A Heart That Continually Grows Forever or Infinitely
By Sophia Zhanissov (grade 6) and Maria Droujkova
Some math activities are so rich in mathematical connections, inspirations, and energy! If mathematical routines were people, Growing Patterns would be a pillar of the community. Growing Patterns routine connects several key ideas in computer science, algebra, geometry, and beyond. Moreover, all that is doable at the pre-algebra level; you’ll be essentially learning algebraic reasoning from scratch. Here’s what we did over the hours of the project, in a nutshell:
- Admired Fawn Nguyen’s (last name is pronounced “win”) whimsical collection of hundreds of growing patterns at VisualPatterns.org and decided to add to it!
- Made our own visual growing patterns using graph paper, colored pens, and wooden cubes.
- Using spreadsheets, scripted recursive formulas for a few starter number sequences: counting numbers, even numbers, odd numbers, and alternating numbers (1, -1, 1, -1…).
- Scripted recursive formulas for the three patterns we’d like to add to the collection.
- Wrote stories explaining the three patterns.
- Scripted closed formulas for n-th entries in the three patterns.
- Looked at the corresponding functions using a grapher (Desmos).
The drawings, stories, functions, formulas, and scripts are all different representations of the same infinite, growing sequences. They help us understand the mathematical nature of the sequence and express its growth in a single closed formula. That formula, for us, evokes many geometric, algebraic, and computer-science connections, kind of like e=mc^2 packs a vast amount of physics for those who know how to derive it.
A Very Long Pool
A very long pool is my pattern, where the water is some blue squares(or opposite), the Rim is the green squares making an equal shell around the water(or opposite once again), and the stairs is the singular green square on the right side of the pool. The “very long pool” is essentially a pool that grows by the second, making it (technically) infinite or just very long.
The number of squares and recursive formulas for the water, rim, stairs, and the total space of the pool:
Closed formula for the Nth total space: N*6+4.
Heart
Growing emotions is one of my patterns, it is a heart that continually grows forever or infinitly. The yellow squares inside is the inner heart, where it beats and grows in size, and the purple squares outside is the outer heart, that protects the inner heart from dangers. Given that is grows infinitly, it isn’t strong enough to be on it’s own, and is very sensitive(emotional) and prone to damage. Together they grow and grow to the point where there is no stopping point.
The number of squares and recursive formulas for the inner heart, the outer heart, and the total number of squares:
Closed formula for the Nth total space, outer heart: N*4+6; inner heart 2*N^2+4*N-3.
Road to… Nothing
The road to nothing is my pattern, where there is a road, a very long road, or you could just say, infinite. Most patterns go infinite if there is no stopping point, so does this pattern. This road leads to, you guessed it, nothing… If there is no stopping it, there is no end which means it leads to nothing. It may come across one or two things, but it doesn’t stop going, it leads no one, once again. It has no purpose, it is useless you may call it, it has no emotions, unlike the growing emotions pattern, so it has no feelings, so say whatever you want to it, as it won’t even respond. But anyways, the yellow part is some holes in the road, and the purple is some stepping pads that make up the road.
The number of squares and recursive formulas for the green parts, pink parts, and total squares in the road:
Note that in this example, the closed formula for the total number of squares in the road to nothing is quite easy. But alternating recursive patterns by color took some doing, the way we did them!
Closed formula for the Nth total bricks in the road: N*3.
Posted in Make & Grow
Summer camps with Pomo and Navajo math friends: June 2024 news
Hello, math friend! I’m Dr. Maria Droujkova, director of Natural Math and co-president of the Natural Math Alliance. Please write me any time at reach.out@naturalmath.com if you have a question or would like to talk about learning advanced mathematics in kind ways.
Our partner Alliance of Indigenous Math Circles organized two summer camps this June. The first was at the beautiful and impressively curated California Indian Museum and Cultural Center in Santa Rosa. The second was at the Navajo Preparatory School, a high-quality academic and outreach hub at Farmington, NM.
The camps included a math festival, a problem-solving game called Math Wrangle, many deep-dive math sessions on topics from cryptography to mathematics of braids, and of course a lot of cultural activities. We will share more materials as we organize them for publication. Meanwhile, here are photos from the camps and two art projects for you, which we created with Pomo and Navajo people.
A weaver’s hands.
Donna Fernandez, director of AIMC, and campers. “It doesn’t matter what tribe on the planet you are from; you all still made baskets. Everyone already has a basket in them and I’m just helping them to bring it out.” —Pomo basket weaver Corine Pearce.
A camper with her stick-and-bean game, exploring probability.
James Taylor, leader of Math Amigos, with the campers playing tic-tac-toe on a torus to explore the game of Set. “I learned that mathematical engagement has a “shape.” When a group of participants is thoroughly “hooked” by a problem, it is as if the center of the table they circle becomes a black hole, drawing them in, arching their backs as if in a deep huddle. When I see that shape, I know that students are hooked, so I stand back to give them space to let joyful mathematics progress.” Klein, B. (2019). Math unbounded: A transcultural experiment.
Another familiar shape of collaborative mathematics: a big math circle by Navajo hogan house (see below).
Desert sunset.
Math friends sharing a solstice. The optics of our camera caught the rays of the rising sun perfectly aligning with the walls of this Great Kiva, a Puebloans building from about 1000 years ago.
The Navajo are Indigenous people of the Southwestern United States. Explore hogans, traditional Navajo dwellings known for their sturdy, sustainable, and comfortable designs. Learn to construct geometric shapes with hogan builders’ tools. Part 1 compares tools and axioms of different geometries. Part 2 focuses on the Navajo geometry. Click these links to download the materials.
The Pomo are Indigenous Californians. They are world-famous for their basket art. You can find gorgeous Pomo baskets in many major museums. They are also used for practical purposes in modern households. In this set of activities, you can learn about fractals and fractal dimensions using designs from Pomo artists. Click the link to download the materials.
