The post Spring 2023 at FUNDAPROMAT: Calculus for 5-Year-Olds Materials appeared first on Natural Math.

]]>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

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]]>The post Q&A: Multiplication and Jo Boaler? appeared first on Natural Math.

]]>*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 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*

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]]>The post Sunflower Bluebird Pi Day 2023 appeared first on Natural Math.

]]>Pi Day is all about circles. Here are two Math Trek printouts from our Inspired by Calculus series. The series is for children ages five and up and their families. This is literally ELI5 Calculus (explain-like-I’m-5). You can also use these short embodied activities for a warm-up with a group at any level. Happy Pi Day!

MATH TREK Integrals: Shapes out of Shapes

MATH TREK Integrals: Time-Lapse Vision and Shapes of Revolution

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]]>The post Pi Day Jokes for Natural Math Circles appeared first on Natural Math.

]]>Mirror photo by Envision

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]]>The post Calm Math Playlist: Learn to lead online groups appeared first on Natural Math.

]]>**When**: Wednesday April 1 from noon to 1 PM EDT**Math topic**: start with symmetry (algebraic geometry) and bridge to more**Education topic**: emotional support via mathematics for children ages 5 and up**Who**: parents, teachers, and math circle leaders**Where**: Natural Math Zoom**Supplies**: plain paper, graph paper (print here if you need it), scissors, colored pencils or markers, reliable internet, microphone

After you register, you will receive an email from reach.out@naturalmath.com with a link to login instructions. If you can’t find the email, send a message to that address and we’ll figure it out. Please log into Zoom on the device you will use for the event and check your audio and video systems at least fifteen minutes in advance.

Natural Math makes advanced mathematics accessible to everyone in kind ways. How? At Natural Math, families with toddlers do projects on symmetry and tessellations; four-year-olds design function machines; and six-year-olds build fractal models of infinity. Our motto: “Math is what you make of it.” Natural Math has more than two decades of proven track record in curriculum development, experience design, and publishing.

Dr. Maria Droujkova focuses her research and development efforts on learning communities, informal education, online education, advanced mathematics for young children, and game design. She holds a Ph.D. in Mathematics Education from NCSU, and M.S. in Mathematics from Tulane. Maria is the founder of Natural Math, an educational design, consulting, and publishing organization started in 1996. Her approach to teaching focuses on the easy complexity (such as calculus for five-year-olds), openness, and kindness. She co-authored Moebius Noodles and Avoid Hard Work, popular books with innovative math activities for parents, teachers, and math circle leaders.

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]]>The post Ying is here; Five Fabulous Activities is next – some math for every age! Newsletter February 2020 appeared first on Natural Math.

]]>Next up! Tenth book to be published by Natural Math, *Five Fabulous Activities for Your Math Circle*, is up for crowdfunding!

A math circle is any group of people gathering to explore mathematics—it could be in your home, in school, or even online. Math circle activities are often interactive, exploratory, flexible, open-ended, and social. How does one make it happen? *Five Fabulous Activities* * *by Samuel Coskey, Paul Ellis, and Japheth Wood, veteran math circle leaders, will help show you how. This book is a guide and a collection of recipes for anyone who wants to help others discover joyful and challenging math.

The book is primarily for middle and high school students. Each chapter also includes activities and explanations intended for younger math friends. The underlying mathematics can be of interest even to adult learners, including the authors!

The funding campaign quickly reached its goal. At the current stretch goal, for every $100 we raise, we will donate a copy of the book to a math circle serving groups of people who are underrepresented in STEM fields.

Our math friends shared these function machines. Can you guess what they do?

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]]>The post Ying is 72% funded; try magic square activities – Newsletter August 13 appeared first on Natural Math.

]]>There are some recent additions to the campaign page. Check out the new variable prize, where supporters can choose the number of books they want. Thank you for the suggestion, Tina G! There are also new answers to several questions, such as, “How do you go about using a story book in a math classroom?”

**Question** (mind the book spoiler): – The emperor eventually invites Ying to join the team working out engineering solutions to the annual floods. Does solving abstract puzzles like magic squares really help with applied mathematics?

(This is a page layout with Ying and her, sadly empty, fishing pole, from when the river flooded.)

**Answer**: Solving puzzles such as magic squares leads to uncovering the general structure of a problem type. There are properties that are true of all magic squares that can be unpacked. At first glance they are simple, but their beauty is in the complexities of understanding how it is that they “work”.

When we change one number in the magic square, that affects the row and the column and ultimately the rest of that SYSTEM. We can use that property to make a point about system thinking. We can’t solve the magic square cell by cell, but have to consider it as a whole. Likewise, rivers, rains, and the weather are systems that have many components all affecting one another.

Ying demonstrates for readers the qualities that are required for engineers and designers, such as perseverance and being intellectually brave. Young children can also start their STEM journeys with puzzles. The beauty and challenge of problem solving will help them develop lifelong skills necessary in the STEM fields.

