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

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.

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!**

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!

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.**

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.

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.

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

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

We regularly receive emails with great questions about various approaches to math education. Dr. Maria answers them by email. You can ask your question here. We edit some of the answers to share at this blog. Names and personal details are removed to protect anonymity.

*Q: I am curious about your thoughts on Jo Boaler’s work. She does not believe in memorizing formulas or even the times table. I think it was very helpful for my child to memorize the times table. I am unsure of how sound her theories are, especially moving to higher math. Many of my homeschooling friends love her work though. *

A: I like the parts of Jo Boaler’s work that I’ve seen so far. I’ve read “Experiencing School Mathematics,” the book that came from her dissertation research in the 1990s, and followed some of her further developments. Dr. Boaler works with school systems. Since the school systems are, unfortunately, caught in political turmoil, her work is constantly subject to disinformation and other attacks. This means we can’t trust all sources about Dr. Boaler’s work.

So it’s better to read Jo Boaler’s writing directly to know where she stands. To quote her: “It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact.” Dr. Boaler emphasizes that memorizing does not equal understanding. She opposes timed tests and memorizing instead of understanding. Don’t only memorize times tables and call that “learning multiplication.” You can read more of her words on the topic here: https://www.youcubed.org/evidence/fluency-without-fear/

For my part, I also believe it’s a good idea to be fluent in times tables. There are multiple (ha!) paths to fluency, and memorizing is a good one. In the essay linked above, Jo Boaler says she never memorized all times tables. I did, but I never memorized addition facts; 8 plus 4 still goes, *8 => 10 and 2 extra => 12* in my mind, albeit lightning-fast. (I used to play and win math and science Olympiads as a kid; I needed speed.) **Different learners achieve fluency differently**. My math friends and I developed a system for memorizing times tables efficiently, and with an eye on supporting future concepts, including algebra. Our system offers choices that cover different kinds of learners. It is a part of this course: https://naturalmath.com/multiplication-explorers/

I celebrate when I see a fluent teen with a home-field advantage on the multiplicative conceptual field: multiplication, division, factorization, proportions, and so on. But here is something else to keep in mind: memorizing may not be the best START for learning multiplication. Most students first need to learn what multiplication is and what it means, where in life you multiply, and how to see patterns within times tables. Based on that connected understanding, students can *then* memorize the times tables well. This way, multiplication becomes a cornerstone for algebra.

Together with my colleagues, I developed an at-your-own-pace course called Multiplication Explorers that supports both deep understanding and efficient memorizing. The page has three “math sparks,” sample activities from this course that you can try with your child. Two are aimed at understanding what multiplication is, and the third helps you to see patterns in the multiplication tables. https://naturalmath.com/multiplication-explorers/

We also have a multiplication poster with 12 examples of where in life you multiply, such as computing areas, using symmetry, or counting the number of combinations. You can view it online or get a printed copy. https://naturalmath.com/multiplication-models-poster/

Here are some other Natural Math multiplication resources that you might find helpful.

Download a card that teaches an ancient merchant’s multiplication trick first recorded around the 15th century. Back then, merchants used finger reckoning to calculate prices and profits. A calculating device took half a room, while finger math was easy to take on a trip. Merchants twiddled their fingers away from the prying eyes of their competitors: that’s where the phrase “under the table” came from – https://naturalmath.com/s/wp-content/uploads/2019/05/AncientMultiplicationTrick_NaturalMath.pdf

From Buttons to Multiplication Blind Spots, a blog post about a no-prep activity that explorers multiplication – https://naturalmath.com/2015/05/from-buttons-to-the-multiplication-blind-spots/

Explore the commutative property of multiplication and add your ideas on whether 2×3 is really the same as 3×2 – https://naturalmath.com/2013/04/what-would-you-rather-have-commutative-game/

*Editing: Yelena McManaman *

*Proofreading: Emilie Desmarais*

Posted in Grow, Make & Grow