Posted in Newsletter
Book Clubs and Community-Responsive Math Circles
Hello, math friend! I’m Dr. Maria Droujkova, director of Natural Math and co-president of the Natural Math Alliance. Please write any time at reach.out@naturalmath.com if you have a question or would like to talk about learning advanced mathematics in kind ways. Meanwhile, here are some opportunities for learners of all ages to make math together.
Geometry and Trig Reading Circle
Join an online math circle for students ages 12 to 15 in March and April. Participants will explore geometry, proof, and the basics of trigonometry. Sue VanHattum, editor of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers and life-long fan and teacher of math, is writing a new book series. In four young adult novels, Althea and her friends explore some of the mysteries of mathematics. We need folks to test the books out before publication.
In Althea and the Mysteries of Triangles, Circles, and Pi, Althea and friends, with the help of Althea’s mom, explore geometry and proof in order to then learn the basics of trigonometry. Sue will lead an online circle, in which participants will explore these mysteries along with Althea and her friends. Five to ten participants will join Sue for nine weekly sessions. The sessions form a deep and friendly math course as well as a unique book club, with the author refining the story based on student reactions.
Do you know any students who enjoy or want to enjoy math, know a bit of algebra, and want to volunteer as user testers? Invite them to check out the details and sign up!
K-6 Storybook
Here is an opportunity for anyone who loves puzzles and for K-6 learners to develop their number sense. Sue Looney is the author of Ying and the Magic Turtle – in this photo, she is reading her book in a classroom. Dr. Looney wrote another storybook based on a perplexing classic riddle. Sue writes:
I am in search of a handful of Beta Readers who will help me out as I finalize edits to my upcoming storybook. If you’re interested in reading my new math adventure with children, whether they are your own children or they are your students, drop me an email, and I will send you more information about what being a Beta Reader entails. You’ll be helping me improve my story, and you’ll be an important part of bringing my book to life. If this sounds fun to you, please reach out and let me know. Your feedback is invaluable!
Community-Responsive Math Circles: Get a Taste, Join, or Volunteer
For about 28 years, Natural Math has been designing and helping to organize math circles: informal learning spaces where people make mathematics their own. More recently, my colleagues developed a shared vision of community-responsive math circles. What does that mean?
A community-responsive math circle focuses on the needs of a community and the benefits to participants and their families. It is accountable to the community it serves. We’ve been helping with diverse mathematical needs, from gathering materials for a mission to post-earthquake Türkiye to working with Diné (Navajo) teachers, mathematicians, and builders to define a set of geometric axioms embedded in the Hogan house traditions. One of the best parts? Many communities generously share their mathematical gifts! Get a taste and grab an activity you can try at home or in class. Want to participate, volunteer, or start a project for your community? Drop us a line and let’s talk.
Spanish/English Family Math Circles in Cary, North Carolina
Children from multilingual families get huge math boosts if they do mathematics in all the languages spoken at home. Natural Math collaborates with the non-profit Semillas de Amor (Seeds of Love) in Cary, North Carolina, to organize a weekly math circle on Wednesday evenings. If you are in town and Spanish is one of your home languages, sign up here.
Download and Try:
Bucles Infinitos y Secuencias Infinitas | Infinite Loops and Infinite Sequences
These are 3-page PDF flyers in Spanish and English with activities for the whole family.
Bluebird Math Circle at the Alliance of Indigenous Math Circles
The mission of the Alliance of Indigenous Math Circles (AIMC) is to create mathematical opportunities for Indigenous students and to build community among math teachers of Indigenous students while respecting Indigenous culture. You are welcome to visit our online math circles and teacher events.
Download and Try:
Fractal Dimensions of Pomo Baskets
This is a 2-page PDF newsletter, school teacher guide, and colorful slides with activities at middle and high school levels.
Math Adventures for the Very Young and Their Families
Young children are open to mathematical work and play. What kind of math person can I be? What mathematical powers do I want? What flavors of mathematics do I prefer? Early math experiences can help children feel that they belong with math and that math belongs to them. They grow awe-struck with infinity, are amused by logical paradoxes, relax with tessellating patterns, and share their math passions in stories and art. Young math circles inspire grown-ups in two ways: to help their children learn and to restore their own relationships with mathematics. Yet only 5% of math circles recently surveyed by the National Association of Math Circles welcome five-year-old participants. We have one such rarity in Chapel Hill, North Carolina. Check it out and join if you are in town.
Download and Try:
Dihedral Groups and Snowflakes
This is a 4-page PDF flyer with some group theory and algebra accessible for 5-year-old children and their young-at-heart math friends.
Posted in Make & Grow, Newsletter
Spring 2023 at FUNDAPROMAT: Calculus for 5-Year-Olds Materials
It was my pleasure to present for the wonderful math friends at the Panamanian Foundation for the Promotion of Mathematics (FUNDAPROMAT). Their mission is to change the world’s perception so that one and all can experience mathematics as accessible, relevant and inherently joyous. Check out FUNDAPROMAT events and projects!
I presented about making calculus radically accessible. As in, actually doing calculus activities with five-year-old children. Why? Because young calculus goes to show that everyone can learn advanced mathematics in kind ways! #ELI5
I shared a 3-minute presentation introducing Natural Math. Then we did some activities from the math circles that I run with young mathematicians and their families. Below are the flyers that I shared, and maybe a few more relevant ones, to give context to the activities.
This is a middle-school version of the activities; the flyer for my youngest group is coming up soon.
Fractal Dimension of Pomo Baskets
Integrals: Shapes out of Shapes
Integrals: Time-Lapse Vision and Shapes of Revolution
Posted in A Math Circle Journey
