How to Think like a School Math Genius

Posted on by Denise Gaskins
Teen student thinking

“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.

      Continue reading How to Think like a School Math Genius

      Notice–Wonder–Discover: The Foundation of Learning Well

      Posted on by Denise Gaskins
      Notebook on desk, with the words "Notice. Wonder. Discover."

      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 new 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!

      Find More Information

      What the Course Covers

      The course consists of six weekly online workshops, plus an optional bonus session:

      • Session 1: Orientation – Why Learning Feels Hard
        The default scripts we carry from school, and how to overcome them.
      • Session 2: Notice – The First Act of Learning
        Before anything can be learned, it must be seen.
      • Session 3: Structure – Why Look for Structure?
        Structure is the beginning of understanding — and the root of algebraic thinking.
      • Session 4: Arbitrary vs. Necessary
        Not all information deserves equal weight. Here’s how to make space for thinking.
      • Session 5: Wonder – Curiosity That Moves Forward
        Wonder is not optional — it’s the engine of discovery, the heart of orientation, and the builder of wisdom.
      • Session 6: Discover – What Was Always There
        Discovery is our response to insight, how we orient inward and express outward after recognizing something true.
      • Optional Bonus Session
        Mothering as Orientation — A Christian Reflection.

      The course begins October 14: Notice–Wonder–Discover Course.

      More About Sonya

      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:

      And in fact, anyone who is taking one of her homeschool math courses (or signs up in the next few weeks) will get the Notice-Wonder-Discover Course included for free.

      Sonya says:

      “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.”

      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.

      Sign up now, before it’s too late:

      Get the Notice–Wonder–Discover Course

       
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      Featured image above copyright © Sonya Post.

      Mental Math: Advanced Addition

      Posted on by Denise Gaskins
      photo of kids having fun with math

      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…

      Continue reading Mental Math: Advanced Addition

      Mental Math: Early Division

      Posted on by Denise Gaskins
      Boy doing mental math calculation

      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.

      • Use friendly numbers.
      • Estimate, then adjust.

      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.

      Continue reading Mental Math: Early Division

      Mental Math: Early Multiplication

      Posted on by Denise Gaskins
      mother and daughter talking math together

      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.

      Let’s Try an Example

      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?

      Continue reading Mental Math: Early Multiplication

      FAQ: How To Start a Homeschool Math Club

      Posted on by Denise Gaskins
      Denise Gaskins reading math with preschoolers

      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…

      Continue reading FAQ: How To Start a Homeschool Math Club

      Mental Math: Early Subtraction

      Posted on by Denise Gaskins
      mother and child doing math homework

      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.

      • Use friendly numbers.

      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.

      • Estimate, then adjust.

      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.

      Continue reading Mental Math: Early Subtraction

      New Membership Tiers: One-on-One Mentorship

      Posted on by Denise Gaskins
      Two women chatting over tea

      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.

      Here’s How It Works

      (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.

      Interested?

      These new tiers are limited to a few members each—first paid, first in. Check out all the details on my Patreon home page.

      Join me on Patreon

       
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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.

      Mental Math: Early Addition

      Posted on by Denise Gaskins
      child counting on fingers

      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.

      • Use friendly numbers.

      For early single-digit addition, the most important friendly numbers are 5 and 10, the pairs of numbers that make 10, and the doubles.

      • Estimate, then adjust.

      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.

      Continue reading Mental Math: Early Addition

      Tell Children Interesting Things

      Posted on by Denise Gaskins
      quote by John Conway

      “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:

      Back to School Sale 2025