If Not Methods: Reasoning About Subtraction

Posted on by Denise Gaskins
Father and son reasoning about subtraction

We’ve been examining the fact that, while there may be only one right answer to a math problem, but there’s never only one right way to get that answer.

What matters in math is the journey. How do your children make sense of the problem and reason their way to that answer?

As always, real math is not about the answers but the thinking.

But if we don’t want to give our children a method, how can we teach? What if we pose a problem and the child doesn’t know how to solve it?

What if our children get stumped on a subtraction calculation like 431 – 86?

Continue reading If Not Methods: Reasoning About Subtraction

Podcast: How to Transform Math Lessons without Changing your Curriculum

Posted on by Denise Gaskins
Homeschooling math together - photo

I have a new podcast interview, and I think you’ll enjoy it!

Check out Pam Barnhill’s 10 Minutes to a Better Homeschool on your favorite podcast app, or listen on the website:

Go to the podcast ❱

Here’s a couple of excerpts…

Continue reading Podcast: How to Transform Math Lessons without Changing your Curriculum

Musings: If Not Methods, Then What?

Posted on by Denise Gaskins

Last week, I quoted Pam Harris calling out a foundational myth of math education, the idea that we need to teach kids the methods that work on even the most difficult math problems.

“We have a misconception in math education that we think we need to teach methods so that kids can answer the craziest kind of a particular problem.

    “We would be far better served to teach kids to think about the most common kinds of questions WELL, and let technology handle the crankiest.”

    —Pam Harris

    Since many of us grew up in schools that taught these methods, they may feel like the only sensible approach to math. Without the standard procedures, how will our kids learn to do math?

    If we don’t teach subtraction with borrowing/renaming, how can students figure out calculations like 431 − 86? If we don’t teach fraction rules, how will they handle problems like 1 1/2 ÷ 3/8?

    Continue reading Musings: If Not Methods, Then What?

    Celebrating Spring with Playful Math Carnival 172

    Posted on by Denise Gaskins
    Playful Math Carnival 172

    Welcome to the 172nd edition of the Playful Math Blog Carnival, a buffet of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.

    The carnival went on hiatus for a couple of months due to unexpected life issues facing our volunteer hosts. But we’re back now, and ready to celebrate!

    By tradition, we start the carnival with a puzzle in honor of our 172nd edition. But if you’d rather jump straight to our featured blog posts, click here for the Table of Contents.

    Try This: Lazy Caterers and Clock-Binary Numbers

    172 is a lazy caterer number: Imaging a caterer who brought a single large pie to serve the whole party. He needs to cut it into as many pieces as he can, using the fewest (straight) cuts he can get away with.

    • If each guest gets one piece of pie, what sizes of parties (numbers of people) can the lazy caterer serve?
    • Can you find a pattern in the lazy caterer sequence?

    But for those of you who have followed the carnival for years, you may remember we played with the lazy caterer back in Playful Math 106. (That time, the caterer was serving pizza.) So here’s a bonus activity we’ve never done before…

    The first several stages of a pattern are as follows:

    Clock Binary pattern image

    • What do you notice about this pattern of shapes?
    • What is the next shape in the sequence?
    • Can you figure out how the shape below fits into the pattern?

    Clock Binary puzzle image

    This pattern sequence was named clock binary by its creator, noelements-setempty.

    • What questions can you ask about this sequence?
    • How are these shapes like the binary numbers?
    • How are they different?

    Click here for all the mathy goodness!

    Musings: A Common Misconception

    Posted on by Denise Gaskins
    Father and son thinking together about a math problem

    One of my favorite podcasts to listen to is Pam Harris’s Math Is Figure-Out-Able because she puts so many of my thoughts into words.

    For example:

    “We have a misconception in math education that we think we need to teach methods so that kids can answer the craziest kind of a particular problem.

      “We would be far better served to teach kids to think about the most common kinds of questions WELL, and let the cranky ones go to ChatGPT. Because they’ll recognize the sense of the answer.

        “Let technology handle the crankiest, and REASON about the rest of them.”

