving dogs treats.

Conditional probability is fun. It’s full of questions easy to present and contradicting intuition to solve. Wayne Chadburn’s Big Question explores one of them. It’s based on a problem which went viral a couple years ago, called “Hannah’s Sweet”. I missed the problem when it was getting people mad. But Chadburn explores how to think through the problem.

Photograph of a large carousel, seen by night. The four horses in front are each at a different position despite being in the same column: the white horse nearest the camera is several feet behind the tan horse farther from it, and a grey and a black horse are scattered forward and back of those.
A column of horses at Cedar Point (Ohio)’s Cedar Downs, a racing merry-go-round. The horses move forward and backward in those slots. Also the carousel moves fast, which makes it much better. (October 2019.)

Paul Godding’s 7 Puzzle Blog gives a string of recreational mathematics puzzles. Some include factoring, some include making expressions equal to particular numbers. They’re all things you can do when Slylock Fox printed the Six Differences puzzle too small for your eyes.

FractalKitty has a cute cartoon, No 5-second rule … about how the set of irrational numbers interacts with rationals in basic arithmetic.

Carnivals

Now to some deeper personal interests. I am an amusement park enthusiast: I’ve ridden at least 250 different roller coasters at least once each. This includes all the wooden Möbius-strip roller coasters out there. Also all three racing merry-go-rounds. The oldest roller coaster still standing. And I had hoped, this year, to get to the centennial years for the Jackrabbit roller coaster at Kennywood Amusement Park (Pittsburgh) and Jack Rabbit roller coaster at Seabreeze Park (Rochester, New York). Jackrabbit (with spelling variants) used to be a quite popular roller coaster name.

So plans went awry and it seems unlikely we’ll get to any amusement parks this year. No county fairs or carnivals. We can still go to virtual ones, though. Amusement parks and midway games inspire many mathematical questions. So let’s take some in.

Michigan State University’s Connected Mathematics Program set up set up a string of carnival-style games. The event’s planners figured on then turning the play money into prize raffles but you can also play games. Some are legitimate midway games, such as plinko, spinner wheels, or racing games, too.

Resource Area For Teaching’s Carnival Math offers for preschool through grade six a semi-practical carnival game. There’s different goals for different education levels.

Hooda Math’s Carnival Fun offers a series of games, many of them Flash, a fair number HTML5, and mostly for kindergraden through 8th grade. There are a lot of mathematics games here, along with some physics and word games.

Photograph of a couple of midway games: Knock It Off (a milk-cans game), Hot Shots (basketball game), Cat Rack (knocking over cat figures), and Balloon Darts (a pop-the-balloons game). No one is playing them yet and it's not clear they're attended.
Some midway gaves on offer at Seabreeze Park in Rochester, New York (2019). It was a slow day and the park had just opened minutes before.

I found interesting the talk about Math Midway, a touring exhibition meant to make mathematics ideas tactile. I’m not sure it’s still a going concern, though. Its schedule lists it as being at the Singapore Science Centre from February 2016 to present. But it’s not mentioned on the Singapore Science Centre’s page. (They do have a huge Tesla coil, though. Also they at least used to have an Albert Einstein animatronic, forever ascending and descending a rope. I enjoyed visiting it, although I would recommend going to the Tiger Balm Gardens as higher prioerity.) Still, exploring this did lead me to The National Museum of Mathematics, located in New York City. It has a fair number of exhibits and its events online.

Rides

But enough of the carnival as a generic theme. How about specific, actual rides and games? Theme Park Insider, one of the web’s top amusement-park-industry news, published Master the Midway: The Theme Park Insider Guide to Winning Carnival Games several years ago. The take from midway games is an expression of the Law of Large Numbers. The number of prizes won and their value will fluctuate day to day, but the averages will be predictable. And what players can do to better their chances is subject to reason.

Specific rides, though, are always beautiful and worth looking at. Ann-Marie Pendrill’s Rotating swings—a theme with variations looks at rotating swing rides. These have many kinds of motion and many can be turned into educational problems. Pendrill looks at some of them. There are other articles recommended by this, which seem relevant, but this was the only article I found which I had permission to read in full. Your institution might have better access.

Lin McMullin’s The Scrambler, or A Family of Vectors at the Amusement Park looks at the motion of the most popular thrill ride out there. (There are more intense rides. But they’re also ones many people feel are too much for them. Few people in a population think the Scrambler is too much for them.) McMullin uses the language of vectors to examine what path the rider traces out during a ride, and what they say about velocity and acceleration. These are all some wonderful shapes.

