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…
“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)
* * *
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.
From the very beginning of a child’s experience with math, we want to focus on reasoning, making sense of numbers, thinking about how they relate to each other and how we can use these relationships to solve problems.
The basic idea of addition is putting like things together: combining parts to make a whole thing, putting together sets to make a collection, or starting with an original amount and adding the increase as it grows. Connecting two numbers in relationship with a third number we call the sum.
When you work with young children learning addition, remember the two key mental-math strategies I mentioned in the previous post.
For early single-digit addition, 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.
“You don’t educate people by telling them useful things; you educate people by telling them interesting things.”
— John Conway
If you want help educating your children with interesting things about math, check out Denise Gaskins’ Playful Math store.
We’re currently running a huge back-to-school sale on ALL of my playful math ebooks, problem-solving activities, math journaling task cards, and math art projects.
So many great ways to play with math!
The 20% discount will automatically apply when you check out. No discount code required.
Check it out:
Denise Gaskins' Let's Play Math 