        —Pam Harris,
        the Math is Figure-out-able Fractions Challenge

        Well, I do think she’s wrong about the AI chatbot, because ChatGPT comes up with the strangest bald-faced nonsense about math problems. Wolfram Alpha is a much more reliable resource.

        But Harris’s main point stands. This misconception, this math-education myth, drives much of what happens in our classrooms and home schools today.

        Continue reading Musings: A Common Misconception

        Musings: A Philosophy of Education

        Posted on by Denise Gaskins

        I’ve tried a few times over the years to express my philosophy of teaching math. Back when I first started doing workshops for homeschooling parents, I told them:

        “Instead of drudgery, mathematics should be a game of discovery. It should give children the same ‘Eureka!’ thrill that sent Archimedes running through town in his birthday suit. I call this the ‘Aha!’ factor, the delight in solving a challenging puzzle.”

        Years later, as the internet developed and much of life moved online, I started a blog about playing with math. And since all good blogs need an “About Me” page, I had another chance to sum up my thoughts:

        “Math is like ice cream, with more flavors than you can imagine — and if all your children ever see is textbook math, that’s like feeding them broccoli-flavored ice cream.”

        But over the years, some people got the impression that my goal was all about playing games. They asked, “How can we make math fun for our kids?” — as if gamification adds a candy coating to make the disgusting medicine more palatable.

        And of course, I do write a lot of books about games. I think games serve much better than worksheets for practicing basic math skills.

        Still, I wanted people to see that the ideas of math themselves are tasty tidbits worth playing with.

        Continue reading Musings: A Philosophy of Education

        Playful Math Store Grand Opening Sale

        Posted on by Denise Gaskins
        Family hiking in the mountains

        I’ve just launched the Denise Gaskins’ Playful Math Store!

        The Grand Opening Sale runs through April 20 — add the discount code GRAND_20 at checkout to save 20% on your first order.

        We have plenty of fun mathy goodness on hand, so I hope you’ll take some time to browse our collections. Don’t miss the math art collection, or the journaling resources, or my classic Math You Can Play series. And of course, there’s the merch.

        Even better, we have both digital and print books, which is something my previous store couldn’t handle.

        So many ways to celebrate the joy of mathematics.

        May your family always enjoy the adventure of learning!

        Shop Now

        Calendar with sale dates marked

        Why Thinking Thursday?

        Posted on by Denise Gaskins
        Thinking Thursday math journal prompt

        There’s a new math journaling prompt this week.

        Have your kids tried it yet?

        This week’s prompt features one of my favorite quotations to get kids thinking (and writing) about the value of mistakes in learning. Or you might prefer last week’s prompt, featuring a classic math brainteaser and encouraging students to create their own related puzzles.

        Or, if you’re reading this post later and missed those, there’s another great new prompt this week for you to explore.

        Check it out:

        Visit Thinking Thursday

        Continue reading Why Thinking Thursday?

        Middle School Math Proof

        Posted on by Denise Gaskins

        Homeschool Memories…

        Kitten (my daughter) and I sat on the couch sharing a whiteboard, passing it back and forth as we took turns working through our prealgebra book together.

        The chapter on number theory began with some puzzles about multiples and divisibility rules.

        Continue reading Middle School Math Proof

        Gameschooling Math

        Posted on by Denise Gaskins

        Games are fun, building a positive attitude toward math. They give students a refreshing break from textbook work and make kids willing to practice their math. Games make math practice enjoyable, something children want to do. We can happily work through many more calculations during a game than anyone would ever want to do on a homework page.

        Benefits of Math Games

        But more important than the fun, math games push children to think about what numbers mean and how they work. The numbers in a math game are not just meaningless abstractions, but tools that players can use to gain an advantage over their opponent.

        A good math game reinforces the idea that math is about reasoning, using the things you know to figure out what you need. Math is not just about getting the right answer. It’s about what goes on in your head on the way to that answer. The answer itself is merely a side-effect. of what really matters, your thinking.

        A good math game helps students develop flexibility, the ability to adapt, applying what they have learned to new situations, finding a way to work out the things they haven’t mastered yet. All these add up to a more robust type of mathematical fluency than what many people imagine possible.

        Continue reading Gameschooling Math