Numberless Word Problems

Posted on by Denise Gaskins

As I mentioned yesterday, my new book includes links to online resources to help you play with word problems. So this week, I’m sharing a few of my favorites.

Visit the Kickstarter

Today we examine a time-tested method to help kids reason about math: Leave out the numbers.

First up, there’s Brian Bushart’s numberless problem bank for young students. Then we’ll look at Farrar Williams’s modern revision of a math teaching classic with problems for upper-elementary and middle school students.

Have fun thinking math with your kids!

Word Problem Bank

Word problems are commonplace in mathematics classrooms, and yet they regularly confound students and lead to frustrated teachers saying things like:

  • “They just add all the numbers! It doesn’t matter what the problem says.”
  • “They don’t stop to think! They just start computing as soon as they’re done reading the problem.”

Brian Bushart offers a collection of ready-to-go slide presentations that walk through the steps of making a word problem make sense.

Visit the Site

Math With No Numbers

Discover Farrar Williams’s book Numberless Math Problems: A Modern Update of S.Y. Gillian’s Classic Problems Without Figures, available in ebook or paperback.

Williams writes: “In order to answer the question, they’ll have to explain it, because the problem doesn’t give you anything to calculate with. The only way to answer is by explaining your process. See how sneaky a numberless problem is? It makes students really think about the process of solving the problem.”

Find Out More

“When students face a word problem, they often revert to pulling all the numbers out and “doing something” to them. They want to add, subtract, multiply, or divide them, without really considering which operation is the right one to perform or why.

    “When you don’t have numbers, it sidesteps that problem.

      “For students who freeze up when they see the numbers, this can be a really good way to get them to think about their process with math.”

      —Farrar Williams, Math With No Numbers

      CREDITS: Feature photo (top) by saeed karimi via Unsplash.com.

      Exciting New Homeschool Math Program

      Posted on by Denise Gaskins

      Homeschooling friends, check out this new homeschool math program that’s fun, rigorous, and engaging — a delightful, hands-on course that helps parents (and their children) understand math.

      Introduction to Cuisenaire Rod Structures Course

      I had the privilege of previewing this class as Sonya and Lacy put it together. I highly recommend it to anyone who struggles with math, or who wants to take a non-traditional approach.

      By focusing on making sense of number relationships, and by teaching algebra before arithmetic, this course provides a stress-free path to rich mathematical mastery.

      And for all they provide, including weekly live workshops and a slew of printable math journal pages that prompt deep thinking, the price is a steal!

      Continue reading Exciting New Homeschool Math Program

      Free Number Sense Resources from Steve Wyborney

      Posted on by Denise Gaskins

      If you teach children in the primary grades, you’ll enjoy this new series from the wonderful Steve Wyborney. Every day for the rest of the school year, Steve will post a new estimation or number sense resource for grades K–8 (or any age!) at his blog:

      “This is my way of providing support and encouragement – as well as bringing math joy to your classroom… I’m going to stick with you all year long.”

      —Steve Wyborney

      Click to visit Steve’s blog

      Playing Math with Michael and Nash

      Posted on by Denise Gaskins

      Michael and Nash have been creating and posting new math games with astonishing regularity throughout the pandemic. Their YouTube channel is a great resource for parents who want to play math with elementary-age children.

      Today’s entry: Closest to Ten, a quick game for addition and subtraction fluency with a tiny bit of multiplication potential.

      And here’s one of my favorites for older players: Factor Triangles, a card game for 2-digit multiplication.

      Check out their channel, and have fun playing math with your kids!

      Visit Michael and Nash on YouTube

      Math with Young Children

      Posted on by Denise Gaskins

      The question came up again:

      “What is the best curriculum for my children? They are four and six years old, and I’m afraid of letting them fall behind.”

      I remember being a young parent, eager to start homeschooling. I used to get mad (without letting it show, like a true introvert) when people told me, “They are young. Just let them play.”

      Now I see the wisdom in it.

      The most important thing for your children right now, by far, is for them to enjoy learning. The joy of learning is a child’s natural state. As a parent, your primary job is to keep yourself from stomping it out.

      But our parental fears can push us into joy-trampling before we realize it.

      And our own experience of school makes it hard for us to see how much of our children’s play really is learning. We expect education to look like schoolwork, but natural learning looks nothing like that.

      Continue reading Math with Young Children

      Playing with a Hundred Chart #35: The Number Grid Game

      Posted on by Denise Gaskins

      This is a pretty simple game, but it makes a nice variation on the Race-to-100 game for young children who need to work on counting by tens from any number.

