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!

    Thinking Thursday: Anatole France

    Posted on by Denise Gaskins
    Thinking Thursday math journal prompt

    Writing to Learn Math: What did the author mean? Put the thought in your own words. Do you agree or disagree? Why?

    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.

    Continue reading Thinking Thursday: Anatole France

    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 Education Carnival 171: Modern Math Artists

        Posted on by Denise Gaskins

        Welcome to the 171st edition of the Playful Math Education Blog Carnival — a smorgasbord 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.

        Bookmark this post, so you can take your time browsing over the next week or so.

        There’s so much playful math to enjoy!

        By tradition, we start the carnival with a puzzle/activity in honor of our 171st edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

        Try This Puzzle/Activity

        171 is a triangular number, the sum of all the numbers from 1 to 18:

        • 1 + 2 + 3 + … + 17 + 18 = 171.
        • Can you think why a number like this is called “triangular”?
        • What other triangular numbers can you find?

        Also, 171 is a palindrome number, with the same digits forward and backward. It’s also a palindrome of powers:

        • 171 = 52 + 112 + 52
        • 171 = 23 + 43 + 33 + 43 + 23

        So in honor of our 171st Playful Math Carnival, here is a palindrome puzzle that leads to an unsolved question in math:

        • Does every number turn into a palindrome eventually?

        palindrome number activity

        Click here for all the mathy goodness!

        Living Books for Math

        Posted on by Denise Gaskins

        What is a “living book”? English education reformer Charlotte Mason introduced this term for any book that brings the reader directly into contact with the major ideas that have fascinated humans across the ages.

        We know that reading aloud helps build our children’s love for books. But did you know it works for math as well?

        And that it can transform the parent’s attitude as well as the child’s?

        A playful math book fleshes out the bones of abstract math,
        brings it alive,
        makes it human, relatable,
        interesting to readers of all ages,
        opening our eyes to the wonderful world of big ideas,
        where concepts meet and topics intertwine
        in a beautifully intricate dance
        of understanding.

        We live in an age of abundance, with more new creative math books being published every year, so many that I can’t keep track of them all, not to mention the older classics, some out of print, that can still be found in public libraries.

        Here are a few of my favorite books of playful, living math, both old and new:

        Continue reading Living Books for Math

        Podcast: Cultivating Math Curiosity and Reasoning in Kids

        Posted on by Denise Gaskins

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

        Check out Learning Is Disruptable on your favorite podcast app, or listen on the website:

        Go to the podcast ❱

        Here’s an excerpt…

        “I think the most important thing that we need to change…we need to radically change what our idea is of what it means to learn math.

          “Our biggest failure, both in the classroom and in homeschool settings, is that we’ve given our children a totally wrong idea of what math is all about.

          Continue reading Podcast: Cultivating Math Curiosity and Reasoning in Kids