Evening photograph of a Scrambler, yellow and lit by its neon tube lighting. The greater focus is on the Googie-style Scrambler ride sign, self-illuminated in panels of rectangles rising and falling around the baseline and in a gorgeous thin typeface.
You may have wondered on a Scrambler ride how long it takes to get back to the same ground position. The answer is that it depends on just how the pieces rotate. (Lakeside Park, Denver, visited in June 2018.)

And Amusement Parks

Many amusement parks host science and mathematics education days. In fact I’ve never gone to the opening day of my home park, Michigan’s Adventure, as that’s a short four-hour day filled with area kids. Many of the parks do have activity pages, though, suggesting the kinds of things to think about at a park. Some of the mathematics is things one can use; some is toying with curiosity.

Here’s The State Fair of Texas’s Grade 6 STEM games. I don’t know whether there’s a more recent edition. But also imagine that tasks like counting the traffic flow or thinking about what energies are shown at different times in a ride do not age.

Photograph of a Dentzel antique carousel, focused on a black horse with red-and-white rose-themed saddle and leading the gryphon-themed chariot.
Dorney Park’s antique carousel, which at one time turned in the small Lake Lansing Amusement Park. Photo from August 2014.

Dorney Park, in northeastern Pennsylvania, was never my home park, but it was close. And I’ve had the chance to visit several times. People with Kutztown University, regional high schools, and Dorney Park prepared Coaster Quest – Geometry. These include a lot of observations and measurements all tied to specific rides at the park. (And a side fact, fun for me: Dorney Park’s carousel used to be at Lake Lansing Amusement Park, a few miles from me. Lake Lansing’s park closed in 1972, and the carousel spent several decades at Cedar Point in Ohio before moving to Pennsylvania. The old carousel building at Lake Lansing still stands, though, and I happened to be there a few weeks ago.)

And I have yet to make it to Six Flags America, but their Math & Science In Action page offers a similar roster of activities tied to that park. Six Flags America is their park in Maryland; the one in Illinois is Six Flags Great America.

Math Word Problems Solved offers a booklet of Amusement Park Word Problems Starring Pre-Algebra. These tie in to no particular amusement park. They do draw from real parks, though. For example it lists the highest point on the tallest steel roller coaster as 456 feet; it doesn’t name the ride, but that’s Kingda Ka, at Great Adventure. The highest point on the tallest wooden roller coaster is given as 218 feet, which was true at its 2009 publication: Son of Beast at Kings Island. Sad to say Son Of Beast closed in 2009, and was torn down in 2012. The current record heights in wooden coasters are T Express at Everland in South Korea, and Wildfire at Kolmården in Sweden. (Too much height is not really that good for wooden roller coasters.)

A 2018 posting on Social Mathematics asks: Do height restrictions matter to safety on Roller Coasters? Of course they do, or else we’d have more roller coasters that allowed mice to ride. The question is how much the size restriction matters, and how sensitive that dependence is. So the leading question is a classic example of applying mathematics to the real world. This includes practical subtleties like if a person 39.5 inches tall could ride safely, is it fair to round that off to 40 inches? It also includes the struggle to work out how dangerous an amusement park is.

Speaking from my experience as a rider and lover of amusement parks: don’t try to plead someone’s “close enough”. You’re putting an unfair burden on the ride operator. Accept the rules as posted. Everybody who loves amusement parks has their disappointment stories; accept yours in good grace.


Three roller coasters, one a tall skinny steel coaster, one a chunky white racing wooden coaster, and one a sleek modern wooden roller coaster, seen side-by-side.
Kingda Ka, Rolling Thunder, and El Toro, side by side. Rolling Thunder, itself a racing roller coaster, has since been torn down. Rolling Thunder’s greatest height was 96 feet, on both sides of the train. (Photo from July 2013.)

This leads me into planning amusement park fun. School Specialty’s blog particularly offers PLAY & PLAN: Amusement Park. This is a guide to building an amusement park activity packet for any primary school level. It includes, by the way, some mention of the historical and cultural aspects. That falls outside my focus on mathematics with a side of science here. But there is a wealth of culture in amusement parks, in their rides, their attractions, and their policies.

And to step away from the fun a moment. Many aspects of the struggle to bring equality to Americans are reflected in amusement parks, or were fought by proxy in them. This is some serious matter, and is challenging to teach. Few amusement parks would mention segregation or racist attractions or policies except elliptically. (That midway game where you throw a ball at a clown’s face? The person taking the hit was not always a clown.) Claire Prentice’s The Lost Tribe of Coney Island: Headhunters, Luna Park, and the Man Who Pulled Off the Spectacle of the Century is a book I recommend. It reflects one slice of this history.


