As our children grow and develop their math skills, the mental math strategies grow with them.
The basics of mental math don’t change:
But we have new ways to help children do math in their heads as the numbers get bigger and the problems more challenging.
For example, how might kids figure out a multi-digit subtraction like 67 − 38?
First, we need to adjust our mindset…
I’ve been updating my old book files to a new publishing program, and in the process discovering anew how often websites change and disappear. So frustrating!
Thankfully, the Internet Archive keeps some things that we can look back at. It doesn’t always work, but today it came to the rescue and let me recover the wonderful workbook from the Metropolitan Museum of Art, Islamic Art And Geometric Design, plus some of the pattern pages at the old School of Islamic Geometric Design.
I reference these pages in my Geometric Coloring Designs series (available at my Playful Math Store), so I was glad to find they weren’t lost forever.
Math Art = a great way to spark energy when your students hit the mathematical doldrums.
If you’d like to play around with math art patterns, each image below links to Eric Broug’s step-by-step instructions for creating the design. Also, Broug made printable pages to get students started, which you can download here. No compass construction required!
There are two more patterns on the old SIGD site, but I can’t get those instructions to load. That might be my antique rural internet connection, however, so feel free to do some exploring on your own.
And have fun drawing math with your kids!
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Are you looking for more creative ways to play math with your kids? Check out all my books, printable activities, and cool mathy merch at Denise Gaskins’ Playful Math Store. Or join my email newsletter.
This blog is reader-supported. If you’d like to help fund the blog on an on-going basis, then please join me on Patreon for mathy inspiration, tips, and an ever-growing archive of printable activities.
“Geometric Math Art, Recovered” copyright © 2025 by Denise Gaskins. Image at the top of the post copyright © Denise Gaskins.
“The true joy in mathematics, the true hook that compels mathematicians to devote their careers to the subject, comes from a sense of boundless wonder induced by the subject.
“There is transcendental beauty, there are deep and intriguing connections, there are surprises and rewards, and there is play and creativity.
“Mathematics has very little to do with crunching numbers. Mathematics is a landscape of ideas and wonders.”
—James Tanton
James Tanton has a new website. It looks cool, and it’s a great place to discover the things he’s working on these days.
But his wonderful, old-fashioned site full of great insights and interesting problems is gone.
😞 I hate it when some part of the internet that I love disappears. So here’s my attempt to recover one tiny bit of the old site, five tips for creative problem solving through intellectual play.
Most of us were never taught how to teach. And we certainly weren’t taught what to do when NOTHING is working.
My friend Sonya Post is offering a free orientation course that will help you rethink how learning actually works, how you can stop second-guessing yourself and start seeing real growth.
I’ve taken the earlier iterations of her course, and I’d recommend it to all parents.
Truly wonderful insights!
The NWD Orientation Course is an introduction to the Notice, Wonder, Discover framework and the underlying habits that develop the underlying thinking skill necessary for learning. Rather than focusing on curriculum tricks or information delivery, the course helps parents understand how attention, observation, curiosity, reasoning, and response shape the learning process itself.
Inside the course, parents will explore how children develop understanding through guided discovery, pattern recognition, conversation, and relationship.
You’ll learn how to create a calmer and more connected learning environment, how to recognize common teaching habits that unintentionally shut down thinking, and how to support the development of flexible, confident thinking across every subjects.
You’ll also learn how all subjects are connected.
The Orientation Course changes the way you see teaching, learning, and formation at home. It lays the foundation for the rest of Sonya’s community and courses by helping parents develop a shared language and vision for learning rooted in attention, wonder, meaning, and formation.
You may have heard me mention Sonya before. She created The Best Math Game Ever, and she teaches the math course I wish my kids and I could have taken:
“I don’t just build these resources — I use them. I’ve walked the road of frustration, math tears, and feeling lost about how to teach well. This course exists because I don’t want you to walk that road alone. This is the framework I wish I’d had years ago, and I can’t wait to share it with you.”
—Sonya Post
I really can’t praise Sonya’s work enough. If you’re struggling at all with your teaching or family life, she’ll turn you around and give you new perspective on how to move forward with grace.
Check out her free course:
Get the Notice, Wonder, Discover Orientation Course
(affiliate link)
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Featured image above copyright © Sonya Post.
Mental math is doing calculations with our minds, and perhaps with the aid of scratch paper or a whiteboard to jot down notes along the way.
But we cannot simply transfer the standard pencil-and-paper calculations to a mental chalkboard. That’s far too complicated.
We still want to follow our basic strategies of using friendly numbers, estimating, and adjusting the answer. So how can we help children do math in their heads as the numbers get bigger and the problems more challenging?