There are 10 days left on the book’s crowdfunding campaign, with about 73% of the target goal gathered by the supporters – THANK YOU! If you contribute, you receive the book plus special prizes, and your name in the book’s dedication pages. You can also dedicate your contribution under an alias, list your whole family, or put your math circle, school, or business name into the book.

We are very grateful for each and every contribution. Starting at, and including, the $1 level, the contributions do add up and make the book possible. Each contribution is a voice of support on a long and difficult journey from the concept to the finished book. Each is a vote of confidence that warms the hearts of the book’s team. Each supporter sparks hope for the future authors who contemplate if they should write a math book one day.

In addition to the story, the book has materials for teachers and parents to explore magic squares with children, and to help children pose their own problems. Here are some bonus activities to try.

Smaller numbers make for faster puzzles, accessible to younger children. Arrange these numbers:

1, 1, 1, 2, 2, 2, 3, 3, 3

in the nine boxes below so that each row and column has the same sum.

Magic squares are a mathematical puzzle that has fascinated learners throughout time, including capturing the attention of Benjamin Franklin. Franklin created his Magic Square in 1771 and later wrote, *“I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so unentertaining that I was induc’d to amuse myself with making magic squares or circles”* (Franklin 1793). Here is his famous Franklin Square:

What do you notice about the Franklin square? What do you wonder? What patterns do you see? Send your ideas, magic squares, and questions our way to share with the Natural Math community!

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]]>The post Ying and the Magic Turtle math story book: newsletter July 25 appeared first on Natural Math.

]]>We just started the book’s crowdfunding campaign. Visit the campaign to watch a video invitation, see sample illustrations and layouts, and learn how this book can enhance your children’s mathematics at home, math circle, or school. Please contribute to the campaign and help us spread the word. Thanks to parents and teachers like you, who support Natural Math authors, this small indie publisher can keep offering unique materials with the open Creative Commons license. Of course, there are special prizes for every supporter, from your name in the book to group packages.

We at Natural Math were thrilled when Dr. Sue Looney, a veteran mathematics educator and provider of professional development, approached us with the book’s concept: retell an ancient legend about magic squares as a children’s story. We want to see many more mathematical story books like *Life on the Infinite Farm* (infinity), *Anno’s Magic Seeds* (exponential growth), and *Funville Adventures* (functions and functionals).

“Long ago in the land of China, there were many rain storms … and the land of China was slowly sinking into the sea. This is the story of how a wise emperor, an observant girl, and a magic turtle saved the villages of China from the great flood.” So begins the story. I wish I read it as a young child, so I could pretend-play to be Ying, saving my land – with math!

Children ages 5 and up, parents, and teachers can enjoy the book for its rich beauty in mathematics and as an ancient legend. It is the kind of story to revisit over and over again. This book is perhaps best experienced *with *someone, as a read-aloud or read-together. When reading, we learn of Ying’s trouble, and we root for her to find her solution. We find ourselves drawn into the life problems that Ying is facing, but also drawn into the inherent mathematics of the story. It is through the beauty of the pattern of the dots on the turtle’s shell that the solution is finally found and the land is saved.

We can appreciate each scene as we read, and then pause and predict what might come next. We can play with the mathematics, solving right alongside Ying. We can delve deeper into the power of magic squares by working with puzzles presented at the end of the story. Best yet, there are unsolved problems in number theory even a young child can try, such as finding all the possible magic squares of a given size.

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]]>The post Kids Build Together – Math Readiness in Early Childhood appeared first on Natural Math.

]]>We are lying on our backs looking up at the ceiling, searching for shapes. We see squares, rectangles. The smoke detector is a circle. One of the kids points out that the joins between the ceiling panels make crosses, and excitedly: “Parallel lines!”

When my kids were small, I ran a play & learning center in a small town outside Sydney, Australia. It held mostly building blocks and was a place where parents and kids could come and build whatever they liked. Inspired by Papert’s constructionism, it was a space where people could learn by constructing items meaningful to them, and sharing them with others. That’s why I called it **Kids Build Together.**

On Monday mornings I ran a math readiness group. It had mostly 2 to 4 year-olds. There were three main areas of focus – enjoying patterns, enjoying problems, and enjoying the process. We read a lot of math-themed picture books, tussle with simple problems. I tried to always end with a math-themed poem. Here’s a sample of one of our sessions:

“Today we’ve sung “Round about the circle” and passed a bag around with different shapes. Now the kids are going to try to fit them together. There’s no right answer, and it’s fun to see the different ways that the kids turn and feel and explore the properties of their shapes.

Next we read a gorgeously illustrated book about spirals in nature called “Swirl by Swirl” by Joyce Sidman. Afterwards, the kids use different colored stones to make their own spirals. They’re beautiful.