Let me resume the fun, by looking to imaginary amusement parks. TeachEngineering’s Amusement Park Ride: Ups and Downs in Design designs and builds model “roller coasters”. This from foam tubes, toothpicks, masking tape, and marbles. It’s easier to build a ride in Roller Coaster Tycoon but that will always lack some of the thrill of having a real thing that doesn’t quite do what you want. The builders of Son Of Beast had the same frustration.

The Howard County Public Schools Office published a Mathatastic Amusement Park worksheet. It uses the problem of finding things on a park map to teach about (Cartesian) coordinates in a well-motivated way.

The Brunswick (Ohio) City Schools published a nice Amusement Park Map Project. It also introduces students to coordinate systems. This by having them lay out and design their own amusement park. It includes introductions to basic shapes. I am surprised reading the requirements that merry-go-rounds aren’t included, as circles. I am delighted that the plan calls for eight to ten roller coasters and a petting zoo, though. That plan works for me.

Cheryl Q Nelson and Nicole L Williams, writing for Mathematics Teacher, published the article Sprinklers and Amusement Parks: What Do They Have To Do With Geometry? Both (water) sprinklers and amusement park vendors are about covering spaces without waste. Someone might wonder at their hypothetical park where the bumper cars are one of the three most popular rides. I recommend a visit, when possible, to Conneaut Lake Park, in northwestern Pennsylvania. Their bumper cars are wild. Their roller coaster’s pretty great too.

And finally a bit of practical yet light news. Dickinson University was happy to share how The Traveling Salesman Problem Finds A Novel Application in Summer Student-Faculty Research Project. The Traveling Salesman Problem is the challenge to find the most efficient way to any set of points. It’s a problem both important and difficult. As you try to get to more points the problem (typically) gets far more difficult. I hadn’t seen it applied to amusement park itineraries before, but that’s a legitimate use. I am disappointed the press release did not share their work on most efficient routes around Hersheypark and Disney World. They did publish a comparison of ways to attack the problem.


Evening photograph showing the outlines of Kingda Ka and El Toro roller coasters against night clouds. A streak of pinpoint lights go over the top of Kingda Ka.
Kingda Ka and El Toro seen by night. It looks like there’s a train going right over the top of El Toro. (Photo from July 2013.)

And this closes the carnival, for today. If you’d like to follow this blog, please click the “Follow NebusReseearch” button the page. Or you can add the articles feed to your favorite RSS reader. My Twitter account @Nebusj is all but moribund. For whatever reason Safari often doesn’t want to let me see it. I am also present and active on Mathstodon. This is the mathematics-themed instance of Mastodon, as @Nebusj@mathstodon.xyz. I would be glad to have more people to chat with there. Thank you as ever for reading.

So I’m hosting the 141th Playful Math Education Blog Carnival


I mentioned yesterday Iva Sallay’s hosting of the 140th Playful Math Education Blog Carnival. This is a collection of pieces of educational, recreational, or otherwise just delightful mathematics posts. I’d said I hoped I might have the energy to host one again this year and, you know? Denise Gaskins, who organizes this monthly event, took me up on the offer.

So, if you write, or read, or are just aware of a good mathematics or mathematics-related blog, please, leave me a comment! I’ll need all the help I can get finding things worth sharing. Anything that you’ve learned from, or that’s delighted you, is worth it. It’ll teach and delight other people too.

And if you have a blog and would like to try out hosting it, please do! There are always months available, and it’s a neat different sort of blogging challenge.

FindTheFactors hosts the 140th Playful Math Education Blog Carnival


Iva Sallay’s Find the Factors recreational mathematics puzzle blog is once again host to the Playful Math Education Blog Carnival. It’s the 140th installment of this blog series, this month, and you might find several pieces of mathematics instruction or trivia or recreation or just play worth reading. It’s been a while since I had the energy to host one but I hope to be able to again this year. Enjoy, I hope.

The Playful Math Education Blog Carnival #136


Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.

Playful Math Education Blog Carnival banner, showing a coati dressed in bright maroon ringmaster's jacket and top hat, with multiplication and division signs sitting behind atop animal-training podiums; a greyscale photograph audience is in the far background.
Banner art again by Thomas K Dye, creator of Newshounds, Infinity Refugees, Something Happens, and his current comic strip, Projection Edge. You can follow him on Patreon and read his comic strip nine months ahead of its worldwide publication. The banner art was commissioned several weeks ago when I expected I would be in a more playful mood this week.

