Writing to Learn Math: Number play doesn’t have to follow school math methods.
Do you want your children to develop the ability to reason creatively and figure out things on their own?
Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.
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.
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.
“We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.”
— Pam Harris, Math Is Figure-Out-Able Podcast
Whether you work with a math curriculum or take a less-traditional route to learning, do not be satisfied with mere pencil-and-paper competence. Instead, work on building your children’s mental math skills, because mental calculation forces a child to understand arithmetic at a much deeper level than is required by traditional pencil-and-paper methods.
Traditional algorithms (the math most of us learned in school) rely on memorizing and rigidly following the same set of rules for every problem, repeatedly applying the basic, single-digit math facts. Computers excel at this sort of step-by-step procedure, but children struggle with memory lapses and careless errors.
Mental math, on the other hand, relies on a child’s own creative mind to consider how numbers interact with each other in many ways. It teaches students the true 3R’s of math: to Recognize and Reason about the Relationships between numbers.
The techniques that let us work with numbers in our heads reflect the fundamental properties of arithmetic. These principles are also fundamental to algebra, which explains why flexibility and confidence in mental math is one of the best predictors of success in high school math and beyond.
Your textbook may explain these properties in technical terms, but don’t be intimidated by the jargon. These are just common-sense rules for playing with numbers.
On May 5, we celebrate one of the rarest math holidays: Square Root Day, 5/5/25.
Here are a few ideas for playing math with squares and roots.
Five is the square root of twenty-five, which means it is the number we can “square” (multiply times itself) to get 25.
The root is the base number from which the square grows. In physical terms, it is the side of the square.
Imagine a straight segment of length 5, perhaps a stick or a piece of chalk. Now lay that segment down and slide it sideways for a distance equal to its length. Drag the stick across sand, or pull the chalk across paper or a slate.
Notice how this sideways motion transforms the one-dimensional length into a two-dimensional shape, a square.
The area of this shape is the square of its root: 5 × 5 = 25.
What do you think would happen if you could drag the square through a third dimension, or drag that resulting shape through a fourth dimension?
How many shapes do you suppose might grow from that original root of 5?
There’s still time to check out my Math Journaling Adventures project and discover how playful writing activities will help your students learn mathematics. Preorder your books today!
Meanwhile, here’s a math puzzle to share with your kids…
Write down any whole number. It can be a single-digit number, or as big as you like. For example:
64,861,287,124,425,928
Now, count up the number of even digits (including zeros), the number of odd digits, and the total number of digits your number contains. Write those counted numbers down in order, like this:
64,861,287,124,425,928
even 12, odd 5, total 17
This is the last post (for now, at least) in our If Not Methods series about how to help children figure out tough calculations.
By the time students reach the topic of multiplying fractions, they have become well-practiced at following rules. After some of the complex procedures they’ve learned, a simple rule like “tops times tops, and bottoms times bottoms” comes as a relief.
But we know that relying on rules like that weakens understanding, just as relying on crutches weakens physical muscles.
If we want our students to think, to make sense of math, to figure things out, what can we do with a problem like 5/6 × 21 ?
Continuing our series on teaching the tough topics of arithmetic…
Our own school math experiences led many of us to think that math is all about memorizing and following specific procedures to get right answers. But that kind of math is obsolete in our modern world.
The math that matters today is our ability to recognize and reason about numbers, shapes, and patterns, and to use the relationships we know to figure out something new.
But what if our children get stumped on a mixed-number calculation like 2 5/12 + 1 3/4?
We’re continuing our series of posts on how to build robust thinking skills instead of forcing our children to walk with crutches.
When we say, “Use this method, follow these steps,” we teach kids to be mathematical cripples.
If your student’s reasoning is, “I followed the teacher’s or textbook’s steps and out popped this answer,” then they’re not doing real math. Real mathematical thinking says, “I know this and that are both true, and when I put them together, I can figure out the answer.”
But what if our kids get stumped on a fraction calculation like 7/8 − 1/6?
We’ve been exploring the many ways to help children reason about tough math problems, without giving them rules to follow.
As always, real math is not about the answers but the thinking.
But what about division with scary, big numbers? What if our kids get stumped on a calculation like 3840 ÷ 16?
We can teach without crippling children’s understanding if we follow the Notice-Wonder-Create cycle:
“Notice, Wonder, Create” is not a three-step method for solving math problems. It’s the natural, spiraling cycle by which our minds learn anything.
Denise Gaskins' Let's Play Math 