How might kids figure out a multi-digit addition like 87 + 39?
Here are three useful strategies…
Mental math is doing calculations with our minds, though we can use scratch paper or whiteboards to make notes as we work.
Doing mental math, children use the basic principles of arithmetic to simplify problems so they can think about number relationships, mastering the basic structures of how numbers work, the same structures that underlie algebraic reasoning.
As always, we rely on two key mental-math strategies.
Division is the mirror image of multiplication, the inverse operation that undoes multiplication, which means we are scaling numbers down into smaller parts. Important friendly numbers include halves, thirds, and tenths, plus the square numbers and any multiplication facts the student happens to remember.
Children learn best through interaction with others, and mental math prompts can lead to fascinating conversations, listening as our kids apply their creativity to the many ways numbers interact.
With mental math, students master the true 3R’s of math: to Recognize and Reason about the Relationships between numbers.
And these 3Rs are the foundation of algebra, which explains why flexibility and confidence in mental math is one of the best predictors of success in high school math and beyond.
Multiplication involves scaling one number by another, making it grow twice as big, or three times as much, or eightfold the size. Multiplication by a fraction scales the opposite direction, shrinking to half or a third or five-ninths the original amount.
The key friendly numbers for multiplication and division are the doubles and the square numbers. As with addition and subtraction, students can estimate the answer using any math facts they know and then adjust as needed.
How many ways might children think their way through the most-missed multiplication fact, 8 × 7?
The question hits my inbox whenever parents start planning for a new school year:
“Hello! I am on the board of a homeschool co-op. We have had requests for a math club and wondered if you have any tips for starting one. We service children from K-10th and would need to try to meet the needs of as many ages as possible.”
There are several ways you might organize a homeschool math club, depending on the students you have and on your goals. I think you would have to split the students by age groups — it is very hard to keep that wide of a range of students interested. Then decide whether you want an activity-oriented club or a more academic focus.
When I started my first math club, I raided the math shelves in the children’s section at my library (510-519) for anything that interested me. I figured that if an activity didn’t interest me, I couldn’t make it fun for the kids. Over the years we have done a variety of games, puzzles, craft projects, and more — always looking for something that was NOT like whatever the kids would be doing in their textbooks at home.
Let’s look at the possibilities by grade level…
By doing mental math, we help our children use the basic principles of arithmetic to simplify problems so they can think about number relationships, mastering the basic structures of how numbers work.
And the more our children practice these structures in mental math, the better prepared they will be to recognize the same principles in algebra.
The basic idea of subtraction is finding the difference between two quantities: comparing a larger amount to a smaller one, figuring out what’s left when you remove a part, or finding the distance between two measurements (or two points on the number line).
When you work with young children learning subtraction, remember our two key mental-math strategies.
For early subtraction with numbers less than 20, the most important friendly numbers are 5 and 10, the pairs of numbers that make 10, and the doubles.
When children apply their creative minds to reasoning about math, they can use friendly numbers to get close to an answer, and then tweak the result as needed.
Over the years, in my public writing and in personal correspondence, I’ve tried to help younger parents by sharing the things I wish someone would have told me forty years ago. Oftentimes, I will answer a question but never hear back what the person ended up doing or how their family got along.
It leaves me feeling as though I could have done more to encourage them along the way.
In the past few years, I’ve watched a professional writer I admire organize several more formal mentorships for younger indie authors to grow their writing and publishing careers. I think a similar program could be helpful for homeschooling families.
So now I’m offering two new Patreon membership tiers that give you a direct one-on-one connection with me through email or email-plus-video.
My goal is to help you think about math as a creative, playful investigation where you and your children explore concepts together using the math program or curriculum of your choice. This is not a tutoring arrangement. I will help you understand and teach math, but I will not do the teaching for you.
(1) You email me every Sunday—or pick the day that works best for you, at least once every two weeks—to keep me up to date with what you are doing: good times, rough spots, progress or no progress, plans for the coming week. You can also ask any questions about math or education (or indie book publishing).
(2) I’ll write back with encouragement and tips, and I’ll answer questions to the best of my ability. Caveat: I’ve forgotten everything except the broad brushstrokes of calculus and beyond.
(3) If you signed up for a video mentorship, include in your first email a list of several dates and times when you would be available for our monthly one-on-one chat. I’ll compare that with my schedule to find a time we can meet.
These new tiers are limited to a few members each—first paid, first in. Check out all the details on my Patreon home page.
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Thank you for reading!
“New Membership Tiers: One-on-One Mentorship” copyright © 2025 by Denise Gaskins. Image at the top of the post copyright © deagreez1 / Depositphotos.
Denise Gaskins' Let's Play Math 