In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.

Kreig’s writing drew the attention of another kind contributor to my harvesting. Symmetry and Multiplying Negative Numbers explores one of those confusing things about negative numbers: how can a negative number times a negative number be positive? One way to understand this is to represent arithmetic operations as geometric operations. Particularly, we can see negation as a reflection.

That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …

This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.

Fermat’s Two-Squares Theorem, discussed here, is not the famous one about a^n + b^2 = c^n . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.

What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.

Iva Sallay also brought up Jenna Laib’s Making Meaning with Arrays: More Preschooler Division which similarly sees numerical truths revealed through geometric reasoning. Here, particularly, by the problem of baking muffins and thinking through how to divide them up. A key piece here, for a particular child’s learning, was being able to pick up and move things around. Often in shifting between arithmetic and geometry we suppose that we can rearrange things without effort. As adults it’s easy to forget that this is an abstraction that we need to learn.

Sharing of food, in this case cookies, appears in Helena Osana’s Mathematical thinking begins in the early years with dialogue and real-world exploration. Mathematic, Osana notes, is primarily about thinking. An important part in mathematics education is working out how the thinking children most like to do can also find mathematics.

I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?

Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.

Speaking of the game, David Coffey’s Creating Positive Change in Math Class was written in anticipation of the standardized tests meant to prove out mathematics education. Coffey gets to thinking about how to frame teaching to more focus on why students should have a skill, and how they can develop it. How to get students to feel involved in their work. Even how to get students to do homework more reliably. Coffey’s scheduled to present at the Michigan Council of Teachers of Mathematics conference in Grand Rapids this July. This if all starts going well. And this is another post I know of thanks to Goldenoj.

These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.

Mathematical study starts small, though. Often it starts with games. There are many good ones, not least Iva Sallay’s Find the Factors puzzles.

Besides that, Dads Worksheets has provided a set of Math Word Search Puzzles. It’s a new series from people who create worksheets for many grade levels and many aspects of mathematics. They’re on Twitter as @dadsworksheets.

Mr Wu, of the Singapore Math Tuition blog, has also begun a new series of recreational mathematics puzzles. He lays out the plans for this, puzzles aimed at children around eight to ten years old. One of the early ones is the Stickers Math Question. A more recent one is The Secret of the Sweets (Sweet Distribution Problem). Mr Wu can be found on Twitter as @mathtuition88.

Denise Gaskins, on Twitter as @letsplaymath, and indefatigable coordinator for this carnival, offers the chance to Play Math with Your Kids for Free. This is an e-book sampler of mathematics gameplay.

I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.

The Guys and Good Health blog presented Happy Pi Day on the 14th, with — in a move meant to endear the blog to me — several comic strips. This includes one from Grant Snider, who draws lovely strips. I’m sad that his Incidental Comics has left GoComics.com, so I can’t feature it often during my Reading the Comics roundups anymore.

Virtual Brush Box, meanwhile, offers To Celebrate Pi Day, 10 Examples of Numbers and 10 Examples of Math Involved with Horses which delights me by looking at π, and mathematics, as they’re useful in horse-related activities. This may be the only blog post written specifically for me and my sister, and I am so happy that there is the one.

There’s a bit more, a bit of delight. It was my greatest surprise in looking for posts for this month. That is poetry. I mean this literally.

Whimsy-Mimsy wrote on Pi Day a haiku.

D Avery, on Shift N Shake, wrote the longer Another Slice of Pi Day, the third year of their composing poems observing the day.

Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.

Blind On The Light Side similarly wrote Pi poems, including a Pi-Ku, for March 12, 2020. These poems don’t reach long enough to deal with the zero-syllable line, but we can forgive someone not wanting to go on that long.

As a last note, I have joined Mathstodon, the Mastodon instance with a mathematics theme. You can follow my shy writings there as @nebusj@mathstodon.xyz, or follow a modest number of people talking, largely, about mathematics. Mathstodon is a mathematically-themed microblogging site. On WordPress, I do figure to keep reading the comics for their mathematics topics. And sometime this year, when I feel I have the energy, I hope to do another A to Z, my little glossary project.

And this is what I have to offer. I hope the carnival has brought you some things of interest, and some things of delight. And, if I may, please consider this Grant Snider cartoon, Hope.

Life Through A Mathematician’s Eyes is scheduled to host the 137th installment of the Playful Math Education Blog Carnival, at the end of April. I look forward to seeing it. Good luck to us all.