We’ve been reading “Measuring Penny” by Loreen Leedy and using different ways to measure things. Today the kids use big blocks to build towers as tall as themselves. They’re predicting how many blocks they’ll need, and I write it down on the board. They’re surprised to discover it only needs 15 blocks, rather than 100, to reach their height! I give them a measuring tape as another way to check their height, and 100 cm is close – so their predictions weren’t far off, after all.

To finish off I read a poem about time “Twenty-Four Hours” by Charles Causley. The kids love listening to poetry, the rhythm captures them instantly.

When I think back to learning math when I was at school, music, stories and games were the things I wished we used (I remember trying to design my own curriculum back in high school!) Finding the incredible mathematical patterns in nature, from fractals to Fibonacci spirals, has been a wonderful discovery of adulthood, as well as learning the stories of the mathematicians themselves. I hoped to share the pleasures of discoveries like these with the kids and their parents, and let them in on a secret – that math is more than memorizing numbers, and that readiness for math can be readiness for enjoying life.

Now my two kids are at school, and my center is no more. I help run a math class for the little ones at the local school, and include some Math in Your Feet dance activities there. I also tutor some children with literacy and numeracy challenges. It’s been fun sharing some of the resources with the older ones, such as the book about Fibonacci and the ideas of fractals.

It’s now I realize why all the work on symmetry in the younger years is essential – so much of mathematics is symmetry and balance! There’s not much time to cover it in school, a lesson or so. The ones who’ve spent years building with blocks are immediately advantaged. I encourage the parents of the older ones to bring the Lego back out, and build together; to walk in nature and observe the wonderful math in flower petals and tree branches. The secret hasn’t changed with their ages; math is still more than memory, and is still an important part for enjoying life.

*Enjoyed this story and feel inspired to try these math activities with your preschoolers? Let us know how it goes. Do you have a math story to share? We can’t wait to hear from you!*

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]]>The post Natural Math authors invite you for beta testing – Newsletter January 16 appeared first on Natural Math.

]]>Math Punday is Monday: the day to share math jokes on social media. I love Zeno’s paradoxes, and hope this recent post makes you and your children smile:

Math pun by JoeGP.com

All activities in Natural Math books are play-tested many times, not only by the authors, but also by our *beta readers*. These adventurous and caring early reviewers help authors make books better. Beta readers are parents, teachers, and math circle leaders who want their math materials to be beautiful, useful, enjoyable, and human-readable. As a beta reader, what do you get?

- You influence math materials, making them
**fit your needs**. The book is customized by and for you, and your children. - You and your children are the earliest users of the newest books even before they are off the press. It’s like
**test-driving**concept cars before they are even announced to the market. - You spend
**quality time with interesting people**. In the context of the book, you talk with authors about your children and your ideas on learning. Imagine the possibilities! - In the book, the authors will
**thank you**by your name or an alias of your choice.

Read on for two upcoming beta reader opportunities, one by email and one live.

Photo: Bard Math Circle

Meet new Natural Math authors Sam Coskey, Paul Ellis, and Japheth Wood. The working title of their book is *The Fantastic Five*. The book is for teachers, parents, and math circle leaders with students ages 10 and up as well as younger friends. It offers rich and original problem-solving activities in the contexts such as a trip to Mars or traditional Sona storytelling from Angola/Congo area. The authors write:

We are collaborating on a book of five math circle activities that we plan to publish with Natural Math. We would love it if you would agree to be a “beta-reader” of one chapter (or more, if you have time). If you have an appropriate forum in which to try the activity (such as a math circle or family math at home) that would be wonderful. If not, we would be happy if you would just read the chapter and provide feedback by email.

You are welcome to provide feedback in any form that you like. However, if you wish to have some guidance, we’ll suggest questions that we brainstormed together.

Sam Coskey, Boise Math Circle

Paul Ellis, Westchester Area Math Circle

Japheth Wood, Bard Math Circle

Alfreda Poteat is a Natural Math author who is passionate about the roots of modern mathematics in the world history, multiple cultures, and people’s stories. On **January 23 at 7 – 8:30 PM Eastern** (New York) time, Alfreda will lead a **live online workshop**. She invites parents and teachers of middle school children to join the workshop and play-test activities from her upcoming book with the working title *Math Talk: Math Roots*.

Alfreda’s thoughtful, playful, and kind book is especially suitable for a struggling learner. Children who’ve had some math grief or math anxieties are welcome to try accessible math activities. The book invites children to meet and pretend-play with the friendly geeks from the cave-people to modern-day scientists.

In the workshop, participants will improve on Euclid’s work, go on a math scavenger hunt in a cute board game, and explore mathematics of clay pottery.

*See you online!*

*Dr. Maria Droujkova, and the Natural Math crew*

Questions? Ideas? Email reach.out@naturalmath.com

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