      See the Number Grid Game (PDF)

      How to Play

      You’ll need a 6-sided die, a hundred chart (printables here), and a small token to mark each player’s square. A crumpled bit of colored construction paper works well as a token.

      Take turns rolling the die. If you roll:

      • 1: Move either 1 or 10 squares, your choice.
      • 2: Move either 2 or 20 squares.
      • 3–6: Move that number of squares.

      The first player to reach the final square by exact count wins the game.

      Variation #1: For a shorter game, the first player to move off the board wins. You don’t have to hit the final square by exact count.

      Variation #2: For a longer game, if you cannot move your full roll forward, you must move backward. Rolling 6 is a “wild card” — you can move any number from one to ten.

      Variation #3: Count down. Start at the highest number on your chart and subtract each roll, moving toward zero. If you have a chart like the original shown above, a player whose move goes past zero into negatives will add the number on their next roll.

      More Ways to Play on a Hundred Chart

      A hundred chart can provide mathematical play from preschool to high school. The list on my blog began many years ago with seven activities, games, and logic puzzles.

      Wow, has it grown!

      Discover 30+ Things To Do with a Hundred Chart

      Math Journals for Elementary and Middle School

      Posted on by Denise Gaskins

      This fall, my homeschool co-op math class will play with math journaling.

      But my earlier dot-grid notebooks were designed for adults. Too thick, too many pages. And the half-cm dot grid made lines too narrow for young writers.

      So I created a new series of paperback dot-grid journals for my elementary and middle school students.

      I hope you enjoy them, too!

      Click here for more information

      Math Journaling Prompts

      So, what can your kids do with a math journal?

      Here are a few ideas: 

      I’m sure we’ll use several of these activities in my homeschool co-op math class this fall.

      Noticing and Wondering

      Learning math requires more than mastering number facts and memorizing rules. At its heart, math is a way of thinking.

      So more than anything else, we need to teach our kids to think mathematically — to make sense of math problems and persevere in figuring them out.

      Help your children learn to see with mathematical eyes, noticing and wondering about math problems.

      Whenever your children need to learn a new idea in math, or whenever they get stuck on a tough homework problem, that’s a good time to step back and make sense of the math.

      Kids can write their noticings and wonderings in the math journal. Or you can act as the scribe, writing down (without comment) everything child says.

      For more tips on teaching students to brainstorm about math, check out these online resources from The Math Forum:

      Problem-solving is a habit of mind that you and your children can learn and grow in. Help your kids practice slowing down and taking the time to fully understand a problem situation.

      Puzzles Are Math Experiments

      Almost anything your child notices or wonders can lead to a math experiment.

      For example, one day my daughter played an online math game…

      a math experiment
      Click the image to read about my daughter’s math experiment.

      A math journal can be like a science lab book. Not the pre-digested, fill-in-the-blank lab books that some curricula provide. But the real lab books that scientists write to keep track of their data, and what they’ve tried so far, and what went wrong, and what finally worked.

      Here are a few open-ended math experiments you might try:

      Explore Shapes
      • Pick out a 3×3 set of dots. How many different shapes can you make by connecting those dots? Which shapes have symmetry? Which ones do you like the best?
      • What if you make shapes on isometric grid paper? How many different ways can you connect those dots?
      • Limit your investigation to a specific type of shape. How many different triangles can you make on a 3×3 set of dots? How many different quadrilaterals? What if you used a bigger set of dots?
      Explore Angles

      • On your grid paper, let one dot “hold hands” with two others. How many different angles can you make? Can you figure out their degree without measuring?
      • Are there any angles you can’t make on your dot grid? If your paper extended forever, would there be any angles you couldn’t make?
      • Does it make a difference whether you try the angle experiments on square or isometric grid paper?
      Explore Squares
      • How many different squares can you draw on your grid paper? (Don’t forget the squares that sit on a slant!) How can you be sure that they are perfectly square?
      • Number the rows and columns of dots. Can you find a pattern in the corner positions for your squares? If someone drew a secret square, what’s the minimum information you would need to duplicate it?
      • Does it make a difference whether you try the square experiments on square or isometric grid paper?

      Or Try Some Math Doodles

      Create math art. Check out my math doodling collection on Pinterest and my Dot Grid Doodling blog post. Can you draw an impossible shape?

      How Would YOU Use a Math Journal?

      I’d love to hear your favorite math explorations or journaling tips!

      Please share in the comments section below.

      P.S.: Do you have a blog? If you’d like to feature a math journal review and giveaway, I’ll provide the prize. Send a message through my contact form or leave a comment below, and we’ll work out the details.

      Funville Adventures: Blake’s Story

      Posted on by Denise Gaskins

      Today we have a guest post — an exclusive tale by Sasha Fradkin and Allison Bishop, authors of the new math storybook Funville Adventures. Enjoy!

      Funville Adventures is a math-inspired fantasy that introduces children to the concept of functions, which are personified as magical beings with powers.

      Each power corresponds to a transformation such as doubling in size, rotating, copying, or changing color. Some Funvillians have siblings with opposite powers that can reverse the effects and return an object to its original state, but other powers cannot be reversed.

      In this way, kids are introduced to the mathematical concepts of invertible and non-invertible functions, domains, ranges, and even functionals, all without mathematical terminology.

      We know about Funville because two siblings, Emmy and Leo, were magically transported there after they went down an abandoned slide.

      When they came back, Emmy and Leo shared their adventures with their friends and also brought back the following manuscript written by their new friend Blake.

      Continue reading Funville Adventures: Blake’s Story

      How to Talk Math With Your Kids

      Posted on by Denise Gaskins

      A friend shared this video, and I loved it! From Kent Haines, a father who happens to also be a math teacher…

      “I hope that this video helps parents find new ways of interacting with their kids on math topics.”

      —Kent Haines

      More from Kent Haines

      Advice and Examples of Talking Math with Kids

      Danielson-Talking Math

      If you enjoyed Kent’s video, you’ll love Christopher Danielson’s book and blog.

      It’s a short book with plenty of great stories, advice, and conversation-starters. While Danielson writes directly to parents, the book will also interest grandparents, aunts & uncles, teachers, and anyone else who wants to help children notice and think about math in daily life.

      “You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.
       
      “You don’t need to love math. You don’t need to have been particularly successful in school mathematics. You just need to notice when your children are being curious about math, and you need some ideas for turning that curiosity into a conversation.
       
      “In nearly all circumstances, our conversations grow organically out of our everyday activity. We have not scheduled “talking math time” in our household. Instead, we talk about these things when it seems natural to do so, when the things we are doing (reading books, making lunch, riding in the car, etc) bump up against important mathematical ideas.
       
      “The dialogues in this book are intended to open your eyes to these opportunities in your own family’s life.”

      — Christopher Danielson
      Talking Math with Your Kids

      CREDITS: “Kids Talk” photo (top) by Victoria Harjadi via Flickr (CC BY 2.0). “Parent Rules” by Kent Haines.

      Even a Math Workbook Can Be a Game

      Posted on by Denise Gaskins

      Homeschooling Memories…

      My youngest daughter wanted to do Singapore math. Miquon Red was her main math text at the time, but we added a bit of Singapore Primary Math 1B whenever she was in the mood.

      We turned to the lesson on subtracting with numbers in the 30-somethings.

      The first problem was pretty easy for her:

      30 − 7 = _____

      I reminded her that she already knew 10 − 7.

      She agreed, “Ten take away seven is three.”

      Then her eyes lit up. “So it’s 23! Because there are two tens left.”

      Wow, I thought. She’s catching on quickly.

      Mom Always Talks Too Much

      We went to the next problem:

      34 − 8 = _____

      “Now, this one is harder,” I said. “But you know what ten minus eight is, right? So we could take one of these tens and—”

      She waved at me to be quiet.

      I was just getting started on my standard speech about how to turn a tough subtraction like 34 − 8 into the easy addition of “2 + 4 + two tens left.” But her mind was still on the last problem, specifically on the two tens and the seven.

      “If you have 27,” she said, “and you add three more, you get 30. And four more is 34.”

      “Um, yes, but…” I interrupted.

      She shushed me again.

      “And then you can take away the four. And then you can take away the three. And then you can take away one more…It’s 26!”

      Mom Learns a Lesson

      She continued through the next page that way. For every problem, she started with whatever number struck her fancy, usually containing at least one digit from the problem before. She added enough to get up to the 30-something number in the book.

      Only then would she deign to subtract the number in question.

      I don’t think she ever saw the point of the mental math technique the book and I were trying to teach, but she did have a lot of fun playing around with the numbers.

      In the long run, that’s much more important.

      Feature photo: “Laughing Girl” by ND Strupler via Flickr (CC BY 2.0).